On Rational Functions Related to Algorithms for a Computation of Roots. I

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k-th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

Essential completions of distributive lattices

The salient feature of the essential completion process is that for most common distributive lattices it will yield a completely distributive lattice. In this note it is shown that for those distributive lattices which have at least one completely distributive essential extension the essential completion is minimal among the completions by infinitely distributive lattices. Thus in its setting the essential completion of a distributive lattice behaves in much the some way as the one-point compactification of locally compact topological space does in its setting.

THE PHYSICAL STATES OF ANHYDROUS SODIUM SOAPS

Density measurements and differential cooling curves have been carried out on sodium stearate and sodium oleate over a range of temperatures. For sodium stearate, the genotypical or unidimensional melting point at 70 °C. found by Thiessen et al. has been corroborated. A large transition has been noted at about 100 °C. and it is suggested that this phase transition is due to a melting in a second dimension at right angles to the molecular axis. A further large transition has been found at 125° to 130 °C. leading to a plastic state in the soap, and it is suggested that this transition point denotes the essential completion of bidimensional melting of the lattice. No further considerable phase change was found up to about 200 °C., when disruption of the polar bonds probably occurs to effect complete disruption of the lattice to the liquid crystalline state. Only the latter transition point was found for sodium oleate at about 135 °C.