Georgia Institute of Technology College of Sciences

We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on R^d with a quadratic, conservative nonlinearity B(x, x) constrained to possess various properties common to finite-dimensional fluid models and a linear damping term −Ax that acts only on a proper subset of phase space in the sense that dim(kerA) ≫ 1. Existence of a stationary measure is straightforward if kerA = {0}, but when the kernel of A is nontrivial a stationary measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on B that guarantees the existence of a stationary measure and prove that they hold “generically” within our constraint class of nonlinearities provided that dim(kerA) < 2d/3 and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction dim(kerA) < 2d/3 can be removed if one allows the nonlinearity to change by a small amount at discrete times. In particular, for a Markov chain obtained by evolving our SDE on approximately unit random time intervals and slightly perturbing the nonlinearity within our constraint class at each timestep, we prove that there exists a stationary measure whenever just a single mode is damped.

Finding Cheeger cuts of graphs is an NP-hard problem, and one often resorts to approximate solutions. In the literature, spectral graph theory provides the most popular approaches for obtaining such approximate solutions. Recently, K.C. Chang introduced a novel nonlinear spectral graph theory and proved that the seek of Cheeger cuts is equivalent to solving a constrained optimization problem. However, this resulting optimization problem is also very challenging as it involves a non-differentiable function over a non-convex set that is composed of simplex cells of different dimensions. In this talk, we will discuss an ADMM algorithm for solving this optimization problem and provide some convergence analysis. Experimental results will be presented for typical graphs, including Petersen's graph and Cockroach graphs, the well-known Zachary karate club graph, and some preliminary applications in material sciences.

Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a quasikernel if $Q$ is an independent set and if every vertex of $D$ can be reached from $Q$ by a path of length at most 2.  The Small Quasikernel Conjecture of P.L. Erdős and Székely from 1976 states that every $n$-vertex source-free digraph $D$ contains a quasikernel of size at most $\frac{1}{2}n$.  Despite being posed nearly 50 years ago, very little is known about this conjecture, with the only non-trivial upper bound of $n-\frac{1}{4}\sqrt{n\log n}$ being proven recently by ourself.  We discuss this result together with a number of other related results and open problems around the Small Quasikernel Conjecture.