Series: Grundlehren der mathematischen Wissenschaften, Vol. 325
2016, XXXVIII, 826 p. 52 illus.
Printed book
Hardcover
ISBN 978-3-662-49449-3
* Written by an author who is unquestionably an authority on the
subject as well as a masterly writer
* Revised, up-to-date fourth edition includes new material on the
Euler equations of gas dynamics, and on hyperbolic balance laws
with dissipative source, as well as new applications to elasticity and
differential geometry
* Features an expanded bibliography that now comprises close to two
thousand titles
This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic
conservation laws. It illustrates the essential role of continuum thermodynamics in
providing motivation and direction for the development of the mathematical theory
while also serving as the principal source of applications. The reader is expected to have
a certain mathematical sophistication and to be familiar with (at least) the rudiments
of analysis and the qualitative theory of partial differential equations, whereas prior
exposure to continuum physics is not required. The target group of readers would consist
of
(a) experts in the mathematical theory of hyperbolic systems of conservation laws who
wish to learn about the connection with classical physics;
(b) specialists in continuum mechanics who may need analytical tools;
(c) experts in numerical analysis who wish to learn the underlying mathematical theory;
and
(d) analysts and graduate students who seek introduction to the theory of hyperbolic
systems of conservation laws.
This new edition places increased emphasis on hyperbolic systems of balance laws
with dissipative source, modeling relaxation phenomena. It also presents an account of
recent developments on the Euler equations of compressible gas dynamics. Furthermore,
the presentation of a number of topics in the previous edition has been revised,
expanded and brought up to date, and has been enriched with new applications to
elasticity and differential geometry. The bibliography, also expanded and updated, now
comprises close to two thousand titles.
Series: Undergraduate Texts in Mathematics
1st ed. 2016, XVI, 300 p. 186 illus. in color.
Printed book
Hardcover
ISBN 978-3-319-39798-6
* Can be used as a textbook in elementary and more advanced courses
in differential geometryFocuses on applications of differential
geometry, lending simplicity to more difficult and abstract concepts
* Features full-color text and inserts to distinguish fundamental
definitions and theorems
This is a textbook on differential geometry well-suited to a variety of courses on this
topic. For readers seeking an elementary text, the prerequisites are minimal and include
plenty of examples and intermediate steps within proofs, while providing an invitation
to more excursive applications and advanced topics. For readers bound for graduate
school in math or physics, this is a clear, concise, rigorous development of the topic
including the deep global theorems. For the benefit of all readers, the author employs
various techniques to render the difficult abstract ideas herein more understandable and
engaging.
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts
in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems
help to visually organize the mathematical content. Color is even used within the text to
highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in
its application to cartography. Evolutes, involutes and cycloids are introduced through
Christiaan Huygens' fascinating story: in attempting to solve the famous longitude
problem with a mathematically-improved pendulum clock, he invented mathematics
that would later be applied to optics and gears. Clairaut’s Theorem is presented as a
conservation law for angular momentum. Green’s Theorem makes possible a drafting tool
called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along
a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel
vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics,
which is beyond the scope of this book. The GPS in any car wouldn’t work without general
relativity, formalized through the language of differential geometry. Throughout this
book, applications, metaphors and visualizations are tools that motivate and clarify the
rigorous mathematical content, but never replace it.
Series: C.I.M.E. Foundation Subseries, Vol. 2161
1st ed. 2016, XII, 170 p.
Softcover
ISBN 978-3-319-42308-1
* Contains a clear exposition of the mathematical structure of
isogeometric analysis
* Contains a very illustrative application to the Navier Stokes
equation The format of courses is well-balanced with an excellent
presentation
Providing an introduction to isogeometric methods with a focus on their mathematical
foundations, this book is composed of four chapters, each devoted to a topic of special
interests for isogeometric methods and their theoretical understanding. It contains
a tutorial on splines and generalizations that are used in CAD parametrizations, and
gives an overview of geometric modeling techniques that can be used within the
isogeometric approach, with a focus on non-tensor product splines. Finally, it presents
the mathematical properties of isogeometric spaces and spline spaces for vector field
approximations, and treats in detail an application of fundamental importance: the
isogeometric simulation of a viscous incompressible flow.
Series: Universitext
1st ed. 2016, X, 198 p. 48 illus.
Printed book
Softcover
ISBN 978-1-4471-7285-7
* Provides a concise introduction to ergodic theory and dynamical
systems
* Presents numerous examples in detail
* Technically sound and up-to-date, with an approach that favors
generality
This textbook is a self-contained and easy-to-read introduction to ergodic theory and the
theory of dynamical systems, with a particular emphasis on chaotic dynamics.
This book contains a broad selection of topics and explores the fundamental ideas of
the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms
of dynamical systems, the book then focuses on several chaotic transformations
with hyperbolic dynamics, before moving on to topics such as entropy, information
theory, ergodic decomposition and measurable partitions. Detailed explanations are
accompanied by numerous examples, including interval maps, Bernoulli shifts, toral
endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems,
rational maps on the Riemann sphere and strange attractors.
Ergodic Theory and Dynamical Systems will appeal to graduate students as well as
researchers looking for an introduction to the subject. While gentle on the beginning
student, the book also contains a number of comments for the more advanced reader.
Series: Developments in Mathematics, Vol. 45
1st ed. 2016, XI, 229 p. 35 illus.
Printed book
Hardcover
ISBN 978-3-319-42756-0
* Includes applications of symmetric spaces to Ergodic theory
* Contains a detailed study of Lorentz, Marcinkiewicz and Orlicz spaces
* Provides results with applications in functional analysis
Key definitions and results in symmetric spaces, particularly Lp, Lorentz, Marcinkiewicz
and Orlicz spaces are emphasized in this textbook. A comprehensive overview of the
Lorentz, Marcinkiewicz and Orlicz spaces is presented based on concepts and results of
symmetric spaces. Scientists and researchers will find the application of linear operators,
ergodic theory, harmonic analysis and mathematical physics noteworthy and useful.
This book is intended for graduate students and researchers in mathematics and may be
used as a general reference for the theory of functions, measure theory, and functional
analysis. This self-contained text is presented in four parts totaling seventeen chapters
to correspond with a one-semester lecture course. Each of the four parts begins with
an overview and is subsequently divided into chapters, each of which concludes with
exercises and notes. A chapter called “Complements” is included at the end of the text as
supplementary material to assist students with independent work.