A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimens...

Suppose κ is λ-supercompact witnessed by an elementary embedding j : V →M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F (α)) and (2) α < β =⇒ F (α) ≤ F (β). In this article we address the question: assuming GCH, what additional assumptions are necessar...

We introduce and use the concept of (∈γ ,∈γ ∨qδ)-fuzzy (left, right, bi-) ideals to study the structural properties of a non-associative algebraic ordered structure. We characterize an intra-regular ordered AGgroupoid by these generalized fuzzy ideals. 2010 AMS Classification: 06F05, 20M12, 08A72

Let the Kneser graph K of a distance-regular graph Γ be the graph on the same vertex set as Γ, where two vertices are adjacent when they have maximal distance in Γ. We study the situation where the Bose-Mesner algebra of Γ is not generated by the adjacency matrix of K. Let Γ be a distance-regular graph of diameter d on n vertices. Let Γi be the graph with the same vertex set as Γ where two vert...

We present the first of two articles on the small volume fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. Here we focus attention on the sharp-interface version of the functional and consider a limit in which the volume fraction tends to zero but the number of minority phases (called particles) remains O(1). Using the languag...

We study, by means of Γ-convergence, the asymptotic behaviour of a variational problem modeling a dislocation ensemble moving on a slip plane through a discrete array of obstacles. The variational problem is a two dimensional phase transition type energy given by a non local term and a non linear potential which penalizes non integer values. In this paper we consider a regime corresponding to a...

Suppose Γ′ to be of a subgraph of a graph Γ. We define a sampling of a Γ-design B = (V,B) into a Γ′-design B′ = (V,B′) as a surjective map ξ : B → B′ mapping any block of B into one of its subgraphs. A sampling will be called regular when the number of preimages of any block of B′ under ξ is a constant. This new concept is closely related with the classical notion of embedding, which has been e...

We present new regularity criteria involving the integrability of the pressure for the Navier-Stokes equations in bounded domains with smooth boundaries. We prove that either if the pressure belongs to L x,t with 3/γ + 2/q ≤ 2 and 3/2 < γ ≤ ∞ or if the gradient of the pressure belongs to L x,t with 3/γ+2/q ≤ 2 and 1 < γ ≤ ∞, thenweak solutions are regular. Local regularity criteria in terms of ...

Open problem: Is γ or exp(γ) rational ? Since the regular continued fraction gives best rational approximations, continued fraction computations can give theorems of the form: If x is rational, say x = p/q, then |q| > B for some (very large) bound B. To obtain a result like this with given bound B, we need to compute x with absolute error O(1/B2). Using this method we know that, if γ or exp(γ) ...

Let Ω ⊂ Ω̃ be open sets in R, I ⊂ R be an open neighborhood of the origin and let η be a compactly supported smooth function on Ω× I; we assume that η(·, 0) does not vanish identically. For each x ∈ Ω let t 7→ Γ(x, t) ⊂ Ω̃ be a regular parametrization of a submanifold Mx ⊂ Ω̃ with codimension l. We assume that Γ(x, t) ⊂ Ω if (x, t) ∈ supp η, and that Γ satisfies Γ(x, 0) = x and depends smoothly on...