In this paper we show that, if T is an area-minimizing 2-dimensional integral current with ∂T=Q〚Γ〛, where Γ a C1,α curve for α>0 and Q arbitrary integer, then has unique tangent cone at every boundary point, polynomial convergence rate. The proof simple reduction to the case Q=1, studied by Hirsch Marini (2019).

Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p′ are connected in Γ(G), whenever G has an element of order pp′. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G) = Γ(P ), G has a composition factor isomorphic to P . In [...

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f -, hand γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and γ-vectors involving descent statistics. This includes a combinatorial interpretat...

Let Γ be a graph and let G be a group of automorphisms of Γ. The graph Γ is called G-normal if G is normal in the automorphism group of Γ. Let T be a finite non-abelian simple group and let G = T l with l ≥ 1. In this paper we prove that if every connected pentavalent symmetric T -vertex-transitive graph is T -normal, then every connected pentavalent symmetric G-vertex-transitive graph is G-nor...

In this paper we make a survey of some recent developments of the theory of Sobolev spaces W (X, d,m), 1 < q < ∞, in metric measure spaces (X, d,m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger’s work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakene...

By looking at the Γ-convergence of some easy functionals in terms of underlying Young measures, we emphasize the connection of this issue with the Monge-Kantorovich mass transfer problem. After exploring this relationship, we study a typical example in periodic homogenization to realize the difference between the Monge-Kantorovich problem and the usual cell-problem. We also state a version of t...

We investigate the limiting description for a finite-difference approximation of a singularly perturbed Allen–Cahn type energy functional. The key issue is to understand the interaction between two small length-scales: the interfacial thickness ε and the mesh size of spatial discretization δ. Depending on their relative sizes, we obtain results in the framework of Γ-convergence for the (i) subc...

Let G be a connected non-compact Q-simple real algebraic group defined over Q, that is, the identity component of the group of the real points of a connected Q-simple algebraic group which is R-isotropic. Let Γ ⊂ G(Q) be an arithmetic subgroup of G. As is well known, Γ has finite co-volume in G [BH]. Denote by μG the G-invariant Borel probability measure on Γ\G. Two subgroups Γ1 and Γ2 of G are...

By reparametrization, one may assume that γ is Lipschitz with constant equal to length(γ). We will also make use of the following simple (and well-known) criterion: a compact set Γ is the image of a rectifiable curve if and only if it is connected and H(Γ) < ∞. Indeed, one may choose γ so that length(γ) ≤ CH(Γ); see, for example, [1, 2]. Here and below, H denotes the one-dimensional Hausdorff m...