Series: Springer Monographs in Mathematics
2014, XI, 139 p. 28 illus., 20 illus. in color.
Hardcover
ISBN 978-1-4939-1937-6
Due: November 14, 2014
Includes discussion topics and questions
Informal survey-style benefits, survey courses, and seminar classes focused on discussion and independent projects
Gradual increase in depth as the book progresses - refocuses students and learners on topics covered in earlier chapters
Focuses on particular kind of feature, vortex filaments, in numerous areas of physics rather than a particular area of physics
This book presents fundamental concepts and seminal results to the study of vortex filaments in equilibrium. It also presents new discoveries in quasi-2D vortex structures with applications to geophysical fluid dynamics and magnetohydrodynamics in plasmas. It fills a gap in the vortex statistics literature by simplifying the mathematical introduction to this complex topic, covering numerical methods, and exploring a wide range of applications with numerous examples.
The authors have produced an introduction that is clear and easy to read, leading the reader step-by-step into this topical area. Alongside the theoretical concepts and mathematical formulations, interesting applications are discussed. This combination makes the text useful for students and researchers in mathematics and physics.
Introduction.- Vortex Filaments and Where to Find Them.- Statistical Mechanics.- Parallel Filaments.- Curved Filaments.- Quantum Fluids.- Plasmas.- Computational Methods.- Quasi 2-D Monte Carlo in Deep Ocean Convection.- Conclusion.
Series: Applied Mathematical Sciences, Vol. 190
2014, XIII, 333 p. 3 illus., 1 illus. in color.
Hardcover
ISBN 978-3-319-11085-1
Due: December 14, 2014
Written by well known international researchers based on their lectures between 2007 and 2013
Accessible to broad audience with examples and exercises throughout
Topics are first approached with simpler scalar Helmholtz equations before turning to Maxwell equations
Appendix material includes results from functional analysis, vector calculus, and differential geometry
This book gives a concise introduction to the basic techniques needed for the theoretical analysis of the Maxwell Equations, and filters in an elegant way the essential parts, e.g., concerning the various function spaces needed to rigorously investigate the boundary integral equations and variational equations. The book arose from lectures taught by the authors over many years and can be helpful in designing graduate courses for mathematically orientated students on electromagnetic wave propagation problems. The students should have some knowledge on vector analysis (curves, surfaces, divergence theorem) and functional analysis (normed spaces, Hilbert spaces, linear and bounded operators, dual space).
Written in an accessible manner, topics are first approached with simpler scale Helmholtz Equations before turning to Maxwell Equations. There are examples and exercises throughout the book. It will be useful for graduate students and researchers in applied mathematics and engineers working in the theoretical approach to electromagnetic wave propagation.
Introduction.- Expansion into Wave Functions.- Scattering From a Perfect Conductor.- The Variational Approach to the Cavity Problem.- Boundary Integral Equation Methods for Lipschitz Domains.- Appendix.- References.- Index.
Series: Lecture Notes in Mathematics, Vol. 2122
2015, X, 204 p. 17 illus.
Softcover
ISBN 978-3-319-11336-4
Due: December 14, 2014
An introduction to the techniques of Geometric Invariant Theory via a detailed analysis of the GIT problem for polarized curves
An introduction to the problem of compactifying moduli spaces through an interpretation of the output of the GIT analysis
An introduction to the rich theory of compactified Jacobians for singular curves via three explicit examples
A detailed description of the quotient stacks associated to the different GIT quotients, illustrating the interplay between these two techniques
We investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as d decreases with respect to g. We prove that the first three values of d at which the GIT quotients change are given by d=a(2g-2) where a=2, 3.5, 4. We show that, for a>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3.5<a<4, the Hilbert semistable locus coincides with the Chow semistable locus and it maps to the moduli stack of weakly-pseudo-stable curves. If 2<a<3.5, the Hilbert and Chow semistable loci coincide and they map to the moduli stack of pseudo-stable curves. We also analyze in detail the critical values a=3.5 and a=4, where the Hilbert semistable locus is strictly smaller than the Chow semistable locus. As an application, we obtain three compactications of the universal Jacobian over the moduli space of stable curves, weakly-pseudo-stable curves and pseudo-stable curves, respectively.
Introduction.- Singular Curves.- Combinatorial Results.- Preliminaries on GIT.- Potential Pseudo-stability Theorem.- Stabilizer Subgroups.- Behavior at the Extremes of the Basic Inequality.- A Criterion of Stability for Tails.- Elliptic Tails and Tacnodes with a Line.- A Strati_cation of the Semistable Locus.- Semistable, Polystable and Stable Points (part I).- Stability of Elliptic Tails.- Semistable, Polystable and Stable Points (part II).- Geometric Properties of the GIT Quotient.- Extra Components of the GIT Quotient.- Compacti_cations of the Universal Jacobian.- Appendix: Positivity Properties of Balanced Line Bundles.
Series: Lecture Notes in Mathematics, Vol. 2126
2015, XVII, 508 p. 58 illus., 37 illus. in color.
Softcover
ISBN 978-3-319-11321-0
Due: December 14, 2014
Unique combination of mathematical methods and their applications to analysis of models from natural sciences
Unique blend of analytic and probabilistic methods for evolutionary equation
Comprehensive presentation of main methods of analysis of deterministic fragmentation and coagulation models
Comprehensive exposition of probabilistic methods for long term dynamics of evolution equations
Up to date presentation of pattern formations
Numerics for flows in complex geometries
With the unifying theme of abstract evolutionary equations, both linear and nonlinear, in a complex environment, the book presents a multidisciplinary blend of topics, spanning the fields of theoretical and applied functional analysis, partial differential equations, probability theory and numerical analysis applied to various models coming from theoretical physics, biology, engineering and complexity theory.
Truly unique features of the book are: the first simultaneous presentation of two complementary approaches to fragmentation and coagulation problems, by weak compactness methods and by using semigroup techniques, comprehensive exposition of probabilistic methods of analysis of long term dynamics of dynamical systems, semigroup analysis of biological problems and cutting edge pattern formation theory.
The book will appeal to postgraduate students and researchers specializing in applications of mathematics to problems arising in natural sciences and engineering.
Wilson Lamb: Applying functional analytic techniques to evolution equations.- Adam Bobrowski: Boundary conditions in evolutionary equations in biology.-Ernesto Estrada: Introduction to Complex Networks: Structure and Dynamics.-Jacek Banasiak: Kinetic models in natural sciences.- Philippe Laurencot: Weak compactness techniques and coagulation equations.- Ryszard Rudnicki: Stochastic operators and semigroups and their applications in physics and biology.- Mustapha Mokhtar-Kharroubi: Spectral theory for neutron transport.-Anna Marciniak-Czochra: Reaction-diffusion-ODE models of pattern formation.- Mapundi Kondwani Banda: Nonlinear Hyperbolic Systems of Conservation Laws and Related Applications.