There is a remarkable cubic graph of 28 vertices, discovered by Prof. H.S.M. Coxeter, which has no circuit of fewer than 7 edges and in which all oriented arcs made up of three edges are equivalent under the automorphism group.

The structure of the graph is as follows. There are three disjoint heptagons A(a1a2a3a4a5a6a7a1), B(b1b5b2b6b3b7b4b1) and C(c1c6c4c2c7c5c3c1). We denote the seven remaining vertices by d1, d2,…, d7. For each suffix i the three edges incident with dijoin it to ai, bi and ci. In drawing a diagram of the graph it seems best to show only the heptagons A, B and C, leaving the rest of the figure to the imagination.

The purpose of this note is to establish another property of the graph, that it has no Hamiltonian circuit.

The following conjecture has been investigated recently by Mordell [1]. “Let z1 , …, zn be any set of n complex numbers. Then

1

the equality sign being necessary in the case when the z's are at the vertices of a regular n-sided polygon with center at the origin.”

L'objet de ce travail est de déterminer la structure d'une classe d'anneaux, celle des anneaux bipotents à droite.

1. Anneaux bipotents à droite. Un anneau A est dit bipotent à droite, si l'on a

pour tout a ϵ A, On voit facilement qu'un anneau A est bipotent à droite, si et seulement si pour tout couple a, b ϵ A, il existe x ϵ A tel que l'on ait ab = a2x.

In a previous paper [l] we discussed a property of the complementary sequences

1

where square brackets denote the greatest integer function and α is any positive irrational. We called {un} and {vn} Beatty sequences of argument α.

A well known theorem of van der Waerden [1] states that given any two positive integers k and t, there exists a positive integer m such that in every distribution of the numbers 1,2, …, m into k classes, at least one class contains an arithmetic progression of t + 1 terms. Other proofs and generalizations of this theorem have been given by Griinwald [2], Witt [3] and Lukomskaya [4]. The last mentioned proof appears in the booklet of Khinchin “Three pearls of number theory” in which van der Waerden's theorem plays the role of the first pearl.

In this paper an unsuccessful attempt to classify prime - power groups is described. The attempt consisted in combining some ideas of P.Hall and O. N. Golovin.

Golovin [2] defined nilpotent products of groups and showed that every nilpotent group is a factor group of a nilpotent product of cyclic groups.

In his well known monograph [l] (p. 81) H. Bachmann indicates that two ordinal numbers > 1 always have a common left multiple > 1, but not always have a right multiple (RM). The monograph does not, however, contain any further analysis of right multiples. The purpose of the present note is to supplement this by formulating the following propositions which, despite their simplicity, seem not yet to be known.

In [l], Moser derives a recurrence relation and studies the limiting behaviour of the Expectations En of the following game. “A real number is drawn at random from [0,1]. We may either keep the number selected, or reject it and draw again. We can then either keep the second number chosen or reject it, and draw again, and so on. Suppose we have at most n choices. What stopping rule gives the largest En and how can we estimate En?”

Functions employed for interpolation and curve fitting are for the most part polynomials with numerical coefficients. Indeed these are functions whose values for numerically given arguments can be computed directly without resorting to non-algebraic designs. It is known, however, that there are cases where polynomial interpolation does not yield an adequate approximation to a given function (cf. [4], p. 34).

If the Sturm-Liouville eigenvalue problem

1

is first approached from the standpoint of differential equations theory - as opposed, say, to the calculus of variations, or the theory of integral equations - the extremal properties of the eigenvalues seem to be generally regarded as lying beyond the scope of the theory. Thus, neither in the standard work of Bocher [l], nor in the recent work of Coddington and Levinson [2] is any mention made of this topic. Collatz [3, 166-8] gives an elementary proof of the minimum property of the least positive eigenvalue of (1.1), and a brief indication of how this argument can be extended to the higher eigenvalues. The purpose of this paper is to consolidate this elementary approach, and to extend it to cover the singular cases where either the interval is infinite, or one or more of the coefficients are singular at the end-points.

The summability methods

where bnk=ank+an,k+1+ …,are regarded as the sequenceto-sequence and series-to-sequence forms of the same method, and if sk=uo + … + uk, we speak of the series or the sequence {sk} indifferently, as summable A or B.