This paper studies the asymptotic distribution of the multiparameter eigenvalues of a right definite multiparameter Sturm–Liouville eigenvalue problem. A uniform asymptotic analysis of the oscillation number of solutions of a single Sturm–Liouville type equation with potential depending on a general parameter is given; these results are then applied to the system of multiparameter Sturm–Liouville equations to give the asymptotic eigenvalue distribution for the system as a function of a “multi-index” oscillation number.

The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

Among the many formulæ which show special relations existing between the circular functions and the Bessel-Function Jn(x), when n is half an odd integer, there is one due to Lommel

The homogeneous real linear transformation in n variables is such that, when these variables are used as a set of mutually rectangular coordinates, an n-dimensional sphere is transformed into an n-dimensional ellipsoid; n mutually rectangular radii of the sphere become the n, mutually rectangular, principal radii of the ellipsoid. When these principal radii have not been rotated from their original directions, the transformation is said to be pure, or irrotational. Since these radii are necessarily real, the roots of the n-ic for the determination of the n principal elongations are necessarily real.

In this section we extend the definition of an E-set, so that it includes sets of the type

where the only restriction on the Ei is that they be non-singular. We now consider matrices of the type

where each ei takes independently the values 0, 1, …, n − 1, while the a(ei) are either complex numbers or else matrices of order r, the product a(ei) E(ei), in the latter case, being interpreted as the direct product of the two matrices a(ei) and E(ei).

The following is a simple geometrical demonstration of the well-known theorem that, if matter be distributed over a sphere with a surface-density (i.e., mass per unit of surface) inversely as the cube of the distance from either of two points which are the inversions of each other with respect to the sphere, it will act upon all external masses as if it were collected at the interior point:—and upon all internal masses as if a definite multiple of its mass were concentrated at the exterior point.

General Construction for Refracted Ray. The two-circle method of finding the direction of a ray refracted at a plane surface is very old, but seems to be now almost forgotten. It is particularly convenient when the refraction of several rays is to be determined, and its application to the case of a prism is especially elegant and leads to a simple self-contained proof of the condition for minimum deviation.

Let $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.

We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.

AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25