* A solid introduction to analytic number theory, including full proofs
of Dirichlet's Theorem and the Prime Number Theorem
* Concise treatment of algebraic number theory, including a complete
presentation of primes, prime factorizations in algebraic number
fields, and unique factorization of ideals
* One of the few books to include the AKS algorithm that shows that
primality testing is one of polynomial time
* Many interesting ancillary topics, such as primality testing and
cryptography, Fermat and Mersenne numbers, and Carmichael numbers
Now in its second edition, this textbook provides an introduction and overview of
number theory based on the density and properties of the prime numbers. This unique
approach offers both a firm background in the standard material of number theory, as
well as an overview of the entire discipline. All of the essential topics are covered, such
as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity,
arithmetic functions, and the distribution of primes. New in this edition are coverage
of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in
primality testing.
The user-friendly style, historical context, and wide range of exercises that range from
simple to quite difficult (with solutions and hints provided for select exercises) make
Number Theory: An Introduction via the Density of Primes ideal for both self-study and
classroom use.
* Most of the material is not yet available in book form
* Combines linear and commutative algebra in a novel, unified way
* Every chapter starts with a lively and humorous introduction to the
topic
* Follows the two well-received and well-known volumes
“Computational Commutative Algebra 1+2” by the same authors
* Written by active researchers in computer algebra
* Authors have extensive first-hand experience in developing and
implementing computer algebra methods via the development of
the computer algebra system CoCoA
* CoCoA files for the examples available as Electronic Supplementary
Material
This book combines, in a novel and general way, an extensive development of the theory
of families of commuting matrices with applications to zero-dimensional commutative
rings, primary decompositions and polynomial system solving. It integrates the Linear
Algebra of the Third Millennium, developed exclusively here, with classical algorithmic
and algebraic techniques. Even the experienced reader will be pleasantly surprised to
discover new and unexpected aspects in a variety of subjects including eigenvalues and
eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms,
multiplication maps of zero-dimensional affine algebras, computation of primary
decompositions and maximal ideals, and solution of polynomial systems.
This book completes a trilogy initiated by the uncharacteristically witty books
Computational Commutative Algebra 1 and 2 by the same authors. The material treated
here is not available in book form, and much of it is not available at all. The authors
continue to present it in their lively and humorous style, interspersing core content with
funny quotations and tongue-in-cheek explanations.
Series: Seminaire de Probabilites, Vol. 2168
* Provides a broad insight into current, high level research in
probability theoryContinues the exploration of the subject of
peacocks from previous volumesIncludes new material on harmonic
measures, random fields and loop soups
In addition to its further exploration of the subject of peacocks, introduced in recent
Seminaires de Probabilites, this volume continues the series’ focus on current research
themes in traditional topics such as stochastic calculus, filtrations and random matrices.
Also included are some particularly interesting articles involving harmonic measures,
random fields and loop soups. The featured contributors are Mathias Beiglbock, Martin
Huesmann and Florian Stebegg, Nicolas Juillet, Gilles Pags, Dai Taguchi, Alexis Devulder,
Matyas Barczy and Peter Kern, I. Bailleul, Jurgen Angst and Camille Tardif, Nicolas
Privault, Anita Behme, Alexander Lindner and Makoto Maejima, Cedric Lecouvey and
Kilian Raschel, Christophe Profeta and Thomas Simon, O. Khorunzhiy and Songzi Li,
Franck Maunoury, Stephane Laurent, Anna Aksamit and Libo Li, David Applebaum, and
Wendelin Werner.
1st ed. 2016, X, 108 p. 23 illus., 4 illus. in color.
Softcover
ISBN 978-3-319-44464-2
Series: Springer Texts in Statistics
* A concise treatment and textbook on the most important topics in
Stochastic Processes
* All concepts illustrated by examples and more than 300 carefully
chosen exercises for effective learning
* New edition includes added and revised exercises, including many
biological exercises, in addition to restructured and rewritten
sections with a goal toward clarity and simplicity
* Solutions Manual available for instructors
Building upon the previous editions, this textbook is a first course in stochastic processes
taken by undergraduate and graduate students (MS and PhD students from math,
statistics, economics, computer science, engineering, and finance departments) who have
had a course in probability theory. It covers Markov chains in discrete and continuous
time, Poisson processes, renewal processes, martingales, and option pricing. One can only
learn a subject by seeing it in action, so there are a large number of examples and more
than 300 carefully chosen exercises to deepen the reader’s understanding.
Drawing from teaching experience and student feedback, there are many new examples
and problems with solutions that use TI-83 to eliminate the tedious details of solving
linear equations by hand, and the collection of exercises is much improved, with many
more biological examples. Originally included in previous editions, material too advanced
for this first course in stochastic processes has been eliminated while treatment of other
topics useful for applications has been expanded. In addition, the ordering of topics
has been improved; for example, the difficult subject of martingales is delayed until its
usefulness can be applied in the treatment of mathematical finance.
3rd ed. 2016, VIII, 268 p. 6 illus.
Hardcover
ISBN 978-3-319-45613-3
Series: Theoretical and Mathematical Physics
* Offers a rigorous study of the geometric structure of gauge theories
* Provides a systematic and exhaustive presentation of Dirac operators
* Enriched by an introduction with interesting historical remarks, a
very extensive bibliography, and convenient appendices
The book is devoted to the study of the geometrical and topological structure of gauge
theories. It consists of the following three building blocks:
- Geometry and topology of fibre bundles,
- Clifford algebras, spin structures and Dirac operators,
- Gauge theory.
Written in the style of a mathematical textbook, it combines a comprehensive
presentation of the mathematical foundations with a discussion of a variety of advanced
topics in gauge theory.
The first building block includes a number of specific topics, like invariant connections,
universal connections, H-structures and the Postnikov approximation of classifying spaces.
Given the great importance of Dirac operators in gauge theory, a complete proof of the
Atiyah-Singer Index Theorem is presented.
1st ed. 2017, XVI, 832 p. 15 illus., 2 illus. in color.
Hardcover
ISBN 978-94-024-0958-1
Series: Probability Theory and Stochastic Modelling, Vol. 79
* First book ever published to systematically introduce Yosida
approximations and their applications
* Compiles results from the literature spanning more than 35 years
* Most of the results presented are an outgrowth of the author’s own research
This research monograph brings together, for the first time, the varied literature on Yosida
approximations of stochastic differential equations (SDEs) in infinite dimensions and
their applications into a single cohesive work. The author provides a clear and systematic
introduction to the Yosida approximation method and justifies its power by presenting
its applications in some practical topics such as stochastic stability and stochastic
optimal control. The theory assimilated spans more than 35 years of mathematics, but is
developed slowly and methodically in digestible pieces.
The book begins with a motivational chapter that introduces the reader to several
different models that play recurring roles throughout the book as the theory is unfolded,
and invites readers from different disciplines to see immediately that the effort required to
work through the theory that follows is worthwhile. From there, the author presents the
necessary prerequisite material, and then launches the reader into the main discussion of
the monograph, namely, Yosida approximations of SDEs, Yosida approximations of SDEs
with Poisson jumps, and their applications. Most of the results considered in the main
chapters appear for the first time in a book form, and contain illustrative examples on
stochastic partial differential equations. The key steps are included in all proofs, especially
the various estimates, which help the reader to get a true feel for the theory of Yosida
approximations and their use. This work is intended for researchers and graduate students
in mathematics specializing in probability theory and will appeal to numerical analysts,
engineers, physicists and practitioners in finance who want to apply the theory of
stochastic evolution equations. Since the approach is based mainly in semigroup theory, it
is amenable to a wide audience including non-specialists in stochastic processes.
1st ed. 2016, XIX, 405 p.
Hardcover
ISBN 978-3-319-45682-9