Let (Ω, d) be a metric space where Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension and let B be a partition of Ω. The coherent upper conditional prevision defined as the Choquet integral with respect to its associated Hausdorff outer measure is proven to satisfy the disintegration property and the conglomerative principle on every partition.

We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m + 1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof o...

We investigate the theoretical feasibility of near-optimal, distributed sleep scheduling in energyconstrained sensor networks with pairwise sensor redundancy. In this setting, an optimal sleep schedule is equivalent to an optimal fractional domatic partition of the associated redundancy graph. We present a set of realistic assumptions on the structure of the communication and redundancy relatio...

In this paper, we study how to partition a tree into edge-disjoint subtrees of approximately the same size. Given a tree T with n edges and a positive integer k ≤ n, we design an algorithm to partition T into k edge-disjoint subtrees such that the ratio of the maximum number to theminimum number of edges of the subtrees is at most two. The best previous upper bound of the ratio is three, given ...

A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double R...