Georgia Institute of Technology College of Sciences

I will show a new approach based on the discrepancy of the constraint matrix to verify integer feasibility of polytopes. I will then use this method to show a threshold phenomenon for integer feasibility of random polytopes. The random polytope model that we consider is P(n,m,x0,R) - these are polytopes in n-dimensional space specified by m "random" tangential hyperplanes to a ball of radius R centered around the point x0. We show that there exist constants c_1 < c_2 such that with high probability, the random polytope P(n,m,x0=(0.5,...,0.5),R) is integer infeasible if R is less than c_1sqrt(log(2m/n)) and the random polytope P(n,m,x0,R) is integer feasible for every center x0 if the radius R is at least c_2sqrt(log(2m/n)). Thus, a transition from infeasibility to feasibility happens within a constant factor increase in the radius. Moreover, if the polytope contains a ball of radius Omega(log (2m/n)), then we can find an integer solution with high probability (over the input) in randomized polynomial time. This is joint work with Santosh Vempala.

We discuss some recent generalizations of Euler--Maclaurin expansions for the trapezoidal rule and of analogous asymptotic expansions for Gauss--Legendre quadrature, in the presence of arbitrary algebraic-logarithmic endpoint singularities. In addition of being of interest by themselves, these asymptotic expansions enable us to design appropriate variable transformations to improve the accuracies of these quadrature formulas arbitrarily. In general, these transformations are singular, and their singularities can be adjusted easily to achieve this improvement. We illustrate this issue with a numerical example involving Gauss--Legendre quadrature. We also discuss some recent asymptotic expansions of the coefficients of Legendre polynomial expansions of functions over a finite interval, assuming that the functions may have arbitrary algebraic-logarithmic interior and endpoint suingularities. These asymptotic expansions can be used to make definitive statements on the convergence acceleration rates of extrapolation methods as these are applied to the Legendre polynomial expansions.

There are over 28,000 species of fishes, and a key feature of this remarkable evolutionary diversity is a great variety of propulsive systems used by fishes for maneuvering in the aquatic environment. Fishes have numerous control surfaces (fins) which act to transfer momentum to the surrounding fluid. In this presentation I will discuss the results of recent experimental kinematic and hydrodynamic studies of fish fin function, and their implications for the construction of robotic models of fishes. Recent high-resolution video analyses of fish fin movements during locomotion show that fins undergo much greater deformations than previously suspected and fish fins possess an clever active surface control mechanism. Fish fin motion results in the formation of vortex rings of various conformations, and quantification of vortex rings shed into the wake by freely-swimming fishes has proven to be useful for understanding the mechanisms of propulsion. Experimental analyses of propulsion in freely-swimming fishes have led to the development of a variety of self-propelling robotic models: pectoral fin and caudal fin (tail) robotic devices, and a flapping foil model fish of locomotion. Data from these devices will be presented and discussed in terms of the utility of using robotic models for understanding fish locomotor dynamics.