Bounds on the index of rotationally symmetric self-shrinking tori

Geometriae Dedicata - A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by...     

Interlaboratory comparisons and improvements of methods for acid number determination in used motor oils

The results of two interlaboratory comparisons of acid number determinations in used motor oils are discussed. It is shown that the comparability of the measurement results is not as good as that required by known standards for petroleum products. The problem is motor oil contaminants which accumulated during use, and which are the source of a matrix effect in the acid number determination. The standard methods’ drawbacks are analyzed and some improvements are proposed. Repeatability and accuracy of the improved methods are evaluated. Received: 11 December 2001 Accepted: 15 February 2002     

Lefschetz–Verdier Trace Formula and a Generalization of a Theorem of Fujiwara

The goal of this paper is to generalize a theorem of Fujiwara (Deligne’s conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is more elementary than the original one and stays in the realm of ordinary algebraic geometry, that is, does not use rigid geometry. We also give a proof of the Lefschetz–Verdier trace formula and of the additivity of filtered trace maps, thus making the paper essentially self-contained. The work was supported by the Israel Science Foundation (Grant No. 555/04) Received: May 2005 Accepted: August 2005     

Research on 2, 1, 3-thia-and selenadiazole

Reductive decomposition followed by hydrolysis of 5-(,-dicarethoxy--acetylamino) ethylbenz-2, 1, 3-selenadiazole gives -(3, 4-diaminophenyl)alanine.For Part XLVII see [1].     

Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae

We consider the Weyl asymptotic formula

Studies in the field of 2, 1, 3-Thiadiazole and 2,1, 3-Selenadiazole

Benzo-2, 1, 3-thiadiazole (I) undergoes chloromethylation with dichlorodimethyl ether in the presence of anhydrous aluminum chloride with the predominant formation of 4, 7-di(chloromethyl)benzo-2, 1, 3-thiadiazole (II). Bases and pseudobases (paraformaldehyde, hexamethylenetetramine, dimethylformamide) exert an inhibiting influence on this reaction. In the presence of these substances, a mixture of compound II and 4-(chloromethyl)benzo-2, 1, 3-thiadiazole (III) is formed, or else no reaction takes place. The structures of compounds II and III has been shown by their reductive decomposition to o-diamines described in the literature. The high reactivity of the chlorine in the chloromethyl group has enabled various new derivatives of benzo-2, 1, 3-thiadiazole to be obtained by its replacement with hydroxy, thiocyanato, di(-hydroxyethyl)amino, di(-chloroethyl)amino, formyl, and carboxy groups.For communication XLVIII, see [11].     

Investigations on 2, 1, 3-thia- and selenadiazoles

Nitration and nitrosation of 4-hydroxy-5-methyl-benzo-2, 1, 3-thiadiazole gives 4-hydroxy-5-methyl-7-nitro-and 4-hydroxy-5-methyl-7-nitrosobenzo-2, 1, 3-thiadiazoles. Oxidation of the latter, or of 4,7-diamino-5-methylbenzo-2,1,3-thiadiazole gives 5-methyl-4,7-dihydroxy-2, 1, 3-thiadiazole, forming derivatives with sodium bisulfite or hydroxylamine, and reduced by sodium dithionite to 5-methyl-4, 7-dihydroxybenzo-2, 1, 3-thiadiazole. The latter is also obtained by diazotizing 5-methyl-4-hydroxy-7-aminobenzo-2, 1, 3-thiadiazole, and decomposing the diazonium salt. Nitration of 4-ethoxybenzo-2, 1, 3-thiadiazole with sodium ethoxide gives 4-ethoxy-7-aminobenzo-2, 1, 3-thiadiazole, acetylated to 4-ethoxy-7-acetaminobenzo-2, 1, 3-thiadiazole.For Part XXXVII see [1].     

Researches on 2,1,3-thia-and selenadiazole

Chlorination of benzo-2,1,3-thiadiazole (I) in the presence of iron and paraform gives only 4-chlorobenzo-2,1,3-thiazole. Chlorination of 5,6-dimethylbenzo-2,1,3-thiadiazole (II) under exactly the same conditions gives only 5,6-dimethyl-4,7-dichlorobenzo-2,1,2-thiadiazole. Similarly chloromethylation of II in the presence of paraform gives only 5,6-dimethly-4,7-di(chloromethyl)benzo-2,1,3-thiadiazole. Chlorination and chloromethylation of I proceed through the intermediate formation of monosubstituted compounds which change into disubstimted ones.It is known [2,3] that Chlorination of benzo-2,1,3-thiadiazole (I) and its 5,6-dimethyl derivative (II) with chlorine in the presence of iron gives mainly the 4,7-dichloro substitution products III and IV respectively.It was previously shown [4] that chloromethylation of I with dichlorodimethyl ether in the presence of anhydrous aluminum chloride also gives mainly 4,7-di(chloromethyl)benzo-2,1,3-thiadiazole (V). Bases and pseudo-bases (paraform, urotropine. dimethylformamide) have a retarding effect on chloromethylation. When the reaction is run in the presence of these latter the products comprise besides V, 4-chloromethyl-2,1,3-thiadiazole (VI), or else, if enough base is added, there is no reaction.For Part XLII see [1].     

Annulation of pyrrole ring in 4-acylpyridine-3,5-dicarbonitriles in the presence of ammonia

Chemistry of Heterocyclic Compounds - 4-Acyl-2-amino-6-chloropyridine-3,5-dicarbonitriles underwent heterocyclization in the presence of ammonia in aqueous dioxane medium, involving the...