Georgia Institute of Technology College of Sciences

We analyze two-link (or three-link) 2D snake like locomotions and discuss the optimization of the motion. The snake is modeled as two (or three) identical links connected via hinge joints and the relative angles between the links are prescribed as periodic actuation functions. An essential feature of the locomotion is the anisotropy of friction coefficients. The dynamics of the snake is analyzed numerically, as well as analytically for small amplitude actuations of the relative angles. Cost of locomotion is defined as the ratio between distance traveled by the snake and the energy expenditure within one period. Optimal conditions of the highest efficiency in terms of the friction coefficients and the actuations are discussed for the model.

A discussion of the paper "RNA folding with soft constraints: reconciliation of probing data and thermodynamic secondary structure prediction" by Washietl et al (NAR, 2012).

I will discuss recent work on the stability of linear equations under parametric forcing by colored noise. The noises considered are built from Ornstein-Uhlenbeck vector processes. Stability of the solutions is determined by the boundedness of their second moments. Our approach uses the Fokker-Planck equation and the associated PDE for the marginal moments to determine the growth rate of the moments. This leads to an eigenvalue problem, which is solved using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes into "ladder operators." The results are given in terms of a perturbation expansion in the size of the noise. We have found very good agreement between our results and numerical simulations. This is joint work with L.A. Romero.

A graph $G$ contains a graph $H$ as an immersion if there exist distinct vertices $\pi(v) \in V(G)$ for every vertex $v \in V(H)$ and paths $P(e)$ in $G$ for every $e \in E(H)$ such that the path $P(uv)$ connects the vertices $\pi(u)$ and $\pi(v)$ in $G$ and furthermore the paths $\{P(e):e \in E(H)\}$ are pairwise edge disjoint. Thus, graph immersion can be thought of as a generalization of subdivision containment where the paths linking the pairs of branch vertices are required to be pairwise edge disjoint instead of pairwise internally vertex disjoint. We will present a simple structure theorem for graphs excluding a fixed $K_t$ as an immersion. The structure theorem gives rise to a model of tree-decompositions based on edge cuts instead of vertex cuts. We call these decompositions tree-cut decompositions, and give an appropriate definition for the width of such a decomposition. We will present a ``grid" theorem for graph immersions with respect to the tree-cut width. This is joint work with Paul Seymour.