93 research outputs found
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks
This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems
Constant mean curvature surfaces
In this article we survey recent developments in the theory of constant mean
curvature surfaces in homogeneous 3-manifolds, as well as some related aspects
on existence and descriptive results for -laminations and CMC foliations of
Riemannian -manifolds.Comment: 102 pages, 17 figure
Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term
A general purely crystalline mean curvature flow equation with a nonuniform
driving force term is considered. The unique existence of a level set flow is
established when the driving force term is continuous and spatially Lipschitz
uniformly in time. By introducing a suitable notion of a solution a comparison
principle of continuous solutions is established for equations including the
level set equations. An existence of a solution is obtained by stability and
approximation by smoother problems. A necessary equi-continuity of approximate
solutions is established. It should be noted that the value of crystalline
curvature may depend not only on the geometry of evolving surfaces but also on
the driving force if it is spatially inhomogeneous.Comment: 23 page
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