Let $s(n):= \sum _{d\,|\, n,\,d \lt n} d$ denote the sum of proper divisors $n$. It is natural to conjecture that for each integer $k\ge 2$, equivalence \[ \text {$n$ $k$th powerfree} \iff {$s(n)$ \] holds almost al

Suppose that S ⊆ P (X) is a σ-field of subsets ofX and I ⊆ S is a σ-ideal. If I has the countable chain condition (ccc), i.e., every family of disjoint sets in S \ I is countable, then S/I is a complete boolean algebra. A boolean algebra is atomic iff there is an atom beneath every nonzero element. A function f : X → R is S-measurable iff f(U) ∈ S for every open set U . A sequence of S-measurab...

We characterize d-lattices as those bounded lattices in which every maximal filter/ideal is prime, and we show that a d-lattice is complemented iff it is balanced iff all prime filters/ideals are maximal.

The efficacy of present day antineoplastic regimens depends upon the delivery and penetration of therapeutic agents through the tumor vas cular and interstitial spaces to the tumor cell target. The distribution of relevant molecules or cells in a solid tumor is often poor and heteroge neous and is believed to be due to a number of pathophysiological factors, including elevated interstitial flui...

It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a local...