The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=∑v∈V(Γ)f(d(v)) for a f(x) on non-negative real numbers. In this paper, we determine the extremal graphs with maximum (minimum) vertex degree in set of all n-vertex chemical trees, and connected graphs. We also present Nordhaus–Gaddum-type results Hf(Γ)+Hf(Γ¯) Hf(Γ)·Hf(Γ¯).

Introduction: The inflammatory response requires a coordinated integration of various signaling pathway including cyclooxygenase (COX). COX catalyzes the formation of prostaglandins from arachidonic acid. Among prostaglandins, 15-Deoxy-D12, 14-prostaglandin J2 (15d-PGJ2), an endogenous ligand of Peroxisome proliferator-activated receptor-gamma (PPAR-γ), has been demonstrated to have anti-inflam...

We prove a homogenization theorem for non-convex functionals depending on vector-valued functions, defined on Sobolev spaces with respect to oscillating measures. The proof combines the use of the localization methods of Γ-convergence with a ‘discretization’ argument, which allows to link the oscillating energies to functionals defined on a single Lebesgue space, and to state the hypothesis of ...

Abstract. The Allen–Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen–Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove in two and three space dimensions the corresponding lower bound. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the l...

We characterize the functionals which are Mosco-limits, in the L2(Ω) topology, of some sequence of functionals of the kind Fn(u) := ∫ Ω αn(x)|∇u(x)| dx , where Ω is a bounded domain of RN (N ≥ 3). It is known that this family of functionals is included in the closed set of Dirichlet forms. Here, we prove that the set of Dirichlet forms is actually the closure of the set of diffusion functionals...

From elliptic regularity one immediately has that, should such a minimizer u exist, u ∈ C∞(Ω̄). Then u is in fact a smooth (analytic, even) harmonic function obtaining the boundary value g, i.e. a solution to the classical Dirichlet problem with boundary data g. A standard exercise in the Direct Method of the Calculus of Variations provides for the existence of a minimizer, i.e. ∃u ∈ H g (Ω;R) s...

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy Ec modelling the interactions—at a typical length scale of 1/c— of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy by Γ-convergence in the lim...

The rigorous derivation of lower dimensional models for thin structures is a question of great interest in mechanics and its applications. In the early 90’s a rigorous approach to dimension reduction has emerged in the stationary framework and in the context of nonlinear elasticity. This approach is based on Γ-convergence and, starting from the seminal paper [3, 4], has led to establish a hiera...