Lack of relative monotonicity among various measures of trihedral angles

We consider the various measures on trihedral angles that have appeared in the literature and we show that no two of these measures are monotone with respect to each other. In other words, for any measures f, g, there exist trihedral angles ??, ??, ??, ?? such that f(??) >? f(??), g(??) <? g(??), f(??) >? f(??), g(??) >? g(??). This is done through an elementary and systematic method based on multivariable calculus.     

Conjugacy Classes and Invariant Subrings of R-Automorphisms of R[x]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ?₂ and m ² = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.     

The Gergonne and Nagel centers of an n-dimensional simplex

In contrast with the analogous situation for a triangle, the cevians that join the vertices of a tetrahedron to the points where the faces touch the insphere (or the exspheres) are not concurrent in general. This observation led the present author and P. Walker in [4] to devise alternative definitions of the Gergonne and Nagel centers of a tetrahedron that do not assume the concurrence of such cevians and that coincide with the ordinary definitions in the case of a triangle. They then proved that the Gergonne center exists and is unique for all tetrahedra and that the Nagel center, though unique, exists only for tetrahedra that satisfy certain conditions. In this article, we extend these definitions to simplices of any dimension. By keeping the requirement that the Gergonne center be interior and relaxing such a condition for the Nagel center, we prove that both centers exist and are unique for all simplices, thus polishing the definitions and generalizing the results of the above-mentioned article.     

Novel 5,6-bis-(4-substitutedphenyl)-2H(3)-pyridazinones: Synthesis and reactions

A series of 5,6-bis(4-substitutedphenyl)-2H(3)-pyridazinones 2a–f have been synthesized from the condensation of the corresponding benzil monohydrazones 1 either with ethyl cyanoacetate or diethyl malonate in ethanol. The synthesized pyridazinones were converted to the corresponding 3-chloro derivatives 3a–f by the action of phosphoryl chloride. Reaction of the latter halogenated pyridazines with various aromatic amines led to the formation of new 3-aminoaryl pyridazines (4) in moderate yield. The structures of all new compounds 2b,c,e,f, 3b–e, 4 were fully identified by the analysis of their ¹H and ¹³C NMR and mass spectra. Some of these synthetic heterocyclic compounds were screened for their antimicrobial activities but they were almost negative.     

The sequence of generalized median triangles and a new shape function

Starting with a triangle ABC and a real number s, we let AA _s , BB _s , CC _s be the cevians that divide the sides BC, CA, AB, respectively, in the ratio s : 1 ? s, and we let ${\mathcal{H}_s(ABC)}$ be the triangle whose side lengths are equal to those of AA _s , BB _s , CC _s . We investigate the sequence of (the shapes of) triangles ${\mathcal{H}_s^n(ABC)}$ , n = 1, 2, ... by introducing a new shape function that suits this sequence. We also use this shape function to prove a theorem of C. F. Parry concerning automedian triangles.