Part of Cambridge Texts in Applied Mathematics
Publication planned for: October 2015
format: Hardback
isbn: 9781107098411
Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.
Preface
Part I. Setting the Scene:
1. What are singularities all about?
2. Blow-up
3. Similarity profile
4. Continuum equations
5. Local singular expansions
6. Asymptotic expansions of PDEs
Part II. Formation of Singularities:
7. Drop break-up
8. A numerical example: drop pinch-off
9. Slow convergence
10. Continuation
Part III. Persistent Singularities ? Propagation:
11. Shock waves
12. The dynamical system
13. Vortices
14. Cusps and caustics
15. Contact lines and cracks
Appendix A. Vector calculus
Appendix B. Index notation and the summation convention
Appendix C. Dimensional analysis
References
Index.
The book focuses on the educational perspective of Riemann-Roch spaces and the computation of algebraic structures connected to the Riemann-Roch theorem, which is a useful tool in fields of complex analysis and algebraic geometry. On one hand, the theorem connects the Riemann surface with its topological genus, and on the other it allows us to compute the algebraic function field spaces. In the first part of this book, algebraic structures and some of their properties are presented. The second part shows efficient algorithms and examples connected to Riemann-Roch spaces. What is important, a variety of examples with codes of algorithms are given in the book, covering the majority of the cases.
51 pages
Language:
English
Type of Publication:
Specialist Text
Keywords:
RieRiemann-Roch spaces, Diophantine Equations, Integral Domains
Publication Date:
March 2015
ISBN:
978-3-11-042612-0
ISBN: 978-3-11-037277-9
Product Type: Textbooks
Focuses on the mathematics of cryptographic protocols
Discusses relevant ideas from computer science and complexity theory
Includes chapter on noncommutative algebraic cryptography
The subject of this book is mathematical cryptography. By this we mean the mathematics involved in cryptographic protocols. As the field has expanded, using both commutative and noncommutative algebraic objects as cryptographic platforms, a book describing and explaining all these mathematical methods is of immeasurable value.
Approx. xiv, 388 pages
Language:
English
Type of Publication:
Textbook
to be published May 2015
ISBN 978-3-03719-152-1
March 2015, 165 pages, softcover, 17 x 24 cm.
The topic of this book is systems of points in Coulomb interaction, in
particular, the classical Coulomb gas, and vortices in the Ginzburg-Landau
model of superconductivity. The classical Coulomb and Log gases are classical
statistical mechanics models, which have seen important developments in
the mathematical literature due to their connection with random matrices
and approximation theory. At low temperature, these systems are expected
to “cristallize” to so-called Fekete sets, which exhibit microscopically
a lattice structure.
The Ginzburg-Landau model, on the other hand, describes superconductors.
In superconducting materials subjected to an external magnetic field, densely
packed point vortices emerge, forming perfect triangular lattice patterns,
so-called Abrikosov lattices.
This book describes these two systems and explores the similarity between them. It presents the mathematical tools developed to analyze the interaction between the Coulomb particles or the vortices, at the microscopic scale, and describes a “renormalized energy” governing the point patterns. This is believed to measure the disorder of a point configuration, and to be minimized by the Abrikosov lattice in dimension 2.
The book gives a self-contained presentation of results on the mean field
limit of the Coulomb gas system, with or without temperature, and of the
derivation of the renormalized energy. It also provides a streamlined presentation
of the similar analysis that can be performed for the Ginzburg-Landau model,
including a review of the vortex-specific tools and the derivation of the
critical fields, the mean-field limit and the renormalized energy.
This book includes research on the Lax-Milgram theorem, which can be used to prove existence and uniqueness of weak solutions to partial differential equations and several examples of its application to relevant boundary value problems are presented. The authors also investigate nonlinear control problems for couple partial differential equations arisign from climate and circulation dynamics in the equatorial zone; the integration of partial differential equations (PDE) with the help of non-commutative analysis over octonions and Cayley-Dickson algebras; and the existence and properties of solutions, applications in sequential optimal control with pointwise in time state constraints. (Imprint: Nova)
Preface
The Lax-Milgram Theorem and Some Applications to Partial Differential Equation
(Paul Bracken, Department of Mathematics, University of Texas, Edinburg, TX, USA)
Coupled PDEs and Control Systems Arising in Climate Dynamics: Ocean-Atmosphere Interactions and Tropical Instability Waves
(Aziz Belmiloudi, Institut de Recherche MAthematique de Rennes (IRMAR), Rennes, France)
Integration of PDE with the help of Analysis over Octonions and Cayley-Dickson Algebras
(Sergey V. Ludkovsky, Department of Applied Mathematics, Moscow State Technical University MIREA, Moscow, Russia)
Mixed Boundary-Value Problem for Divergent Hyperbolic PDE: Existence and Properties of Solutions, Applications in Sequential Optimal Control with Pointwise in Time State Constraints
(Vladimir S. Gavrilov and Mikhail I. Sumin, Mechanics and Mathematics Faculty, Nizhnii Novgorod State University, Nizhnii Novgorod, Russia)
Index
Series:
Mathematics Research Developments
Binding: Hardcover
Pub. Date: 2015 - 2nd Quarter
Pages: 6x9 - (NBC-R)
ISBN: 978-1-63482-643-3
Status: AN