A certain n x n determinant, namely that in (1), provides a rather simple expression for the Tchebychev polynomial Tn(x) = cos n(cos-1x). This determinant also leads to an interesting combinatorial interpretation of the coefficients in that polynomial.

Let k be the kernel of a Watson transformation; that is k(x)/x ϵ L2(0,∞), and if x and y are positive,

1

Then if g is the transform of F ϵ L2(0,∞), that is if

2

it i s known that g 2(0,∞), that

3

and that

4

For these results see [1; theorem 79].

In Riemannian geometry the autoparallels associated with the affine connexion coincide with the geodesies which arise from the metric. This is not the case in a modification of Riemannian geometry suggested by Lyra. A sufficient condition that the two classes of curves coincide is obtained.

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

What is a von Neumann algebra? What is a factor (i) of type I, (ii) of type II, (iii) of type III? What is a projection geometry? And finally, what is a continuous geometry?

The questions recall some of the most brilliant mathematical work of the past 30 years, work which was done by John von Neumann, partly in collaboration with F. J. Murray, and which grew out of von Neumann1 s analysis of linear operators in Hilbert space.

It i s well known that the Riemannian ζ-function can be extended into the whole region s≠1 by means of the Euler-MacLaurin summation formula. The author wishes to present a proof of this fact which i s rather more elementary than those he i s acquainted with.

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.