This thesis is on the existence and uniqueness of weak solutions to the Navier-Stokes equations in R3 which govern the velocity of incompressible fluid with viscosity ν. The solution is obtained in the space of tempered distributions on R3 given an initial condition and forcing data which are dominated by...
Statistical mechanics studies the probability that a system is in a certain state given one or more constraints which are usually fixed conserved quantities. It is a particularly useful and powerful approach for problems with a large number of degrees of freedom where a complete knowledge of the system is...
We model a fish population in a spatial region comprising a marine protected area and a fishing ground separated by an interface. The model assumes conservation of biomass density and takes the form of a reaction diffusion equation with a logistic reaction term. At the interface, in addition to continuity...
In this thesis, high resolution ocean models are used to evaluate and forecast coastal ocean variability in two different applications. In the first study, the 2-km resolution ocean circulation model for the Eastern Bering Sea is utilized to understand whether slope-interior exchange along the path of the Aleutian North Slope...
Markov chains have long been used to sample from probability distributions and simulate dynamical systems. In both cases we would like to know how long it takes for the chain's distribution to converge to within varepsilon of the stationary distribution in total variation distance; the answer to this is, called...
The bacteria Myxococcus xanthus clusters into fruiting bodies in the absence of food. The movement of M. xanthus has three aspects: self-propulsion (run), change of direction due to collision (tumble), and Brownian fluctuations in movement. We propose a minimalist PDE model for the concentrations of left and right moving agents...
In this dissertation, we use Fourier-analytic methods to study questions of equidistribution on the compact abelian group Zp of p-adic integers. In particu- lar, we prove a LeVeque-type Fourier analytic upper bound on the discrepancy of sequences. We establish p-adic analogues of the classical Dirichlet and Fejér kernels on R/Z,...
This paper concerns a question that frequently occurs in various applications: Is any dispersal coupling of stable discrete linear systems, also stable? Although it has been known this is not the case, we shall identify a reasonably diverse class of systems for which it is true. We shall employ the...
The topic of statistical mechanics has been studied for over a century, and it is one of the pillars of modern physics. This theory can be applied to the study of the thermodynamic behavior of large systems of interacting particles, in which case it is referred to as equilibrium statistical...
Stein's method initially introduced in 1970 by C. Stein is a powerful technique for bounding the distance between the laws of two real-valued random variables. Stein's method has been used to prove distributional convergence to many standard probability distributions such as normal, multivariate normal, Poisson and Brownian motion approximation. In...