Semiclassical resonances in some simple cases

Small Disks and Semiclassical Resonances

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally s...

Counting phylogenetic invariants in some simple cases.

An informal degrees of freedom argument is used to count the number of phylogenetic invariants in cases where we have three or four species and can assume a Jukes-Cantor model of base substitution with or without a molecular clock. A number of simple cases are treated and in each the number of invariants can be found. Two new classes of invariants are found: non-phylogenetic cubic invariants te...

Eigenvalues of the Reference Operator and Semiclassical Resonances

Fractal Upper Bounds on the Density of Semiclassical Resonances

Nonlinear resonances and multi-stability in simple neural circuits.

This article describes a numerical procedure designed to tune the parameters of periodically driven dynamical systems to a state in which they exhibit rich dynamical behavior. This is achieved by maximizing the diversity of subharmonic solutions available to the system within a range of the parameters that define the driving. The procedure is applied to a problem of interest in computational ne...