S. K. Lucas | Searcharxiv

arXiv2007

Clustering of spectra and fractals of regular graphs

We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labelled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.

math.CO↗

arXiv1995

Capillary Wave Scattering from a Surfactant Domain

The study of capillary wave scattering by a circular region with different interfacial properties from the rest of an otherwise homogeneous interface is motivated by experiments on wave attenuation at a monolayer covered air-water interface where domains of one surface phase are dispersed in a second surface phase. Here the scattering function is calculated for an incident wave of frequency $ω$ (wavevector $k_0$) scattering from an isolated circular domain of radius $a$ with surface tension $σ_{1}$ which is imbedded in an otherwise infinite interface of surface tension $σ_{0}$. The underlying fluid is treated as irrotational and the three-dimensional flow problem coupling the heterogeneous surface to the underlying liquid is reduced to a set of dual integral equations, which are solved numerically. With this solution the scattering amplitudes and the total scattering cross sections are calculated as a function of the surface tension ratio $σ_{0}/σ_{1}$ and incident wavenumber $k_0 a$. The analogous problem of a discontinuous change in bending rigidity is also considered and the solution to the complete viscous problem is outlined in an appendix. Experimental implications of these results are discussed.

cond-mat↗