From the figure we have at once

similarly

In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)

In this paper, we show how to give a geometric interpretation of the modular correspondence T3 on the modular curve X(11) of level 11 using projective geometry. We use Klein's theorem that X(11) is isomorphic to the nodal curve of the Hessian of the cubic threefold Λ defined by V2W + W2X + X2Y + Y2Z + Z2V = 0 in P4(C) and geometry which we learned from a paper of W. L. Edge. We show that the correspondence T3 is essentially the correspondence which associates to a point p of the curve X(11) the four points where the singular locus of the polar quadric of p with respect to Λ meets X(11). Our control of the geometry is good enough to enable us to compute the eigenvalues of T3 acting on the cohomology of X(11). This is the first example of an explicit geometric description of a modular correspondence without valence. The results of this article will be used in subsequent articles to associate two new abelian varieties to a cubic threefold, to desingularize the Hessian of a cubic threefold and to study self-conjugate polygons formed by the quadrisecants of the nodal curve of the Hessian.

An action of a group G on a tree, and an associated Lyndon length function l, give rise to a hyperbolic length function L and a normal subgroup K having bounded action. The Theorem in Section 1 shows that for two Lyndon length functions l, l′ to arise from the same action of G on some tree, L = L′ and K = K′. Moreover for L non-abelian L = L′ implies K = K′. That this is not so for abelian L is shown in Section 2 where two examples of Lyndon length functions l, l′ on an H.N.N. group are given, with their associated actions on trees, for which L = L′ is abelian but K≠K′.

Finding the distribution of stress in earth dams containing cracks is an outstanding problem of soil mechanics. Even the simplest mathematical model, viz., that of a wedge containing a plane crack which is symmetrically situated along the bisector plane of the angle of the wedge, with the plane strain assumption of the infinitesimal theory of elasticity, presents a difficult problem of solving the bi-harmonic equation subject to mixed boundary conditions. While elasticity problems related to wedge-shaped bodies have been investigated, it appears little attention has been paid to the mixed boundary-value problems.As a first step towards the solution of the mixed boundary value problem for the biharmonic equation, we discuss in this paper the solution of Laplace's equation

for wedge-shaped regions subject to mixed type of conditions on the boundary. If we assume that φ does not depend on z, the equation (1.1) is reduced to the equation

Let T2 = {(eix1, eix2):0 ≦ xj<2π, j=1,2} be a two dimensional torus and r, s, t and k be positive integers with k>r+s+t–2. Our main object is to study the approximation and interpolation properties of a class of smooth functions whose restrictions to each triangle of a three direction mesh lie in the linear span of or 0≦μ≦r–1, r+s–l≦μ+ν≦r+s+t–2, or 0≦ν≦s–1, r+s–1≦μ+ν≦r+s+t–2} Where (z1, z2) ∈ T2.

Many problems in mathematical analysis require a knowledge of the asymptotic behaviour of Γ(z + α)/Γ(z + β) for large values of |z|, where α and β are bounded quantities. Tricomi and Erdélyi in (1), gave the asymptotic expansion

where the are the generalised Bernoulli polynomials, see (2), defined by

In this note, we show that if, instead of considering z to be the large variable, we consider a related large variable, (1) can be improved from a computational viewpoint.

In the following paper I give a complete list of the types of covariants belonging to the concomitant system of three quaternary quadrics, where covariant is used in its restricted sense and refers solely to a concomitant involving the variable x alone. A complete list of the types, 62 in number, is given in §1. In §§(6–10) the covariants are determined, and in §§(11–12) a list of the identities used in the reduction of the covariants is given, along with typical examples of the process.