Abstract: for (transient) one dimensional random walk in random environment, conditions are known that ensure an annealed CLT. One then also have a quenched CLT, with a different (environment dependent) centering.
In higher dimensions, annealed CLT's have been derived in the ballistic case by Sznitman. We prove that in dimension 4 or more, annealed CLT's together with a mild integrability condition imply a quenched CLT. The proof is based on controlling the intersections of two RWRE paths in the same environment.
We will discuss a strong law of large numbers, an annealed CLT, and
the limit law of the ``environment viewed from the particle" for transient
random walks on a strip (product of Z with a finite set). The model was
introduced by Bolthausen and Goldsheid and includes in particular RWRE
with bounded jumps on the line as well as some one-dimensional RWRE with a
memory.
I will discuss geometric flows of G_2 structures on manifolds. These are flows of a 3-form on a 7-manifold with a certain non-degeneracy condition. The form determines a Riemmannian metric in a non-linear way. There is an associated tensor, called the torsion of the G_2 structure, which vanishes if and only if the manifold has G_2 holonomy.
In this talk we will survey the development on the Green's functions of the Boltzmann equations. The talk will include the motivation from the field of hyperbolic conservation laws, the connection between the Boltzmann equation
and the hyperbolic conservation laws, and the particle-like and the wave-like duality in the Boltzmann equation. With all these components one can realize a clear layout of the Green's function of the Boltzmann equation. Finally we will present the application of the Green's function the an initial-boundary value problem in the half space domain.
We will discuss the equations of motion for 3d homogeneous isotropic elastic materials, in the compressible and incompressible case. We will present results on global existence of solutions to the initial value problem, under the assumption of small deformations and with appropriate structural conditions.
In this talk, we shall design and analyze additive and multiplicative multilevel methods on adapted grids obtained by newest vertex bisection. The analysis relies on a novel decomposition of newest vertex bisection which give a bridge to transfer results on multilevel methods from uniform grids to adaptive grids. Based on this space decomposition, we will present a unified approach to the multilevel methods for $H^1$, $H(\rm curl)$, and $H(\rm div)$ systems.