Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li by

is again in L1, and

Let there be given, on an algebraic curve C, of genus p, a linear series and an algebraic series of index v, both without fixed points. The number of groups of r + 1 points which are common to a set of and a set of has been shown by Schubert (1) to be

where d is the number of double points of .

In a perfect sequence space α, on which a norm is defined, we can consider three types of convergence, namely projective convergence, strong projective convergence and distance convergence. In the space σ∞, when distance is defined in the usual way, the last two types of convergence coincide and are distinct from projective convergence ((2), p. 316). In the space σ1 all three types of convergence coincide ((2), p. 316). It will be shown in this paper that, if distance convergence and projective convergence coincide, then all three types of convergence coincide. It will not be assumed that the limit under one convergence is also the limit under the other convergence.

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution.

It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions.

Much of the work on the theory of diffraction by an infinite wedge has been for cases of harmonic time-dependence. Oberhettinger (1) obtained an expression for the Green's function of the wave equation in the two dimensional case of a line source of oscillating current parallel to the edge of a wedge with perfectly conducting walls. Solutions of the time-dependent wave equation have been obtained by Keller and Blank (2), Kay (3) and more recently by Turner (4) who considered the diffraction of a cylindrical pulse by a half plane.

A Banach space which is not reflexive may or may not be equivalent (in Banach's sense) to an adjoint space. For example, it is an elementary fact that the space (l), though not reflexive, is equivalent to (co)*, where (co) is the space of all sequences that converge to zero, normea in the usual way. On the other hand, (co) itself is not equivalent to any adjoint space : this can be proved by means of the Krein-Milman theorem, but here we obtain the result by an elementary argument which is scarcely more complicated than the standard proof that (co) is not reflexive.

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.

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution of

such that on z = 0

Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution .

Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).

Let K be a valued field, let v denote its valuation and B its valuation ring. Let P denote the valuation ideal. For each a in B, let ā denote the residue class a + P in the field B/P; for f(x)=∑arxr in B[x], let f(x) denote ∑ārxr in B/P[x]. Let Λp denote the leading coefficient of a polynomial p, and ∂p the degree of a non-zero polynomial.