Series: Fields Institute Communications, Vol. 77
* Collects research papers devoted to topics in different areas of
current research in number theory together in one volume
* Presents concise surveys of leading edge number theory research
* Provides surveys of recent advances in number theory made by
leaders in the field as well as new advances in the subject
The theory of numbers continues to occupy a central place in modern mathematics
because of both its long history over many centuries as well as its many diverse
applications to other fields such as discrete mathematics, cryptography, and coding
theory. The proof by Andrew Wiles (with Richard Taylor) of Fermatfs last theorem
published in 1995 illustrates the high level of difficulty of problems encountered in
number-theoretic research as well as the usefulness of the new ideas arising from its
proof.
The thirteenth conference of the Canadian Number Theory Association was held at
Carleton University, Ottawa, Ontario, Canada from June 16 to 20, 2014. Ninety-nine talks
were presented at the conference on the theme of advances in the theory of numbers.
Topics of the talks reflected the diversity of current trends and activities in modern
number theory. These topics included modular forms, hypergeometric functions, elliptic
curves, distribution of prime numbers, diophantine equations, L-functions, Diophantine
approximation, and many more. This volume contains some of the papers presented
at the conference. All papers were refereed. The high quality of the articles and their
contribution to current research directions make this volume a must for any mathematics
library and is particularly relevant to researchers and graduate students with an interest
in number theory. The editors hope that this volume will serve as both a resource and an
inspiration to future generations of researchers in the theory of numbers.
1st ed. 2015, XVIII, 220 p. 15 illus., 8 illus. in
Hardcover
ISBN 978-1-4939-3200-9
Series: Lecture Notes in Mathematics, Vol. 2133
* Provides a self-contained treatment of nonlinear Markov evolution,
with a focus on (classical and quantum) quadratic operators and the
asymptotic behavior of the dynamical systems they generate
* This is the first book to study both classical and quantum (noncommutative)
nonlinear dynamics using the theory of Markov processes
* Covers the most recent developments in the fields of quadratic
dynamical systems, Markov processes and quantum stochastic processes
* Includes material which has potential applications to quantum
information theory
Covering both classical and quantum approaches, this unique and self-contained book
presents the most recent developments in the theory of quadratic stochastic operators
and their Markov and related processes. The asymptotic behavior of dynamical systems
generated by classical and quantum quadratic operators is investigated and various
properties of quantum quadratic operators are studied, providing an insight into the
construction of quantum channels.
This book is suitable as a textbook for an advanced undergraduate/graduate level course
or summer school in quantum dynamical systems. It can also be used as a reference book
by researchers looking for interesting problems to work on, or useful techniques and
discussions of particular problems. Since it includes the latest developments in the fields
of quadratic dynamical systems, Markov processes and quantum stochastic processes,
researchers at all levels are likely to find the book inspiring and useful.
1st ed. 2015, XIV, 231 p. 1 illus.
Softcover
ISBN 978-3-319-22836-5
Series: Springer Undergraduate Mathematics Series
* Analytical and computational approach to PDEs
* Contains 300 exercises, all with full solutions, starred according to
difficulty
* One chapter dedicated to projects intended for individual or group
study
This volume provides an introduction to the analytical and numerical aspects of partial
differential equations (PDEs). It unifies an analytical and computational approach for
these; the qualitative behaviour of solutions being established using classical concepts:
maximum principles and energy methods.
Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates
and the use of flux-limiters when approximating hyperbolic conservation laws. The
numerical analysis of difference schemes is rigorously developed using discrete maximum
principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter
containing projects, intended for either individual or group study, that cover a range
of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the
approximation of multidimensional advection*diffusion problems.
The underlying theory is illustrated by numerous examples and there are around 300
exercises, designed to promote and test understanding. They are starred according to
level of difficulty. Solutions to odd-numbered exercises are available to all readers while
even-numbered solutions are available to authorised instructors.
1st ed. 2015, XI, 368 p. 106 illus., 1 illus. in
Softcover
ISBN 978-3-319-22568-5
Series: Springer Monographs in Mathematics
* Unique in presenting the modern theory of monotone complete
C*-algebras Gives a self-contained introduction to generic
dynamics Explains the new classification theory introducing
spectroids and classification semigroups
This monograph is about monotone complete C*-algebras, their properties and the new
classification theory. A self-contained introduction to generic dynamics is also included
because of its important connections to these algebras.
Our knowledge and understanding of monotone complete C*-algebras has been
transformed in recent years. This is a very exciting stage in their development, with much
discovered but with many mysteries to unravel. This book is intended to encourage
graduate students and working mathematicians to attack some of these difficult
questions.
Each bounded, upward directed net of real numbers has a limit. Monotone complete
algebras of operators have a similar property. In particular, every von Neumann algebra is
monotone complete but the converse is false.
Written by major contributors to this field, Monotone Complete C*-algebras and Generic
Dynamics takes readers from the basics to recent advances. The prerequisites are a
grounding in functional analysis, some point set topology and an elementary knowledge
of C*-algebras.
1st ed. 2015, Approx. 260 p.
Hardcover
ISBN 978-1-4471-6773-0
* Provides a complete and rigorous axiomatic treatment of Euclidean
geometry
* Proofs for many theorems are worked out in detail
* Takes a modern approach by replacing congruence axioms with a
transformational definition of congruence
In this monograph, the authors present a modern development of Euclidean geometry
from independent axioms, using up-to-date language and providing detailed proofs. The
axioms for incidence, betweenness, and plane separation are close to those of Hilbert.
This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving
metric notions and that explores congruence and isometries by means of reflection
mappings. The authors present thirteen axioms in sequence, proving as many theorems
as possible at each stage and, in the process, building up subgeometries, most notably
the Pasch and neutral geometries. Standard topics such as the congruence theorems for
triangles, embedding the real numbers in a line, and coordinatization of the plane are
included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The
final chapter covers consistency and independence of axioms, as well as independence of
definition properties.
There are over 300 exercises; solutions to many of these, including all that are
needed for this development, are available online at the homepage for the book at
www.springer.com. Supplementary material is available online covering construction of
complex numbers, arc length, the circular functions, angle measure, and the polygonal
form of the Jordan Curve theorem.
Euclidean Geometry and Its Subgeometries is intended for advanced students and mature
mathematicians, but the proofs are thoroughly worked out to make it accessible to
undergraduate students as well. It can be regarded as a completion, updating, and
expansion of Hilbert's work, filling a gap in the existing literature.
1st ed. 2015, XVI, 530 p. 59 illus.
Hardcover
ISBN 978-3-319-23774-9