Georgia Institute of Technology College of Sciences

Join us for a discussion of making professional mathematics diagrams and illustrations with free vector graphics editing software Inkscape. We'll discuss and tinker with Bezier curves, TexTex, and vectorization of scanned images.

We consider Hamiltonian systems with normally hyperbolic manifold with a homoclinic connection. The systems are of the form H_0(I, phi, x,y) = h(I) + P(x,y) ,where P is a one dimensional system with a homoclinic intersection. The above Hamiltonian is a standard normal form for near integrable Hamiltonians close to a resonance. We consider perturbations that are time dependent and may be not Hamiltonian. We derive explicit formulas for the first order effects on the stable/unstable manifolds. In particular, we give sufficient conditions for the existence of homoclinic intersections to the normally hyperbolic manifold. Previous treatments in the literature specify the types of the unperturbed orbits considered (periodic or quasiperiodic) and are restricted to periodic or quasi-periodic perturbations. We do not need to distinguish on the perturbed orbits and we allow rather general dependence on the time (periodic, quasiperiodic or random). The effects are expressed by very fast converging improper integrals. This is joint work with M. Gidea. https://arxiv.org/abs/1710.01849

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications. Joint work with Pascal Maillard.

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will prove the existence of 5-edge configurations in (G, a0, a1, a2, b1, b2). Joint work with Changong Li, Robin Thomas, and Xingxing Yu.