Reactivity and Synthetic Utility of 1,3,2λ5-Dioxaphospholanes

Abstract

The kinetics of Lewis acid-mediated decomposition and definitions of the synthetic applicability of 1,3,2λ⁵-dioxaphospholanes for (i) the syntheses of highly-hindered epoxides, and (ii) the stereospecific functionalizations of stereocenters in 1,2-diol arrays are described.     

Time,gravity and the exterior angle of parallelism

Applying calculus to the horizontal and vertical unit hyperbolas we derive the obtuse exterior and the acute angle of parallelism formulas for the arc length of a unit circle. When the analogous arc length of a hyperbola is figured, we further define metric distance with radian measure. Employing a non-conventional velocity of light formula, we explain several aspects of physics with a logical geometry. The hyperbolic geometry Lobacevskii angle of parallelism is mathematically measured to agree with the physical measure, without using the gravitational constant, but rather the electromagnetic spectrum and geodesics. Space is hereby measured with units of radians and time. Half-angle formulas for slopes and boosts pertain to spinor algebra. The new light velocity is justified when its derivative results in Einstein's mass–energy formula. Relativity is reformulated with these half-angle formulas.     

On a strengthening of Jónssonness for ℵω

We discuss a system of strengthenings of “${\aleph }_{\omega }$ is Jónsson” indexed by real numbers, and identify a strongest one. We give a proof of a theorem of Silver and show that there is a barrier to weakening its hypothesis.