The Kervaire Conjecture is correct if it can be shown to hold for non-singular systems of equations of length 3. In this paper we prove it for the case of equations over a group G where each equation has the form axbx−1cy = 1 for a, b, c ∈ G.

ABC is a triangle, right-angled at A, and X, Y, Z are the centres of the squares described on the sides opposite the angles A, B, C; XD, XM, XN are respectively perpendicular to BC, CA, AB; MY, DY are joined, and DY meets AC at E; NZ, DZ are joined, and DZ meets AB at F.

If in a triangle ABC, points are taken on the sides such that

then the radical axis of the circles PQR, P′Q′R′ passes through the centroid and “S.” points of ABC; and if QR, Q′R′ cut in 1, RP, R′P′ in 2, PQ, P′Q′ in 3, then the equation to the circle 123 is

The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.

Consider uniform spaces X and Y and a separately uniformly continuous real-valued function f on X × Y. The following question arises in the theory of games: under what conditions can f be extended to a separately continuous function on × Ŷ, where , Ŷ are the completions of X and Y respectively? Firstly observe that such an extension is not always possible. If X = Y = (0, 1] with the usual uniform structure and f(x, y) = xy then f is separately uniformly continuous but has no separately continuous extension to × Ŷ = [0, 1]2 since such an extension would satisfy f(0, .) = 0 on Y and f(., 0) = 1 on X and so would necessarily have a discontinuity in one argument at the origin.

For determining the greatest bending moment produced by a rolling load consisting of two concentrated loads at a fixed distance from one another passing over a span, the usual method involves somewhat intricate algebraic discussion. Even for two concentrated loads the discussion is complicated, and for three it would be still more so.

The method here explained, which applies directly to the case when the whole of the rolling load is on the span, uses the shearing force curve only, and gives a very simple geometrical construction for finding the position and magnitude of the greatest B.M.

We will begin by stating the construction, and then proceed to give a geometrical proof.

A graph Γ is said to be locally primitive if, for each vertex α, the stabilizer in Aut Γ of α induces a primitive permutation group on the set of vertices adjacent to α. In 1978, Richard Weiss conjectured that for a finite vertex-transitive locally primitive graph Γ, the number of automorphisms fixing a given vertex is bounded above by some function of the valency of Γ. In this paper we prove that the conjecture is true for finite non-bipartite graphsprovided that it is true in the case in which Aut Γ contains a locally primitive subgroup that is almost simple.

This paper is a sequel to a previous paper (1) on axisymmetric potential problems for one or more circular disks situated inside a coaxial cylinder and applies the method used for these problems to the electrostatic potential problem for a perfectly conducting thin spherical cap situated inside an earthed coaxial infinitely long circular cylinder.

In this note, we examine some of the properties of Hermitian operators on complex unital Banach Jordan algebras, that is, those operators with real numerical range. Recall that a unital Banach Jordan algebra J, is a (real or complex) Jordan algebra with product a ˚ b, having a unit 1, and a norm ∥·∥, such that J, with norm ∥·∥, is a Banach space, ∥1∥ = 1, and, for all a and b in j,

Professor W. P. Milne has shown that if a pencil of plane cubic curves cut in two triads of points which are apolar to the members of the pencil, then the other three points of intersection also form an apolar triad to the pencil. I propose to show how to obtain a simple geometrical construction for the third apolar triad though the cubics in this case are not perfectly general. The method of approach is by means of Grassmann's construction for a cubic curve and the use of apolar theorems established for a curve described in this manner.