Due 2019-10-26
Approx. 150 p. 15 illus. in color.
Printed book
Softcover
ISBN 978-981-32-9068-6
Gives a detailed introduction to ergodicity and symplectic and multisymplectic
structures for stochastic nonlinear Schrodinger equations
Provides the study of ergodic numerical approximations for stochastic
nonlinear Schrodinger equations without strong dissipative terms
Constructs numerical approximations which inherit both dynamical behaviors
and geometric structures even for stochastic nonlinear Schrodinger equation
of conservative type
This book provides some recent advance in the study of stochastic nonlinear Schrodinger
equations and their numerical approximations, including the well-posedness, ergodicity,
symplecticity and multi-symplecticity. It gives an accessible overview of the existence and
uniqueness of invariant measures for stochastic differential equations, introduces geometric
structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrodinger
equations and their numerical approximations, and studies the properties and convergence
errors of numerical methods for stochastic nonlinear Schrodinger equations. This book will
appeal to researchers who are interested innumerical analysis, stochastic analysis, ergodic
theory, partial differential equation theory, etc.
Due 2019-11-20
1st ed. 2019, 385 p.
Printed book
Hardcover
ISBN 978-3-030-26383-6
Highlights an unprecedented number of real-life applications of differential
equations and systems
Includes problems in biomathematics, finance, engineering, physics, and even
societal ones like rumors and love
Includes selected challenges to motivate further research in this field
This book highlights an unprecedented number of real-life applications of differential equations
together with the underlying theory and techniques. The problems and examples presented
here touch on key topics in the discipline, including first order (linear and nonlinear) differential
equations, second (and higher) order differential equations, first order differential systems, the
Runge?Kutta method, and nonlinear boundary value problems. Applications include growth of
bacterial colonies, commodity prices, suspension bridges, spreading rumors, modeling the
shape of a tsunami, planetary motion, quantum mechanics, circulation of blood in blood
vessels, price-demand-supply relations, predator-prey relations, and many more. Upper
undergraduate and graduate students in Mathematics, Physics and Engineering will find this
volume particularly useful, both for independent study and as supplementary reading. While
many problems can be solved at the undergraduate level, a number of challenging real-life
applications have also been included as a way to motivate further research in this vast and
fascinating field
Due 2019-10-22
1st ed. 2019, XVIII, 235 p. 24 illus.
Printed book
Softcover
ISBN 978-3-030-26645-5
Provides a self-contained presentation of the mathematical foundations of
Game Theory including recent advances in dynamics and learning
All results are proved in their full generality using tools and concepts defined in the text
Includes 60 exercises, with solutions, which often contain complementary
results and alternative proofs
This book gives a concise presentation of the mathematical foundations of Game Theory, with
an emphasis on strategic analysis linked to information and dynamics. It is largely selfcontained,
with all of the key tools and concepts defined in the text. Combining the basics of
Game Theory, such as value existence theorems in zero-sum games and equilibrium existence
theorems for non-zero-sum games, with a selection of important and more recent topics such
as the equilibrium manifold and learning dynamics, the book quickly takes the reader close to
the state of the art. Applications to economics, biology, and learning are included, and the
exercises, which often contain noteworthy results, provide an important complement to the text.
Based on lectures given in Paris over several years, this textbook will be useful for rigorous, upto-date
courses on the subject. Apart from an interest in strategic thinking and a taste for
mathematical formalism, the only prerequisite for reading the book is a solid knowledge of
mathematics at the undergraduate level, including basic analysis, linear algebra, and probability.
Due 2019-11-05
X, 291 p. 12 illus.
Printed book
Hardcover
ISBN 978-3-030-26453-6
Presents Grobner bases and quiver theories as providers of computing
models for differential equations and systems
Offers a historical background for a better understanding of how theories developed
Appeals to a wide readership, from graduate students to researchers and scholars
This edited volume presents a fascinating collection of lecture notes focusing on differential
equations from two viewpoints: formal calculus (through the theory of Grobner bases) and
geometry (via quiver theory). Grobner bases serve as effective models for computation in
algebras of various types. Although the theory of Grobner bases was developed in the second
half of the 20th century, many works on computational methods in algebra were published
well before the introduction of the modern algebraic language. Since then, new algorithms
have been developed and the theory itself has greatly expanded. In comparison, diagrammatic
methods in representation theory are relatively new, with the quiver varieties only being
introduced ? with big impact ? in the 1990s. Divided into two parts, the book first discusses
the theory of Grobner bases in their commutative and noncommutative contexts, with a focus
on algorithmic aspects and applications of Grobner bases to analysis on systems of partial
differential equations, effective analysis on rings of differential operators, and homological
algebra. It then introduces representations of quivers, quiver varieties and their applications to
the moduli spaces of meromorphic connections on the complex projective line. While no
particular reader background is assumed, the book is intended for graduate students in
mathematics, engineering and related fields, as well as researchers and scholars.
February 2020
Pages: 200
The aim of the book is to present a precise and comprehensive introduction to the basic theory of derived functors, with an emphasis on sheaf cohomology and spectral sequences. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf theory, and also from algebra.
The first part of the book provides the foundational material: Chapter 1 deals with category theory and homological algebra. Chapter 2 is devoted to the development of the theory of derived functors, based on the notion of injective object. In particular, the universal properties of derived functors is stressed, with a view to make the proofs in the following chapters as simple and natural as possible. Chapter 3 provides a rather thorough introduction to sheaves, in a general topological setting. Chapter 4 introduces sheaf cohomology as a derived functor, and, after also defining ?ech cohomology, develops a careful comparison between the two cohomologies which is a detailed analysis not easily available in the literature. This comparison is made using general, universal properties of derived functors. This chapter also establishes the relations with the de Rham and Dolbeault cohomologies. Chapter 5 offers a friendly approach to the rather intricate theory of spectral sequences by means of the theory of derived triangles, which is precise and relatively easy to grasp. It also includes several examples of specific spectral sequences. Readers will find exercises throughout the text, with additional exercises included at the end of each chapter.
Basic Notions: Category Theory, Homological Algebra
Derived Functors: Injective Objects, Right Derived Functors, Free Resolutions, the Long Exact Sequence of a Derived Functor, Acyclic Resolutions, δ-Functors, Left Derived Functors, Additional Exercises
Sheaves: Presheaves and Sheaves, Etale Space, Direct and Inverse Images, Additional Exercises
Cohomology of Sheaves: ?ech Cohomology, Sheaf Cohomology, Comparing Sheaf and ?ech Cohomology, Fine Sheaves and de Rham Cohomology, Sheaf Cohomology of Schemes, Additional Exercises
Spectral Sequences: Filtered Complexes, the Spectral Sequence of a Filtered Complex, the Spectral Sequences Associated with a Double Complex, Hypercohomology, the Spectral of Hyperderived Functors, Some Applications, Additional Exercises
Appendix: Pushouts and Pullbacks, Snake Lemma, Baer's Criterion, Additional Exercises
Graduate students and researchers interested in derived functors and sheaf cohomology.