The first part of this paper depends on the theorem that if a, b, c are three positive quantities such that

then abc is a maximum when a=b=c; with the corollary that if a, b, c are three negative quantities such that

then abc is a minimum when a = b = c.

Schouten and van Kampen (1) have studied the deformation of a . Applying the methods of that paper to the tangent vectors , which exist by hypothesis at all points of a certain region Vm′ (m′ > m) of Vn, we shall have

Whence we define the differentials

In the application of the to the lower index is treated as an ordinal index only. We shall not be concerned with any extension of the to indices other than those of the general Vb (see 1, equ. 3.24).

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).

We write

and suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).

In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group

In this paper we prove five structure theorems for groups with dihedral 3-normalisers. The interest in these theorems lies not so much in the results themselves as in what can be proved from them. The original versions of the results are contained in our doctoral thesis (1) where they are used to prove the following theorem, of which this paper, together with (2), (3) and other papers in preparation, will constitute a published proof:

The important inequality of which Prof. Chrystal has given so many examples may be called the “Power Inequality.”

There is a simple theorem of the Differential Calculus which is to a general function f(x), what the Power Inequality is to the function xm.

Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.

[The foflowing method of dividing a straight line in medial section was communicated to Mr Munn of the Edinburgh High School, by the Right Hon. Hugh C. E. Childers in January last.]

Let AB be the line. (Fig 56).

Draw AC = ½AB, and at right angles to it. Join CB. Bisect the angle ACB with CD, cutting AB at D. Draw DE at right angles to CB. Then the triangles CAD, CED are equal, and AD = DE. Draw a circle with DA and DE radii, cutting AB at F.

Let X be a rearrangement invariant function space on [0,1] in which the Rademacher functions (rn) generate a subspace isomorphic to ℓ2. We consider the space Λ(R, X) of measurable functions f such that fg∈X for every function g=∑bnrn where (bn)∈ℓ2. We show that if X satisfies certain conditions on the fundamental function and on certain interpolation indices then the space Λ(R, X) is not order isomorphic to a rearrangement invariant space. The result includes the spaces Lp, q and certain classes of Orlicz and Lorentz spaces. We also study the cases X = Lexp and X = Lψ2 for ψ2) = exp(t2) – 1.

By the rank r(S) of a finite semigroup S we shall mean the minimum cardinality of a set of generators ofS. For a group G, as remarked in [3], one has r(G)≦log2|G|, the bound being attained when G is an elementary abelian 2-group. By contrast, we shall see that there exist finite semigroups S for which r(S)≧|S| – 1. In the hope that it will not be considered too whimsical, we shall refer to a finite semigroup S of maximal rank (i.e. for which r(S) = |S|) as royal; a semigroup of next-to-maximal rank (i.e. for which r(S) = |S|–1) will be called noble.

This paper is concerned with the existence and uniqueness of solutions for the Picard boundary value problem

in a real Hilbert space. Our theorems improve corresponding results of Mawhin for |k| large.