In this paper we continue our study of the Frattini p-subalgebra of a Lie p-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which L2 is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal also has trivial Frattini p-subalgebra, and if the underlying field is algebraically closed then so does every subalgebra. Finally we consider Lie p-algebras L in which the Frattini p-subalgebra of every subalgebra of L is contained in the Frattini p-subalgebra of L itself.

Let G be one of the following compact simply connected Lie groups: SU(3), Sp(2), G2. In the first two cases there is a well known stable decomposition of G as Q ∨ Sd where d = dim G and Q is a certain subspace of G. For SU(3), Q is the stunted complex quasiprojective space Σ(ℂP2/ℂP1) which fits into a cofibration sequence S3→Q→S5 with stable attaching map η:S5 → S4 For Sp(2), Q is the quaternionic quasi-projective space ℍℚ1 and fits into a cofibration sequence S3→Q→S7 with stable attaching map 2ν:S7→S4 (Here η and ν are generators of respectively.)

Cayley has shown that if the family of surfaces φ(x, y, z) = λ is one of an orthogonal triad, φ satisfies a differential equation of the third order. If, however, the parameter λ is involved implicitly in the equation of the family, the condition requires modification: for example, although

is one of an orthogonal triad, φ does not satisfy Cayley's equation.

Of late years there has arisen a clique of vector analysts who refuse to admit the quaternion to the glorious company of vectors. There are others again who take exception to some of Hamilton's most fundamental principles, and make corrections as they deem them, which logically revolutionise the whole basis of the calculus.

Just over sixty years ago—in December 1841—a Commission on Weights and Measures made the following proposals towards the establishment of a decimal coinage in this country: (1) the sovereign to be the unit; (2) a coin worth two shillings to be introduced under a distinct name; (3) a coin equal to the hundredth part of a pound to be established; (4) the farthing to be considered as the thousandth part of a pound; (5) other coins bearing a simple relation to these (including the shilling and sixpence) to be circulated.

In 1873, at the Lyons meeting of the French Association for the Advancement of the Sciences, Monsieur Emile Lemoine called attention to a particular point within a plane triangle which he called the centre of antiparallel medians. Since that time the properties of this remarkable point and of the lines and circles connected with it have been investigated by various writers, foremost among whom is Monsieur Lemoine himself. The results obtained by them are so numerous (indeed every month adds to their number) and so widely scattered through the mathematical periodicals of the world that it would be a task of considerable magnitude to make even an undigested collection of them. It is the purpose of the present paper to state those properties of the point which had been discovered previously to 1873. A short sketch of some of them will be found at the end of a memoir read by Monsieur Lemoine at the Grenoble meeting (1885) of the French Association, and in a memoir by Monsieur Emile Vigarié at the Paris meeting (1889) of the same Association. The references given by Dr Emmerich in his Die Brocardschen Gebilde (1891) are very valuable. It is a pity they are not more explicit.

Let G be an affine Kac–Moody group over ℂ, and V∞ an integrable simple quotient of a Verma module for g. Let Gmin be the subgroup of G generated by the maximal algebraic torus T, and the real root subgroups.

It is shown that (the least positive imaginary root) gives a character δ∈Hom(G, ℂ*) such that the pointwise character χ∞ of V∞ may be defined on Gmin ∩ G>1.

The periodic solutions of the linear differential equation

,

which reduce to Mathieu functions when v = 0 or 1, will be known as the associated Mathieu functions. The significance of this terminology will appear in the following section.

We are concerned with the problem of bifurcation of solutions of a non-linear multiparameter problem at a simple eigenvalue of the linearised problem.

Let X and Y be real Banach spaces, and let A, Bi, i = 1, …, n∈B(X, Y). Let : Rn × X → Y be a non-linear mapping. We consider the equation

where

and λ=(λ1, λ2,…,λn) ∈ Rn is an n-tuple of spectral parameters.

Suppose X is a Banach space and J is the interval −∞<t<∞. For 1 ≦ p<∞, a function is said to be Stepanov-bounded or Sp-bounded on J if

(for the definitions of almost periodicity and Sp-almost periodicity, see Amerio-Prouse (1, pp. 3 and 77).

In a recent number of the American Mathematical Monthly (December 1914), Professor E. V. Huntington calls attention to the inconclusive, and in some cases erroneous, discussion of the uniplanar motion of a rigid body as it is presented in the more elementary books on mechanics. In the simpler problems considered the solution is usually found by use of the rule that the rate of change of moment of momentum about some convenient point is equal to the torque or turning moment about the same point. In many cases there is no difficulty choosing this convenient point; but apparently there is confusion as to the points which can be legitimately chosen.

In a paper by Mr Arthur Berry, M.A., in the Proceedings of the Cambridge Philosophical Society, Volume X. Pt. I., “On the Evaluation of a certain Determinant which occurs in the mathematical theory of statistics and in that of elliptic geometry of any number of dimensions,” a remark is made that in the case of n = 3 this determinant was readily evaluated by me by means of the formulæ of spherical trigonometry. I have thought that it might be of interest to show this evaluation, but I shall merely state the determinant at once of order 3, and leave the reader to refer to the paper quoted for the general determinant.

An obvious question occurs at the very start of equivariant homotopy theory. What is the relationship between maps equivariant up to homotopy and strictly equivariant maps? This question has been studied by various people, usually away from the group order ([8, 11, 22, 25, 26]). We consider the problem stably and answer it by giving a spectral sequence proceeding from homotopy equivariant to strictly equivariant information. The form of the spectral sequence is not surprising, but there are three distinctive features of our approach: (1) we show that the spectral sequence may be viewed as an Adams spectral sequence based on nonequivariant homotopy, (2) we show how to exploit the product structure, and (3) we give a treatment showing how Dress's algebra of induction theory [13] applies to give non-normal subgroups equal status. As a spinoff from (3) we also obtain spectral sequences for calculating homology and cohomology of universal spaces (3.5).

In this paper we prove that if the strong dual of an echelon space fulfils the Mackey convergence condition the echelon space is quasi-normable. Also we give a characterisation of the quasi-normable echelon spaces and we deduce that every non-quasi-normable echelon space is the strong dual of a non-complete (LB)-space.