In this note we intend to discuss the method of A. Córdoba and R. Fefferman of using covering lemmas to control maximal functions, and make some simplifications which allow us to obtain alternative proofs of some of their results.

The structure of various classes of annihilator algebras has been known for some time. Bonsall and Goldie (1) considered semi-simple Banach algebras with the properties

(i)r(L)≡{x:xε,yx=0(yεL)} ≠(0) for each proper closed left ideal L of ,

(ii)l(K)≡{x:xε,xy=0(yεK)}≠(0) for each proper closed left ideal K of ,

At a recent meeting of the Royal Society of Edinburgh, Professor Tait proposed and solved the following problem:—

To calculate the number of Partitions of any number that can be made by taking any number from 2 up to another given number.

Let us denote by the number of partitions of r obtained by taking any of the numbers 2, 3, 4,……(n − 1), n. In the particular case n = 7, r = 10, the actual partitions are 3 + 7, 4 + 6, 5 + 5; 2 + 2 + 6, 2 + 3 + 5, 2 + 4 + 4, 3 + 3 + 4; 2 + 2 + 2 + 4, 2 + 2 + 3 + 3; 2 + 2 + 2 + 2 + 2; ten in all. Hence =10.

When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the form

where x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.

Since the publication of the memoir of Mathieu on the transverse vibrations of an elliptic membrane, the subject has been discussed by many authors from different points of view. But the corresponding problem of the plate has received but little attention. Mathieu discussed the problem as early as 1869, but the method adopted is different from that followed in the present paper, the main object of which is to apply Whittaker's solutions of Mathieu's Equation to the problem of the elliptic plate. These solutions are really better suited for numerical calculations than the evaluation of infinite determinants.

Let E be a nuclear space provided with a topology different from the weak topology. Let {Ai: i ∈ I} be a fundamental system of equicontinuous subsets of the topological dual E' of E. If {Fi: i ∈ I} is a family of infinite dimensional Banach spaces with separable predual, there is a fundamental system {Bi: i ∈ I} of weakly closed absolutely convex equicontinuous subsets of E'such that is norm-isomorphic to Fi, for each i ∈ I. Other results related with the one above are also given.

1. Figure 54. If ABC be a triangle, P any point, then the system of forces PA, PB, PC is equivalent to the system PH, PK, PL, where H, K, L are the middle points of BC, CA, AB.

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on S separates the points of S. If X is a Banach space and T : X → F is a continuous linear operator, then T is quasi-convex if is quasi-convex, where U is the unit ball of X.

Let C*(F∞) be the full C*-algebra associated to the free group of countably many generators and SnC*(F∞) be the class of all n-dimensional operator subspaces of C*(F∞). In this paper, we study some stability properties of SnC*(F∞). More precisely, we will prove that for any E0, E1 in SnC*(F∞), the Haagerup tensor product E0⊗hE1 and the operator space obtained by complex interpolation Eθ are (1 + ∈)-contained in C*(F∞) for arbitrary ∈>0. On the other hand, we will show an extension property for WEPC*-algebras.

Let K be a field, G a finite group. Let V be an (irreducible) KG-module, where KG is the group algebra consisting of all formal sums . The action of on α = ∑aθg on element ν ∈ V obeys the rule If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.

A combinatorial hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This combinatorial hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups.

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.