The homogenization of $3 \times 3$ system of differential equations related to the Coriolis and Lorentz forces are studied. It generates memory effects. The memory (or nonlocal) kernel is described by the Volterra integral equation. When the coefficient is independent of time, the memory kernel can be characterized explicitly in terms of Young’s measure. The kinetic formulation of the homogenized equation is also obtained.

A conjecture on lifting to Siegel cusp forms of half-integral weight $k - 1/2$ of degree two from each pair of cusp forms of $SL_{2}(\mathbb{Z})$ of weight $2k - 2$ and $2k - 4$ is given with a conjectural relation of the $L$ functions and numerical evidences. We also describe the space of Siegel modular forms of half-integral weight, its “plus subspace” and Jacobi forms of degree two by explicitly given theta functions.

We introduce the stochastic integration with respect to the infinitedimensional fractional Brownian motion. Using the techniques of the anticipating stochastic calculus, we derive an Itô formula for Hurst parameter bigger than $\frac{1}{2}$.

In this paper, we study the minimality of the boundary of a Coxeter system. We show that for a Coxeter system $(W, S)$ if there exist a maximal spherical subset $T$ of $S$ and an element $s_{0} \in S$ such that $m(s_{0}, t) \geq 3$ for each $t \in T$ and $m(s_{0}, t_{0}) = \infty$ for some $t_{0} \in T$, then every orbit $W\alpha$ is dense in the boundary $\partial \Sigma (W, S)$ of the Coxeter system $(W, S)$, hence $\partial \Sigma (W, S)$ is minimal, where $m(s_{0}, t)$ is the order of $s_{0}t$ in $W$.