In the first paper of this series [1] which will be designated I, particular solutions of various kinds have been found for the iterated equation of generalized axially symmetric potential theory (GASPT) which, in the notation defined in I, is (1) where the operator is defined by

This paper deals with the study of a particular md-class of sets. The underlying theory was introduced and studied by J. C. E. Dekker in [4]. We shall assume that the reader is familiar with the terminology and main results of this paper; in particular with the concepts of md-class of sets, gc-class of sets, gc-set, gc-function and the RET of a gc-class of sets. We also use the following notations of [4]: ε = the set of all non-negative integers (numbers), R = Req (ε).

Let Fp be the residue field modulo a prime number p. The mappings of Fp, into itself are viewed as functions in one variable over Fp. When the mapping is onto, the function is a permutation.

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

Let S be a compact semigroup and f a continuous homomorphism of S onto the (compact) semigroup T. What can be said concerning the relations among S, f, and T? It is to one special aspect of this problem which we shall address ourselves. In particular, our primary considerations will be directed toward the case in which T is a standard thread. A standard thread is a compact semigroup which is topologically an arc, one endpoint being an identity element, the other being a zero element. The structure of standard threads is rather completely determined e.g. see [20]. Among the standard threads there are three which have a rather special rôle. These are as follows: A unit thread is a standard thread with only two idempotents and no nilpotent element. A unit thread is isomorphic to the usual unit interval [14]. A nil thread again has only two idempotents but has a non-zero nilpotent element. A nil thread is isomorphic with the interval [½, 1], the multiplication being the maximum of ½ and the usual product — or, what is the same thing, the Rees quotient of the usual [0, 1] by the ideal [0,½ ]. Finally there is the idempotent thread, the multiplication being x o y = mm (x, y). These three standard threads can often be considered separately and, in this paper, we reserve the symbols I1I2 and I3 to denote the unit, nil and idempotent threads respectively. Also, throughout this paper, by a homomorphism we mean a continuous homomorphism.

In his paper (4), Mahler established several strong quantitative results on approximation in algebraic number fields using the geometry of numbers. In the present paper I derive analogous results for algebraic function fields of one variable using an analogue of the geometry of numbers.

Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.

It is well-known that the real number system can be characterised as a topological space [1], [3], as an ordered set [2], and as an ordered field [4]. It is the aim of this note to give two characterisations of the system purely as a field (see Theorems 4 and 9) without any extra notion of order, topology, et cetera.

A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication. (The latter phrase will generally be abbreviated to CAM and the multiplication in a semigroup will be denoted by juxta position unless the contrary is made explicit.)

The iterated equation of generalized axially symmetric potential theory (GASPT) [1] is defined by the relations (1) where (2) and Particular cases of this equation occur in many physical problems. In classical hydrodynamics, for example, the case n = 1 appears in the study of the irrotational motion of an incompressible fluid where, in two-dimensional flow, both the velocity potential φ and the stream function Ψ satisfy Laplace's equation, L0(f) = 0; and, in axially symmetric flow, φ and satisfy the equations L1 (φ) = 0, L-1 (ψ) = 0. The case n = 2 occurs in the study of the Stokes flow of a viscous fluid where the stream function satisfies the equation L2k(ψ) = 0 with k = 0 in two-dimensional flow and k = −1 in axially symmetric flow.

The iterated equation of generalized axially symmetric potential theory (GASPT), in the notation of the first paper of this series [1] which will be designated I, is the equation where the operator Lk is defined by

In [1], J. M. Howie considered the semigroup of transformations of sets and proved (Theorem 1) that every transformation of a finite set which is not a permutation can be written as a product of idempotents. In view of the analogy between the theories of transformations of finite sets and linear transformations of finite dimensional vector spaces, Howie's theorem suggests a corresponding result for matrices. The purpose of this note is to prove such a result.

In 1926, I. J. Schur proved the following theorem on partitions [3].

The number of partitions of n into parts congruent to ±1 (mod 6) is equal to the number of partitions of n of the form 1 + …+bs = n, where bi–bi+1 ≧ 3 and, if 3 ∣ bi, then bi–bi+1 > 3.

Schur's proof was based on a lemma concerning recurrence relations for certain polynomials. In 1928, W. Gleissberg gave an arithmetic proof of a strengthened form of Schur's theorem [2]; however, the combinatorial reasoning in Gleissberg's paper becomes very intricate.

If E is a subset of the real line of positive measure, then the associated Hilbert transform H = HE,

where the integral is a Cauchy principal value, is a bounded self-adjoint operator on L2(E) (cf. Muskhelishvili [4]). In case E = (-∞, ∞) the transformation is also unitary with a spectrum consisting of 1 and -1, each of infinite multiplicity (Titchmarsh [10]). If E is a inite interval the spectral representation of H has been given by Koppelman and Pincus [3]; see also Putnam [6]. In particular the spectrum of H is in this case the closed interval [-1, 1]. Moreover, according to Widom [11], the spectrum of H is [-1, 1] whenever E ≠ (-∞, ∞), that is, whenever

According to Bourbaki [1, pp. 62–63, Exercise 11], a left (resp. right) A-module M is said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].