In this note we consider the set of metric spaces which are the limits with respect to Lipschitz distance dL of compact connected C°°-Riemannian manifolds of curvature uniformly bounded above and below. We call this set "bounded curvature closure" (BCC). It is well known that the limit spaces need not be C-Riemannian manifolds [P, Example 1.8]. Hence, the problem arising is to give a geometrica...

For a given graph G and a sequence P1,P2, . . . ,Pn of additive hereditary classes of graphs we define an acyclic (P1,P2, . . . ,Pn)colouring of G as a partition (V1, V2, . . . , Vn) of the set V (G) of vertices which satisfies the following two conditions: 1. G[Vi] ∈ Pi for i = 1, . . . , n, 2. for every pair i, j of distinct colours the subgraph induced in G by the set of edges uv such that u...

The basic solutions of the linear equation Ax = b , are the solutions of subsystems corresponding to maximal nonsingular submatrices of A. The convex hull of the basic solutions is denoted by C = C(A,b). The residual r(x) of a vector x is r := Ax − b. Given 1 ≤ p ≤ ∞, the lp-approximate solutions of Ax = b, denoted x , are minimizers of ‖r(x)‖p. Given M ∈ Dm, the set of positive diagonal m×m ma...

Let M be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any (p1, v1) and (p2, v2) in T M , is it possible to connect p1 to p2 by a curve γ in M with arbitrary small geodesic curvature such that, for i = 1, 2, γ̇ is equal to vi at pi? In this article, we bring a positive answer to the question if M verifies one of the following three conditions:...

let z2 = {0, 1} and g = (v ,e) be a graph. a labeling f : v → z2 induces an edge labeling f* : e →z2 defined by f*(uv) = f(u).f (v). for i ε z2 let vf (i) = v(i) = card{v ε v : f(v) = i} and ef (i) = e(i) = {e ε e : f*(e) = i}. a labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. the vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. in this paper ...

Given an undirected graph G = (V,E) and vertices slytl;s2,t2 , the problem is to determine whether or not G admits two vertex disjoint paths Pl and P2 ) connecting sl with tl and s2 with t2 respectively. This problem is solved by an O(n.m) algorithm (n = IV1 , m = IEI) , An important by-product of the paper is a theorem that states that if G is )+-connected and non-planar, then such paths pl an...

(4) Let G be a Go-board, p be a point of E2 T, and i, j be natural numbers. Suppose 1 ≤ i and i + 1 ≤ lenG and 1 ≤ j and j + 1 ≤ widthG. Then p ∈ Intcell(G, i, j) if and only if the following conditions are satisfied: (i) (G◦ (i, j))1 < p1, (ii) p1 < (G◦ (i+1, j))1, (iii) (G◦ (i, j))2 < p2, and (iv) p2 < (G◦ (i, j +1))2. (5) For every non constant standard special circular sequence f holds BDD ...

We prove that the product replacement graph on generating 3-tuples of An is connected for n ≤ 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold’s Conjecture, usually stated in terms of T -systems (see [P2]). Our results confirm Wiegold’s Conjecture ...

Given a clique-width expression of a graph G of clique-width k, we provide 2O(k) ·nO(1) time algorithms for connectivity constraints on locally checkable properties such as Connected Dominating Set, Connected Perfect Dominating Set or Node-Weighted Steiner Tree. We also propose an 2O(k) ·nO(1) time algorithm for Feedback Vertex Set. The best running times for all the considered cases were eithe...