An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation $\theta \in (0,1]$ if $\sum_{i=1}^n \frac{1}{x_i} < \theta$. A greedy algorithm constructs $\theta$. For some but not all numbers $\theta$, the gives a unique best for $n$. infinite set rational constructed which underapproximations are best, and also studied.

abstract a study was conducted to evaluate the effect of incubation time, sewage sludge and clay content on the distribution of nickel and cadmium in different fractions of selected calcareous soils. to meet this purpose, an experiment with two different soils (4 and 30 percent caco3), two different textures (8 and 40 percent clay), two levels of nickel (0 and 100 mg.kg-1), two levels of cadmiu...

We find, in the form of a continued fraction, the generating function for the number of (132)avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern. We find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan. the electronic journa...

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a p...

An Egyptian fraction is a sum of reciprocals of distinct positive integers, so called because the ancient Egyptians represented rational numbers in that way. In an earlier paper, the author [8] showed that every positive rational number r has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator. More precisely, there exists a p...