We obtain a series representation for ζ(s) valid in Res> – k(<0). This representation is obtained by a sequence of regrouping of the series 1–2−s + 3−s–4−s + …( = (1—21−s)ζ(s), Re s > 0). We can obtain asymptotic relations like

as an application of our series representation for ζ(s).

What does a simple ring with unity, a topological T0-space and a graph that has at most one loop but may possess edges, have in common? In this note we show that they all are Brown–McCoy semisimple. Suliński has generalised the well-known Brown–McCoy radical class of associative rings (cf. [1]) to a category which satisfies certain conditions. In [3] he defines a simple object, a modular class of objects and the Brown–McCoy radical class as the upper radical class determined by a modular class in a category which, among others, has a zero object and kernels. To include categories like that of topological spaces and graphs, we use the concepts of a trivial object and a fibre. We then follow Suliński and define a simple object, a modular class of objects and then the Brown–McCoy radical class as the upper radical class determined by a modular class.

1. Prof. E. T. Copson has discussed the well-known problem of a circular disc kept at a constant potential Vo in an external field of potential Φ by reducing it to the solution of two integral equations. Tho solution is however fairly simple if we use oblate spheroidal co-ordinates. This is due to the fact that in this system of coordinates the disc can be represented in terms of one co-ordinate only. This method is applied to the above problem and Copson's results are obtained. The solution when Vo is not constant, but any surface function of the disc, is also obtained.

The use of geodesic polar coordinates in the intrinsic geometry of a surface leads to the concept of a geodesic circle, i.e. the locus of points at a constant distance from the pole 0 along the geodesics through 0. A geodesic hypersphere is the obvious generalisation of this for a Riemannian Vn. We propose to consider more general central quadric hypersurfaces of Vn, which we define as follows. Let xi (i = 1, 2, …, n) be a system of coordinates in Vn, whose metric is gijdxidxj, and let aij be the components in the x's of a symmetric covariant tensor of the second order, evaluated at the point 0, which is taken as pole.

Hermite, in 1864 (Comptes Rendus, vol. 58) introduced into analysis the polynomials defined by the relation

where n is a positive integer. He showed that they satisfied the differential equation

that they were orthogonal functions, and that an arbitrary function f(x) could be expanded in the form

It has been well known for many years (2) that if Fμ(t) is the Fourier-Stieltjes transform of a bounded measure μ on the real line R, which is bounded away from zero, it does not follow that [Fμ(t)]−1 is also the Fourier-Stieltjes transform of a measure. It seems of interest (as was remarked, in conversation, by J. D. Weston) to consider measures on the half-line R+ = [0, ∞[, instead of on R.

The following proof of the well-known result, n being any number, a, b, c the different prime factors that singly, or multiply, compose it

seems worthy of notice.

Let E be n-dimensional (n≧2) real vector space with a nondegenerate symmetric scalar product (.|.):E × E → R1 with an arbitrary signature (p, n–p). Let us consider a second order partial differential equation (P.D.E.) of the form:

where φ is a given function of two variables, v is an unknown function (defined on an open subset 0 ⊂E), |∇ν|2: =(∇ν|∇ν) is the square of the gradient ∇ν of the function ν and ∇2, denotes the Laplace-Beltrami operator.

Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such that

provided 0<p≦q.

The object of this paper is to essay an analytical statement of the reduction of the integration of a canonic system of differential equations (into which time does not enter explicitly) to that of the partial differential equation of Jacobi and Hamilton; and to illustrate the principle of duality by an outline of the solution for the problem of two bodies both by the standard form of the equation referred to and by the analogous form which that principle involves. Most statements of the reduction are verifications and somewhat obscure the symmetry of the canonic form. The shortest procedure, of course, is by means of the well known theorem of Jacobi, and this verificatory method is followed by Tisserand, Charlier and Appell. Poincaré gives a proof depending on a simple form given by him to the conditions for a canonical change of variables, but again the statement lacks analytical form. The essentials of this proof will be given here, but in an entirely different way. An analytical treatment of the subject has been given by Professor L. Becker in his class lectures at Glasgow, but it has not been published.

The application of the hodograph method in problems in fluid dynamics dates back to the time of Helmholtz and Kirchhoff. The underlying principle is simple. It is in effect to rewrite the governing differential equations with the roles of the original dependent and independent variables reversed. Such a procedure is not uncommon in problems depending upon ordinary differential equations. For example, if the velocity of a particle in rectilinear motion is prescribed as a function of distance from a fixed point, the problem of finding the relation between its position and the time t can be solved by one quadrature if t is regarded as the dependent variable.