Perfect Modules

Erika Mares introduced the concepts of semi-perfectness and perfectness for projective modules, generalised the structure theorems of H. Bass and obtained results on the endomorphism rings of such modules. The present author has carried out an extensive study of endomorphism rings of various types of modules with two of his collaborators [3], [9]. In particular the concept of semi-perfectness was extended to modules not necessarily projective and results similar to those of Erika Mares obtained for quasi-projective semi-perfect modules. The object of the present paper is to extend the concept of perfectness to modules which are not necessarily projective and obtain results similar to those of Erika Mares, Roger Ware etc. concerning these modules.     

Partitioning multi-edge graphs

We introduce a new kind of graph called multi-edge graph which arises in many applications such as finite state Markov chains and broadcasting communication networks. A cluster in such a graph is a set of nodes such that for any ordered pair of nodes, there is a path of multi-edges from one to the other such that these edges remain within the same set. We give two algorithms to partition a multi-edge graph into maximal clusters. Both these algorithms are based on the depth-first search algorithm to find strongly connected components of the directed graph. We also discuss some applications of clustering in multi-edge graphs.     

Extraction behavior of uranium(VI), plutonium(IV), zirconium(IV), ruthenium(III) and europium(III) with γ-pre-irradiated solutions of N,N′-methylbutyl substituted amides in n-dodecane

The extraction of plutonium(IV), uranium(VI), zirconium(IV), europium(III) and ruthenium(III) with -pre-irradiated n-dodecane solutions of methylbutyl substituted hexanamide (MBHA), octanamide (MBOA) and decanamide (MBDA) from 3.5M HNO₃ has been studied as a function of absorbed dose up to 184×10⁴ Gray. The distribution ratios (K_d) of uranium(VI) decreased gradually up to a dose of 50×10⁴ Gray and became almost constant thereafter, while ruthenium(III) and europium(III) were not extracted in the entire dose range studied. The K_d values of Pu(IV) decreased gradually up to 10×10⁴ Gray, for MBOA, and 30×10⁴ Gray for MBHA and MBDA and then increased up to a dose of 72×10⁴ Gray, indicating the synergistic effect of radiolytic products at higher doses. The extraction of zirconium(IV) was found to increase gradually up to 72×10⁴ Gray. However, the steep fall in K_d values of plutonium(IV), zirconium(IV) beyond a dose of 72×10⁴ Gray was atrributed to third phase formation. The radiolytic degradation of amides was monitored by quantitative IR spectroscopy and was found to follow the order MBOA>MBDA>MBHA at 184×10⁴ Gray having the amines and carboxylic acids as the main radiolytic products.     

Heteropolyacid-catalyzed synthesis of chloromethyl methyl ether

An efficient (in terms of experiment and time) synthetic procedure for chloromethyl methyl ether (MOM-Cl) is described using heteropolyacids as catalysts.     

Facile Synthesis and Unusual Methanesulfonylation Reaction of (2R,3R)‐1,4‐Dimethoxy‐1,1,4,4‐tetrasubstituted‐2,3‐butanediols

A facile method for the synthesis of (2R,3R)‐1,4‐dimethoxy‐1,1,4,4‐tetrasubstituted‐2,3‐butanediols involving oxidative cleavage of benzylidene acetal as a key step is described. These sterically hindered diols unusually formed cyclic sulfites as the major product under methanesulfonylation reaction conditions.     

Notes on Euler’swork on divergent factorial series and their associated continued fractions

Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective.     

RRF rings which are not LRF

Define a ringA to be RRF (respectively LRF) if every right (respectively left)A-module is residually finite. We determine the necessary and sufficient conditions for a formal triangular matrix ring to be RRF (respectively LRF). Using this we give examples of RRF rings which are not LRF.