We deal with a class of reaction-diffusion equations, in space dimension d > 1, perturbed by a Gaussian noise ∂wδ/∂t which is white in time and colored in space. We assume that the noise has a small correlation radius δ, so that it converges to the white noise ∂w/∂t, as δ ↓ 0. By using arguments of Γ-convergence, we prove that, under suitable assumptions, the quasi-potential Vδ converges to the...

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Γ-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either ho...

We prove the stability of a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort and Marigo in [22]. Then we give an application to the quasistatic evolution of cracks in composite materials.

We give results for the Γ-limit of a scaled elastic energy of a film as the thickness h > 0 converges to zero. The elastic energy density models materials with multiple phases or variants and is thus non-convex. The model includes an interfacial energy that allows sharp interfaces between the phases and variants and is proportional to the total variation of the deformation gradient.

Let Σ be a (connected) surface of “complexity” κ; that is, Σ may be obtained from a sphere by adding either 1 2 κ handles or κ crosscaps. Let ρ ≥ 0 be an integer, and let Γ be a “ρ-representative drawing” in Σ; that is, a drawing of a graph in Σ so that every simple closed curve in Σ that meets the drawing in < ρ points bounds a disc in Σ. Now let Γ be another drawing, in another surface Σ of c...

1.1. Aschbacher-Scott Program. Primitive permutation groups have been studied since Galois and Jordan and have applications in many different areas. If Γ is a transitive permutation group with a point stabilizer M then Γ is primitive if and only if M < Γ is a maximal subgroup. So studying primitive permutation groups is equivalent to studying maximal subgroups. In most problems involving a fini...

For the “Hopf bundle” S → S → CH, horizontal lifts of simple closed curves are studied. Let γ be a piecewise smooth, simple closed curve on a complete totally geodesic surface S in the base space. Then the holonomy displacement along γ is given by V (γ) = e where A(γ) is the area of the region on the surface S surrounded by γ; λ = 1/2 or 0 depending on whether S is a complex submanifold or not....

The first class of nonassociative simple Moufang loops was discovered by L. Paige in 1956 [9], who investigated Zorn’s and Albert’s construction of simple alternative rings. M. Liebeck proved in 1987 [7] that there are no other finite nonassociative simple Moufang loops. We can briefly describe the class as follows: For every finite field F, there is exactly one simple Moufang loop. Recall Zorn...

In this paper, we consider the maximum weight ({K1,K2},k,l)-packing problem in a graph. This generalizes number of well-known problems, for example: induced matching, k-separated connected independent set, dissociating k-packing. We show that class cographs, ({K1,K2},k,l)- packing can be computed O(n + m) time. Let Γ graphs and Γ* all simple (with respect to modular decomposition) subgraphs fro...