Johns Hopkins Studies in the Mathematical Sciences – serie

Graph theory is one of the fastest growing branches of mathematics. Until recently, it was regarded as a branch of combinatorics and was best known by the famous four-color theorem stating that any map can be colored using only four colors such that no two bordering countries have the same color. Now graph theory is an area of its own with many deep results and beautiful open problems. Graph theory has numerous applications in almost every field of science and has attracted new interest because of its relevance to such technological problems as computer and telephone networking and, of course, the internet. In this new book in the Johns Hopkins Studies in the Mathematical Science series, Bojan Mohar and Carsten Thomassen look at a relatively new area of graph theory: that associated with curved surfaces. Graphs on surfaces form a natural link between discrete and continuous mathematics. The book provides a rigorous and concise introduction to graphs on surfaces and surveys some of the recent developments in this area.Among the basic results discussed are Kuratowski's theorem and other planarity criteria, the Jordan Curve Theorem and some of its extensions, the classification of surfaces, and the Heffter-Edmonds-Ringel rotation principle, which makes it possible to treat graphs on surfaces in a purely combinatorial way. The genus of a graph, contractability of cycles, edge-width, and face-width are treated purely combinatorially, and several results related to these concepts are included. The extension by Robertson and Seymour of Kuratowski's theorem to higher surfaces is discussed in detail, and a shorter proof is presented. The book concludes with a survey of recent developments on coloring graphs on surfaces.

Primarily intended for research mathematicians and computer scientists, Combinatorics and Partially Ordered Sets: Dimension Theory also serves as a useful text for advanced students in either field. William Trotter concentrates on combinatorial topics for finite partially ordered sets, and with dimension theory serving as a unifying theme, research on partially ordered sets or posets is linked to more traditional topics in combinatorial mathematics -- including graph theory, Ramsey theory, probabilistic methods, hypergraphs, algorithms, and computational geometry. The book's most important contribution is to collect, organize, and explain the many theorems on partially ordered sets in a way that makes them available to the widest possible audience.

Waves occur naturally in a vast number of scientific or engineering situations. Ripples on a pond, the light we see, and the oscillations of bridges and buildings can often be described as solitary or interacting waves. Wave theory is therefore one of the most important branches of pure and applied science. In Modulated Waves: Theory and Applications Lev Ostrovsky and Alexander Potapov consider linear and nonlinear waves such as solitons, waves in inhomogeneous media, and many others. They discuss modulated waves-those characterized by a slow variation of the macroscopic parameters of amplitude, frequency, and profile. Most of the fundamentals of wave theory may be understood by considering this class of waves. Theoretical analysis is supported by examples from different branches of physics: electrodynamics, fluid mechanics, acoustics, optics, and the mechanics of solids.