The clustering coefficient C of a network, which is a measure of direct connectivity between neighbors of the various nodes, ranges from 0 (for no connectivity) to 1 (for full connectivity). We define extended clustering coefficients C(h) of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employed in computing the ordina...

In this paper new concepts countable join property, countable meet property, P–lattice and Pδ –lattice are introduced. We established that P–lattice and Pδ –lattice are measureable partial lattices and characterized partial lattices of a lattice through countable join and meet properties. We also established some interesting result on the injective property of the lattice measurable functio...

The clustering coefficient C of a network, which is a measure of direct connectivity between neighbors of the various nodes, ranges from 0 (for no connectivity) to 1 (for full connectivity). We define extended clustering coefficients C(h) of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employed in computing the ordina...

Verifying that a statistically significant result is scientifically meaningful is not only good scientific practice, it is a natural way to control the Type I error rate. Here we introduce a novel extension of the p-value-a second-generation p-value (pδ)-that formally accounts for scientific relevance and leverages this natural Type I Error control. The approach relies on a pre-specified interv...

We strengthen the revised GCH theorem by showing, e.g., that for λ = cf(λ) > iω, for all but finitely many regular κ < iω, it holds that “λ is accessible on cofinality κ” in some weak sense (see below). As a corollary, λ = 2 = μ > iω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many κ ∈ Reg∩iω. We strengthen previous results on the black box and the m...

We describe the versal deformation of two-dimensional cyclic quotient singularities in terms of equations, following Arndt, Brohme and Hamm. For the reduced components the equations are determined by certain systems of dots in a triangle. The equations of the versal deformation itself are governed by a different combinatorial structure, involving rooted trees. One of the goals of singularity th...

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes Pδ(n, k) and D(n, k) of partitions of n into parts of size at most k in which, for Pδ(n, k), the parts are congruent to one modulo δ and, for D(n, k), the parts are distinct. The change required b...

Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, |Γp| ≥ p r r+2 + , where (p) is any function of p such that (p)→ 0 as p→∞. This is a consequence of two other results. Denote ...