To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >st A) if for every computable function f , for all but finitely many x, mB(x) > f(mA(x)). This settling-time order...

Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any integers k1, k2, k3, k4, k5 there exists a cubic graph G satisfying the following conditions: γ(G)−ir(G) ≥ k1, i(G)−...

We critically review several recent approaches to solving the two cosmological constant problems. The ‘‘old’’ problem is the discrepancy between the observed value of rL and the large values suggested by particle physics models. The second problem is the ‘‘time coincidence’’ between the epoch of galaxy formation tG and the epoch of L domination tL . It is conceivable that the ‘‘old’’ problem ca...

Classical voting rules output a winning alternative (or a nonempty set of tied alternatives). Social welfare functions output a ranking over alternatives. There are many practical situations where we have to output a different structure than a winner or a ranking: for instance, a ranked or non-ranked set of k winning alternatives, or an ordered partition of alternatives. We define three classes...

In 1996, Reed proved that the domination number (G) of every n-vertex graph G with minimum degree at least 3 is at most 3n/8.Also, he conjectured that (H) n/3 for every connected 3-regular (cubic) n-vertex graph H. In this note, we disprove this conjecture. We construct a connected cubic graph G on 60 vertices with (G) = 21 and present a sequence {Gk}∞k=1 of connected cubic graphs with lim k→∞ ...

This paper introduces a class of coalitional games, called pillage games, as a model of Hobbesian anarchy. Any coalition can pillage, costlessly and with certainty, any less powerful coalition. Power is endogenous, so a pillage game does not have a characteristic function, but pillage provides a domination concept that defines a stable set, which represents an endogenous balance of power. Every...

A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k-Dominating Set, or (2) produces a dominating set of size at most g(k), where g(k) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is n...