Georgia Institute of Technology College of Sciences

Variable transformations are used to enhance the normally poor performance of trapezoidal rule approximations of finite-range integrals I[f]=\int^1_0f(x)dx. Letting x=\psi(t), where \psi(t) is an increasing function for 0 < t < 1 and \psi(0)=0 and \psi(1)=1, the trapezoidal rule is applied to the transformed integral I[f]=\int^1_0f(\psi(t))\psi'(t)dt. By choosing \psi(t) appropriately, approximations of very high accuracy can be obtained for I[f] via this approach. In this talk, we survey the various transformations that exist in the literature. In view of recent generalizations of the classical Euler-Maclaurin expansion, we show how some of these transformations can be tuned to optimize the numerical results. If time permits, we will also discuss some recent asymptotic expansions for Gauss-Legendre integration rules in the presence of endpoint singularities and show how their performance can be optimized by tuning variable transformations. The variable transformation approach presents a very flexible device that enables one to write his/her own high-accuracy numerical integration code in a simple way without the need to look up tables of abscissas and weights for special Gaussian integration formulas.

Let K^r_{r+1} denote the complete r-graph on r+1 vertices. The Turan density of K^r_{r+1} is the largest number t such that there are infinitely many K^r_{r+1}-free r-graphs with edge density t-o(1). Determining t(K^r_{r+1}) for r > 2 is a famous open problem of Turan. The best upper bound for even r, t(K^r_{r+1})\leq 1-1/r, was given by de Caen and Sidorenko. In a joint work with Linyuan Lu, we slightly improve it. For example, we show that t(K^r_{r+1})\leq 1 - 1/r - 1/(2r^3) for r=4 mod 6.  One of our lemmas also leads to an exact result for hypergraphs.  Given r > 2, let p be the smallest prime factor of r-1. Every r-graph on n > r(p-1) vertices such that every r+1 vertices contain 0 or r edges must be empty or a complete star.

Under certain conditions, we obtain exact asymptotic expressions for the stationary distribution \pi of a Markov chain.  In this talk, we will consider Markov chains on {0,1,...}^2.  We are particularly interested in deriving asymptotic expressions when the fluid limit of the most probable paths from the origin to the rare event are nonlinear.  For example, we will derive asymptotic expressions for a large deviation along the x-axis (e.g., \pi(\ell, y) for fixed y) when the most probable paths to (\ell,y) initially climb the y-axis before turning southwest and drifting towards (\ell,y).

A plasma is a gas of ionized particles. For a dilute plasma of very high temperature, the collisions can be ignored. Such situations occur, for example, in nuclear fusion devices and space plasmas. The Vlasov-Poisson and Vlasov-Maxwell equations are kinetic models for such collisionless plasmas. The Vlasov-Poisson equation is also used for galaxy evolution. I will describe some mathematical results on these models, including well-posedness and stability issues.