Graded left modular lattices are supersolvable

We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [MT] by combining results of McNamara [Mc] and Liu [Li]. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.Received May 24, 2004; accepted in final form October 12, 2004.     

A Heuristic for the Prime Number Theorem

Conclusion  Might there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/π(x). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.     

Substitution reactions of dicyclooctatetraeneiron. A simple route to zerovalent iron—organophosphorus compounds

Fe(cot)₂ (cot = cyclooctatetraene) reacts with certain phosphines and phosphites to give Fe(cot)L₃ compounds in good yield. Reaction of Fe(cot)₂ with Ph₂PCH₂CH₂PPh₂ under N₂ gives a complex containing coordinated dinitrogen, which is converted into Fe(cot)(Ph₂PCH₂CH₂PPh₂)(CO) on reaction with CO or formate esters.     

A semi-automated and manual method for serum phospholipids

Five methods of acid digestion of amniotic fluid phospholipids for inorganic phosphate release were evaluated at a temperature of 225 °C and a heating time of 30 min. The methods used the following digestants: (A (perchloric acid), B (perchloric acid containing molybdate), C (sulfuric acid, 30% hydrogen peroxide, 5% urea), D (sulfuric-perchloric acids), and E (sulfuric-perchloric acids with vanadium pentoxide). After digestion and dilution with water, a hydrogen ion concentration of about 2.5 mol/ liter was obtained that permitted the use of stannous chloride-hydrazine sulfate as the reducing agent for phosphate color development. Recoveries of the different amniotic fluid phospholipids and aqueous phosphate standards were quantitative for all methods. A comparative study with amniotic fluids showed similar results for all methods. Since all methods were found to be equivalent under the specified conditions described, the choice was dependent upon the user's preference.     

The universe constructed from a sequence of ordinals

We prove that if where is an -sequence of ordinals then the GCH holds. Received January 1, 1993     

Chemical repercussions of orbital interactions through bond and through space. Effect of remote substituents on the addition of nitrenes to bicyclic olefins

The reactivity of a series of bicyclic olefins with nitrenes is profoundly influenced by the nature of remote functional groups. There is a marked lack of reactivity for reactions with carboethoxynitrene as compared to phthalimidonitrene which is distinctly nucleophilic in character. An explanation for the reluctance to form aziridines is offered in terms of orbital interactions between the distant groups and the olefinic bond, making the latter remarkably electron deficient as evidenced by UV-photoelectron spectrosoopy. Because of the complexity of the spectra, identification of the Ip associated with the reactive π-centre was made by recourse to ab initio configuration interaction calculations for key members of the series.