Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements from a uniform reference state grow linearly with position , and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with positi...

In this paper, we show that the Garnier system in n-variables has affine Weyl group symmetry of type B (1) n+3. We also formulate the τ functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice Q(Cn+3) and show that they satisfy Toda equations, Hirota-Miwa equations and bilinear differential equations.

The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and in...

We define hierarchies of differential–q-difference equations, which are q–deformations of the equations of the generalized KdV hierarchies. We show that these hierarchies are bihamiltonian, one of the hamiltonian structures being that of the q–deformed classical W–algebra of slN , defined by Reshetikhin and the author. We also find q–deformations of the mKdV hierarchies and the affine Toda equa...

We consider partially observed stochastic dynamical systems whose state equations are of McKean-Vlasov type SDE and hence contain a measure term where the measure term is also random. Such SDEs are used to model the state dynamics of the agents in Mean Field Games framework with Major and Minor agents. We present nonlinear filtering equations in both normalized and unnormalized forms and, for s...

We find and study a five-parameter family of four-dimensional coupled Painlevé V systems with affine Weyl group symmetry of type D (1) 5 . We then give an explicit description of a confluence from those systems to a four-parameter family of four-dimensional coupled Painlevé III systems with affine Weyl group symmetry of type B (1) 4 . 0. Introduction It is well-known that the Painlevé systems P...

We recall the concept of a Lie algebroid on a vector bundle and the associated notion of Lagrange-type equations. A heuristic calculus of variations approach tells us what a time-dependent generalization of such equations should look like. In order to find a geometrical model for such a generalization, the idea of a Lie algebroid structure on a class of affine bundles is introduced. We develop ...