Bi-amalgamations subject to the arithmetical property

This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.

Bertrand Russell (1872-1970) after twenty years

1990 marks the vicesenary of the death of Bertrand Russell, in his 98th year; and this arithmetical property is sufficient reason to review the historical research that has been published on his life and work during the past 20 years. During his long life he had already become the subject of historical research in many of his activities; but this interest accelerated considerably around the time of his death because in the mid 1960s he had decided to sell the bulk of his manuscripts, to raise money to finance his current projects. One of these was the series of conferences financed by the Canadian industrialist Cyrus Eaton, which began at his birthplace of Pugwash, in Nova Scotia. An alumnus of McMaster University at Hamilton, Ontario, Eaton announced that he would put forward a considerable sum of his own money if the papers went to McMaster. Some deft work by the librarian there secured the rest of the required capital, and the papers were purchased in 1968. Thus was created the ‘Bertrand Russell Archives’, as Russell insisted it be called, rejecting the original appellation of ‘Archive’; it is a major resource for British history of Russell’s time, and for the many other concerns in which he was involved. Soon after its launch in 1972, the first Russell conference at McMaster took place, to commemorate the centenary of his birth; its proceedings were published as a book four years later.

Appendix: Simple Measures of Variation in Lexical Fields

This appendix suggests a number of measures which may be readily calculated in respect of data on lexical usage to provide an indication of the degree of variation in such usage. It introduces and briefly discusses five straightforward measures of variation or consensus which apply to the case where any informant has been constrained to give only one response per stimulus item. None of these measures is, strictly speaking, a statistical measure, although p is almost so in terms of its logic and the D-measures have a form familiar to that of a wide class of elementary statistics. Each measure in fact generates a numerical index which relates to an arithmetical property of the data in question: the measures are not in general estimates of population parameters. The use is heuristic, exploratory and comparative: by systematically reducing the information which comprises the data, the measures should help the investigator to become more aware of the salient characteristics of the data to hand.

XVII.—Invariant Matrices and the Geometry of Numbers

SynopsisWith every matrix representation of the (real) full linear group can be associated a multi-linear mapping of one affine space, Rn, into another, RN. This mapping is studied from the viewpoint of the geometry of numbers of convex bodies, and a general arithmetical property of such mappings is proved. The result generalizes my recent work on compound convex bodies.

Functions with Dominant Polar Singularities on the Circle of Convergence

1. Introduction. In an earlier paper1 the functionwas considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the formwhere the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.