In the study of rational approximations to irrational numbers the following problem presents itself: Let ω be a real irrational number, and let us consider the rational fractions satisfying the inequality

how small can the positive quantity k be chosen with the certainty that there will always be an infinite number of fractions satisfying the inequality whatever the value (irrational) of ω?

The following mnemonics, with one exception, consist of verses or sentences such that if the number of the letters in each word be written down in the order in which the words occur, the desired value will be obtained.

Recall that a norm and monotone if and monotone if If the norm is both absolute and monotone, itis called a Riesz norm. It is easy to show that a norm is Riesz if and only if whenever A Banach lattice is a vector lattice equipped with a complete Riesznorm.

A short proof, based on the Schur complement, is given of the classical result that the determinant of a skew-symmetric matrix of even order is the square of a polynomial in its coefficients.

The problem of the conduction of heat in one dimension is usually concerned with the propagation of a thermal disturbance along a bar or rod of uniform cross section. The solution of the problem is required for a given initial distribution of temperature, and given boundary values, usually at each end of the rod. In most cases this solution is found by assuming a series solution and then proving that the series satisfies the equation of the disturbance well as all the assigned conditions. Other methods, for example the contour integral method developed by Carslaw, also introduce this arbitrary element of choice in choosing the integrand and the contour of integration. The object of the present paper is to develop the application of Heaviside's Operational method to the solution of the problem, and to show that it leads in all cases to solutions equivalent to the known forms, although initially no assumptions are made regarding the nature of the solution.

Let f(z) be represented on its circle of convergence |z| = 1 by the Taylor series

and suppose that its sole singularity on |z| = 1 is an almost isolated singularity at z = 1. In the neighbourhood of such a singularity f(z) is regular on a sufficiently small disk, centre z = 1, with the outward drawn radius along the positive real axis excised. If also in this neighbourhood |f(z)| e−(1/δ)ρ remains bounded for some finite ρ, where δ is the distance from the excised radius, then the singularity is said to be of finite exponential order.

The problem of determining the distribution of stress in the neighbourhood of a Griffith crack, denned by ∣x∣ ≦ 1, y = 0, which is subject to an internal pressure varying along the length of the crack has been solved by Sneddon and Elliott (3) and by England and Green (1). In the former paper a Fourier cosine method is used to arrive at a solution while in the latter paper the problem is reduced to an Abel integral equation by making use of integral representations of the complex potentials given in Green and Zerna (2). Neither paper deals with the calculation of the stress intensity factor

which is extremely important to workers in fracture mechanics.

I. Construction for the Brocard points.

Let ABC be a triangle. Describe a circle touching AB in A and passing through C; draw the chord AP parallel to BC. Join BP meeting this circle in Ω

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.

In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings.

We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup ft with preorder defined by α≺β if β is in the ideal generated by α, and we use the direct sum to construct an Ω-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in Ω when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring S[ft] correspond bijectively to pairs (Φ, Q) with Φ a prime ideal of Ω and Q a prime ideal of S. The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when Ω is an infinite free band, by showing that S[Ω] is prime whenever S is.

When Ω satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of Ω, and also to the direct product of the component rings.

Let M be a positive integer, let a1, a2, …, aM be non-negative reals, and put aM+i = ai for i = 1, 2, 3. Further let each of v1, v2, v3 and δ1, δ2, δ3 be 0 or 1, giving 26 possibilities. This note is concerned with the problem of finding bounds for each of the non-trivial cases out of the 26 cyclic sums

In somes categories, there are structures that look very much like groups, and they usually are. These structures are called group-objects and were first studied by Eckmann and Hilton (1). If our category has an object T such that hom(X, T)= {tx}, a singleton, for each object X ∈ Ob , T is called a terminal object. Our category must have products; i.e. for A1,…, An ∈;. Ob , there is an object A1 × … × An ∈ Ob and morphisms pi: A1 × … × An → Ai so that if fi: X → Ai, i = 1, 2, …, n, are morphisms of , then there is a unique morphism [f1, …, fn]: X → A1 × … × An such that for i = 1, 2, …, n.

In many questions of analysis relating to the theory of plane curves, it is convenient to be able to obtain quickly the expansions of and of These may be obtained as follows:—

By an obvious extension of the theorem of Leibnitz we have

where d1,d2,…dn differentiate y1, y2,…yn respectively.