We investigate both left and right cancellation in the Stone–Čech compactification βS of a discrete semigroup S, obtaining several results for arbitrary semigroups S and others for more restricted semigroups. In particular, if S is the semigroup of injective functions from a set to itself we determine precisely which pairs x, y and S have some p∈βS with px = py. We also obtain several new results about right cancellation in (βℕ, +).

The object of this note is to point out, by a few remarks on a single case, how well worth the attention of younger mathematicians is the full study of certain problems, suggested by physics, but limited (so far as that science is concerned) by properties of matter.

We prove the existence of nontrivial k-type surfaces by constructing k-type immersions of flat tori in 6 which are not product immersions.

This article is intended to enlarge the study of spaces satisfying the countable neighbourhood property and to clarify the incidence of this property in the stability of some locally convex properties of tensor products.

We shall use the standard notations of locally convex spaces as in [17] and [18]. The word space will always mean separated locally convex space. If (£, t) is a space, the set of all continuous seminorms on it will be denoted by cs(E). The linear hull and the absolutely convex hull of a subset C of a space will be written lin(C) and г(C) respectively.

We present some existence results for the “nonresonant” singular boundary value problem a.e. on [0, 1] with Here μ is such that a.e. on [0, 1] with has only the trivial solution.

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].

I had occasion, lately, to consider the following question connected with the Kinetic Theory of Gases:—

Given that there are 3.1020 particles in a cubic inch of air, and that each has on the average 1010 collisions per second; after what period of time is it even betting that any specified particle shall have collided, once at least, with each of the others?

If u and v are vertices of the (finite, connected) graph Γ, let d(u, v) denote the length of the shortest path joining u to v in Γ. The graph Γ is said to be distance-transitive if whenever d(u, v) = d(u′, v′), there exists an automorphism g of Γ such that ug = u′ and if vg = v′. Distance-transitive graphs of valency 3 and 4 were originally classified [2, 11, 12, 13] by using a computer to generate all “feasible intersection arrays” (cf. [1, Chapter 20]). In both cases a classification has since been given by hand [4, 5]. Wecontinue this latter tradition and prove the following theorem—which was recently proved independently by Ivanov et al. using a computer [10].

The methods explained here are applicable to a large number of problems relating to the symmetric algebraic functions of n letters, and the special results here deduced from them are merely specimens to indicate some of the ways of applying these methods.

In the first section the main principle adopted is that of taking the standard form of a symmetric function to be a sum extending over all the cases of the typical term got by permuting the letters involved in all possible ways, whether they are different or not; and the main result reached is an Inequality Theorem arrived at by expressing the excess of the greater over the less in an explicitly positive form.

Stirling numbers of the first and second kind play an important part in many branches of mathematics, in particular in combinatorial analysis and statistics. For their definition and properties we refer to (5) where a whole chapter is devoted to their study. Stirling numbers have been generalized in many ways. One generalization is given in (1). In this paper we generalize the results of (1) to n dimensions. In order to simplify the notation we use methods of linear algebra.

In the full linear theory of thermoelasticity there is a coupling between the thermal and the purely mechanical effects so that not only does a nonuniform distribution of temperature in the solid produce a state of stress but dynamical body forces or applied surface tractions produce variations in temperature throughout the body. In a recent paper (Eason and Sneddon, (2)) an account was given of the calculation of the dynamic stresses produced in elastic bodies, both infinite and semi-infinite, by uneven heating. In this paper we shall consider the propagation of thermal stresses in an infinite medium when, in addition to heat sources, there are present body forces which vary with the time.

It is of some interest to the theory of locally convex *-algebras to know under what conditions such an algebra A is a pre-C*-algebra (the topology of A can be described by a submultiplicative norm such that ‖x*x‖ = ‖x‖2, ∀x∈A). We recall that a locally convex *-algebra is a complex *-algebra A with the structure of a Hausdorff locally convex topological vector space such that the multiplication is separately continuous, and the involution is continuous.

The square described on the hypotenuse of a right-angled triangle is equal to the squares described on the other two sides.

About half a hundred proofs of this theorem have been given, but few of them have been “ocular,” that is, few have shown how the two smaller squares may be decomposed so as to fit into the largest square. One of the most elegant of the ocular proofs is that of Henry Perigal, and was discovered about 1830. A demonstration of its correctness is not difficult to obtain, but the following demonstration is believed to be new. It depends somewhat on algebra, and presupposes a simple lemma.