In this paper, we show that the coupled modified KdV equations possess rich mathematical structures and some remarkable properties. The connections between the system and skew orthogonal polynomials, convergence acceleration algorithms and Laurent property are discussed in detail.

Consider a system of periodic pendulum lattice with analytic weak coupling: i x ̈i +sinxi =−ε 􏰀 ∂xiβα(xj,xj+1,xj+2), xi =xi+N, i∈Z, j =i−2 where N 􏰃 3 is an integer, ε > 0 is a small parameter and the function βα is an analytic function of a certain form. It is shown in this paper that for small enough ε, the system admits motions such that the energy transfers between the pendulums in any predetermined order.

The fundamental problem of nonlinear filtering theory is how to solve robust D-M-Z equation in real time and in memoryless manner. This paper describes a new real time algorithm which reduces the nonlinear filtering problem to off-line computations. Our algorithm gives convergent solutions in both pointwise sense and L/sup 2/ in case that the drift term and observation dynamic term have linear growths. The algorithm presented is slightly better than that given in our previous paper (2000).