We show that in the one-dimensional case the weighted Hardy–Littlewood maximal operator Mμ is bounded on BMO(μ) for arbitrary Radon measure μ, and that this is not the case in higher dimensions.

In this paper we consider Q the class of solvable Lie algebras L with the following property: if A is a subalgebra of L, then Φ(A) ⊆ Φ(L) (where Φ(L) denotes the Frattini subalgebra of L;that is Φ(L) is the intersection of all maximal subalgebras of L). The class Q is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal...

Let 7(G) denote the minimum cardinality of a dominating set of a graph G = (V,E). A longstanding upper bound for 7(G) is attributed to Berge: For any graph G with n vertices and maximum degree A(G), 7(G) <~ n A(G). We eharacterise connected bipartite graphs which achieve this upper bound. For an arbitrary graph G we furnish two conditions which are necessary if 7(G) + A(G) = n and are sufficien...

We present a unified approach to Doob’s Lp maximal inequalities for 1 ≤ p < ∞. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob’s maximal inequalities. These are best possib...

The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted l1 norm over the solution set of a linear complementarity problem ...

An independent dominating set D of a graph G = (V, E) is a subset of vertices such that every vertex in V \ D has at least one neighbour in D and D is an independent set, i.e. no two vertices in D are adjacent. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a s...

We review a characterization of domination perfect graphs in terms of forbidden induced subgraphs obtained by Zverovich and Zverovich [12] using a computer code. Then we apply it to a problem of unique domination in graphs recently proposed by Fischermann and Volkmann. 1 Domination perfect graphs Let G be a graph. A set D ⊆ V (G) is a dominating set of G if each vertex of G either belongs to D ...