Let G = (V,E) be a graph of order n. A closed distance magic labeling of G is a bijection l : V (G) → {1, . . . , n} for which there exists a positive integer k such that ∑ x∈N [v] l(x) = k for all v ∈ V , where N [v] is the closed neighborhood of v. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic la...

Let G be a simple graph with vertex set V(G) and edge set E(G). For each a in V(G), let N(a) = {x E V(G) : ax E E(G)} be the set of neighbours of a. A subset S of V(G) is called a closed set of G if, for each pair of distinct elements u, b in S, N(a) ON(b) s S. Let Z’(G) be the family of closed sets of G, inclusive of the empty set 0. Evidently, Z(G) is closed under arbitrary intersection and i...

We prove a generalization and give a new proof of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group G acting linearly and rationally on a real vector space V . G can be viewed as the real points of a complex reductive group G C which acts on V C := V ⊗ C. In [BHC62] it was shown that G · v ∩ V is a finite union of...

Examples of non-quasiconvex functions that are rank-one convex are rare. In this paper we construct a family of such functions by means of the algebraic methods of the theory of exact relations for polycrystalline composite materials, developed to identify G-closed sets of positive codimensions. The algebraic methods are used to construct a set of materials of positive codimension that is close...

and cross-section g g G ¬ t. w x If p is prime, a p-valued field 8, p. 7 is a valued field, of characteristic zero, in which p has minimal positive value and whose residue field has p Ž . elements. A p-valued field F, ̈ is p-adically closed just in case no algebraic extension F9 of F with valuation ̈ 9 extending ̈ is p-valued. The Ž . class of p-adically closed fields is to Q , ̈ as the class of re...

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G, let gn be the number of graphs in G which have n vertices. The growth constant of G is γ = lim sup(gn/n!) . We study the properties of the set Γ of growth constants of minor-closed classes of graphs. Among other results, we show that Γ does not contai...

For G a closed subgroup of S∞, we provide a precise combinatorial characterization of when the universal minimal flow M(G) is metrizable. In particular, each such instance fits into the framework of metrizable flows developed in [KPT] and [NVT]; as a consequence, each G with metrizable universal minimal flow has the generic point property, i.e. every minimal G-flow has a point whose orbit is co...

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H 2 cb (G, L p (G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).

Let X be a proper hyperbolic geodesic metric space of bounded growth and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not amenable then the second continuous bounded cohomol-ogy group H 2 cb (G, L 2 (G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).