Modelling the dynamics of turbulent floods

Consider the dynamics of turbulent flow in rivers, estuaries and floods. Based on the widely used k-epsilon model for turbulence, we use the techniques of centre manifold theory to derive dynamical models for the evolution of the water depth and of vertically averaged flow velocity and turbulent parameters. This new model for the shallow water dynamics of turbulent flow: resolves the vertical structure of the flow and the turbulence; includes interaction between turbulence and long waves; and gives a rational alternative to classical models for turbulent environmental flows

Completely dissociative groupoids

Consider arbitrarily parenthesized expressions on the $k$ variables $x_0, x_1, ..., x_{k-1}$, where each $x_i$ appears exactly once and in the order of their indices. We call these expressions {\em formal $k$--products}. $F^\sigma(k)$ denotes the set of formal $k$--products. For ${{\bf u},{\bf v}}\subseteq F^\sigma(k)$, the claim, that ${\bf u}$ and ${\bf v}$ produce equal elements in a groupoid $G$ for all values assumed in $G$ by the variables $x_i$, attributes to $G$ a {\em generalized associative law}. Many groupoids are {\em completely dissociative}; i.e., no generalized associative law holds for them; two examples are the groupoids on ${0,1}$ whose binary operations are implication and NAND. We prove a variety of results of that flavor.Comment: 29 page

Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models