The motto of connectivity and superconductivity is that the solutions of the Ginzburg--Landau equations are qualitatively influenced by the topology of the boundaries, as in multiply-connected samples. Special attention is paid to the "zero set", the set of the positions (also known as "quantum vortices") where the order parameter vanishes. The effects considered here usually become important in the regime where the coherence length is of the order of the dimensions of the sample. It takes the intuition of physicists and the awareness of mathematicians to find these new effects. In Connectivity and Superconductivity, theoretical and experimental physicists are brought together with pure and applied mathematicians to review these surprising results. This volume is intended to serve as a reference book for graduate students and researchers in physics or mathematics interested in superconductivity, or in the Schrödinger equation as a limiting case of the Ginzburg--Landau equations.

The motto of connectivity and superconductivity is that the solutions of the Ginzburg-Landau equations are qualitatively in?uenced by the topology of the boundaries. Special attention is given to the "zero set,"the set of the positions (usually known as "quantum vortices") where the order parameter vanishes. The paradigm of connectivity and superconductivity is the Little- Parks e?ect, discussed in most textbooks on superconductivity. This volume is intended to serve as a reference book for graduate students and researchers in physics or mathematics interested in superconductivity, or in the Schr] odinger equation as a limiting case of the Ginzburg-Landau equations. The e?ects considered here usually become important in the regime where the coherence length is of the order of the dimensions of the sample. While in the Little-Parks days a lot of ingenuity was required to achieve this regime, present microelectronic techniques have transformed it into a routine. Mo- over, measurement and visualization techniques are developing at a pace which makes it reasonable to expect veri?cation of distributions, and not only of global properties. Activity in the ?eld has grown and diversi?ed substantially in recent years. We have therefore invited experts ranging from experimental and theoretical physicists to pure and applied mathematicians to contribute articles for this book. While the skeleton of the book deals with superconductivity, micron- works and generalizations of the Little-Parks situation, there are also articles which deal with applications of the Ginzburg-Landau formalism to several fundamental topics, such as quantum coherence, cosmology, and questions in materials science."

A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, physics and engineering. The presentation is lively and up to date, paying particular emphasis to developing an appreciation of underlying mathematical theory. Beginning with basic definitions, properties and derivations of some basic equations of mathematical physics from basic principles, the book studies first order equations, classification of second order equations, and the one-dimensional wave equation. Two chapters are devoted to the separation of variables, whilst others concentrate on a wide range of topics including elliptic theory, Green's functions, variational and numerical methods. A rich collection of worked examples and exercises accompany the text, along with a large number of illustrations and graphs to provide insight into the numerical examples. Solutions to selected exercises are included for students whilst extended solution sets are available to lecturers from [email protected].