The intersection ideal graph Γ(S) of a semigroup S is simple undirected whose vertices are all nontrivial left ideals and two distinct I, J adjacent if only their nontrivial. In this paper, we investigate the connectedness Γ(S). We show that connected, then diameter at most two. Further, classify semigroups in terms such obtain domination number, independence girth strong metric dimension have ...

The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a Γ-convergence analysis, obta...

We study compactness and Γ-convergence for Ginzburg-Landau type functionals. We only assume that the potential is continuous and positive definite close to one circular well, but allow large zero sets inside the well. We show that the relaxation of the assumptions does not change the results to leading order unless the energy is very large.

A homogenization result is given for a material having soft inclusions arranged in a periodic structure, under the requirement that the interpenetration of matter is forbidden. According to the relation between the softness parameter and the size of the microstucture, three different limit models are deduced via Γ-convergence.

This paper is an extended version of the lecture delivered at the Summer School on Differential Equations and Calculus of Variations (Pisa, September 16-28, 1996). That lecture was conceived as an introduction to the theory of Γ-convergence and in particular to the Modica-Mortola theorem; I have tried to reply the style and the structure of the lecture also in the written version. Thus first co...

The eeective energy of a mixture of two elastic materials in a thin lm is characterized using Gamma-limit techniques. For cylindrical shaped inclusions it is shown that 3D-2D asymptotics and optimal design commute from a variational viewpoint. Regularity of local minimizers for the resulting design is addressed.

The generalization to gradient vector elds of the classical double-well, singularly perturbed func-tionals, I" (u;) := Z 1 " W(ru) + "jr 2 uj 2 dx; where W() = 0 if and only if = A or = B, and A ? B is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on W it is shown that I" ?-converge to +1 otherwise, where K is the (constant) interfacial energy per unit area.

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of graph G, if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. It is well known that if e ∈ E(G), then γ(G− e)−1 ≤ γ(G) ≤ γ(G− e). In this paper, as an application of this inequality, we obtain the domination number of some certain graph...

To clone and express the γ-polyglutamic acid (γ-PGA) synthetase gene pgsBCA in Bacillus subtilis, a pWB980 plasmid was used to construct and transfect the recombinant expression vector pWB980-pgsBCA into Bacillus subtilis WB600. PgsBCA was expressed under the action of a P43 promoter in the pWB980 plasmid. Our results showed that the recombinant bacteria had the capacity to synthesize γ-PGA. Th...

Bounds on the number of simple zeros of the derivatives of a function are used to give bounds on the number of distinct zeros of the function. The Riemann ξ-function is defined by ξ(s) = H(s)ζ(s), where H(s) = 2s(s−1)π 1 2 Γ( 1 2 s) and ζ(s) is the Riemann ζ-function. The zeros of ξ(s) and its derivatives are all located in the critical strip 0 < σ < 1, where s = σ+ it. Since H(s) is regular an...