Nous Présentons ici l'étude, dans l'optique exhibée dans [7], d'une catégorie d'algèbres à idempotents, commutatives, non associatives, définies par des relations. On y trouve une structure qui n'est en général ni pondérée ni de Jordan et où, cependant, ces conditions sont équivalentes.

Following a previous paper, we study here a class of commutative and non associative algebras with idempotents and defined by relations. We find a structure which is not, in general, baric or Jordan but where these notions are equivalent.

An orthodox semigroup S is termed quasi-F-orthodox if the greatest inverse semigroup homomorphic image of S1 is F-inverse. In this paper we show that each quasi-F-orthodox semigroup is embeddable into a semidirect product of a band by a group. Furthermore, we present a subclass in the class of quasi-F-orthodox semigroups whose members S are embeddable into a semidirect product of a band B by a group in such a way that B belongs to the band variety generated by the band of idempotents in S. In particular, this subclass contains the F-orthodox semigroups and the idempotent pure homomorphic images of the bifree orthodox semigroups.

Let G be a group having presentation

a braid relator and a commutator respectively, and define the graph of such a group as having vertices labelled x1,…, xn + 1 and such that xi and xj are joined if and only if rij = (xi, xj).

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third order

are permutation matrices. It is convenient to denote them by

where the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

Mathieu's differential equation

is the equation which arises out of those two-dimensional problems Mathematical Physics in which the boundary is an ellipse, such problems, for example, as the vibrations of an elliptic membrane, which was first discussed by Mathieu,* and the scattering electromagnetic waves by a wire of elliptic cross-section. A different use of the same equation is found in Celestial Mechanics the treatment of perturbations and oscillations about periodic orbits, and, in a more mundane connection, it has been shown to be the differential equation of the variety artiste who holds an assistant poised on a pole above his head while he himself is standing on a spherical ball rolling on the ground!

When a student of mathematics commences the study of a subject which involves the assimilation of what are, to him, fundamentally new ideas, his progress is, as a rule, slow at first. And, even after he has become accustomed to these ideas, he may still require a long course of laborious practice, before he can attain to that mastery of the method which would enable him to use it as a powerful aid to research. Thus students, familiar with geometrical methods, when first commencing the study of Cartesian analysis, require much practice before they can call up mentally the geometrical figure corresponding to a given equation. And, the more general the new method is, the greater is the difficulty felt to be. So, in Hamilton's system of quaternions, the difficulty of assimilation is greater than it is in the Cartesian analysis. And it seems as if it were for this reason that, in recent years, attempts have been made, by men of known mathematical ability, to smooth the paths.

One of the proofs of the theorem that the three escribed circles and the inscribed circle of a triangle touch a common circle, depends upon the following well-known property of four circles that touch a common circle:—The common tangents of four such circles satisfy the relation

where 12 denotes the common tangent to the circles 1 and 2, etc. The converse part of the theorem, however, namely, that when the above relation holds good, the four circles touch a common circle, is generally assumed; the object of this note is to supply a demonstration of that part of the theorem. It should be noted that if the circles 1 and 2 touch the fifth circle either both externally or both internally, 12 denotes the direct common tangent; while if one of them touches externally and the other internally, 12 denotes the transverse common tangent. Further, the length of the direct common tangent to two circles remains unaltered if the radius of each be diminished or be increased by the same amount; while the length of the transverse common tangent remains unaltered, if the radius of one be increased and that of the other be diminished by the same amount. It is obvious, moreover, that if two circles, 1 and 2, touch a circle A externally, and if the radii of 1 and 2 be diminished and the radius of A increased by the same amount, the contact still holds good; and all the other cases may be easily considered.

In this note denotes a divergent series of positive, decreasing terms for which are real numbers (convergence factors) such that is convergent. We put

H. Rademacher has shown that

In the proof of the binomial theorem for negative and fractional indices given in many text-books of algebra, and attributed to Euler, one step seems to me to involve a very gross assumption.

The symbol f(m) having been taken to denote the series

it is pointed out that whenever m and n are positive integers we know that f(x)×f(n)≡f(m + n); and the conclusion is drawn that since this is true for all positive integral values of m and n, by the “permanence of equivalent forms” (whatever that may mean) we can conclude that it is true also for negative and fractional values of m and n, whenever f(m) and f(n) are convergent.

We consider spines of spherical space forms; i.e., spines of closed oriented 3-manifolds whose universal cover is the 3-sphere. We give sufficient conditions for such spines to be homotopy or simple homotopy equivalent to 2-complexes with the same fundamental group G and minimal Euler characteristic 1. If the group ring ℤG satisfies stably-free cancellation, then any such 2-complex is homotopy equivalent to a spine of a 3-manifold. If K1(ℤG) is represented by units and K is homotopy equivalent to a spine X, then K and X are simple homotopy equivalent. We exhibit several infinite families of non-abelian groups G for which these conditions apply.

We present in this paper a straight-forward method of deducing integral transform pairs directly from second order differential equations. In (1) this subject has been very thoroughly treated and in view of this we believe that the method described here has its greatest justification in its simplicity.

We prove the following theorem, which settles a conjecture made by J. S. Lew (2, p.379).

A congruence ρ on a semigroup will be called idempotent-separating if each ρ-class contains at most one idempotent. It is shown below that there exists a maximum such congruence µ on every inverse semigroup S. Two characterisations of µ are found, and it is shown (a) that S/µ⋍E, the semilattice of idempotents of S, if and only if E is contained in the centre of S; (b) that µ is the identical congruence on S if and only if E is self-centralising, in a sense explained below.

This note is a sequel to the article by Mine (2) on the same problem.

I described in (1) a notation for indices of powers in non-associative algebra, defined the degree † and altitude of a power or index, and observed that powers can be represented by bifurcating root-trees. For example, the power xx.x is denoted x2 + 1, with index 2 + 1, and is represented by the tree ; the degree (the number of factors, or free knots in the tree) is 3, and the altitude (the height of the tree) is 2. Multiplication being non-commutative or commutative, one maintains or ignores the distinction between left and right in the tree.

The middle points of the diagonals of a complete quadrilateral are collinear.

Let ABCDEF be a complete quadrilateral (see fig. 70), and X, Y, Z, the middle points of its diagonals. To prove X, Y, Z, collinear.