A tournament T is called symmetric if its automorphism group is transitive on the points and arcs of T. The main result of this paper is that if T is a finite symmetric tournament then T is isomorphic to one of the quadratic residue tournaments formed on the points of a finite field GF(pn), pn ≡ 3 (4), by the following rule: If a, b ∈ GF(pn) then there is an are directed from a to b exactly when b – a is a non-zero quadratic residue in GF(pn).

The Hall projective planes of odd order are characterized in terms of their translations, collineations which fix all the points of a Baer subplane, and involutory homologies.

We prove that a translation plane π of odd order is a generalized Hall plane if and only if π is derived from a translation plane of semi-translation class 1–3a. Also, a derivable translation plane of even order and class 1–3a derives a generalized Hall plane. We also show that the generalized Hall planes of Kirkpatrick form a subclass of the class of planes derived from the Dickson semifield planes.

We consider a semigroup S acting as affine continuous maps on a compact convex set X. F denotes the corresponding set of fixed points. Let exX and exF denote the corresponding sets of extreme points. If X is a simplex, conditions are given which ensure that when x ε F, the maximal measure representing x invariant under S. We also prove exF = F ∩ exX under conditions involving extreme amenability of S. Topological properties of exF are also studied.

We give another formulation of Theorem 1 of Ribenboim's paper, Bull. Austral. Math. Soc. 2 (1970), 81–88. Our condition on a module to be totally ordered such that this order extends a given strict order seems to be more useful for applications than condition (2) of Ribenboim. This is shown for example in Corollaries 3 and It of our paper. Corollary 4 is due to Ribenboim. Corollary 3 generalizes the corresponding well known result on abelian groups.

Let n, s, t be integers with s > t > 2. If a family of n different subsets of a set S, with s elements, has the properties, (i) each member belongs to a set of (t+1) members which together have union S, (ii) no member belongs to a set of t members which together have union S, then we prove that n ≤ (t+1)s−t−1. The result is best possible.

A weighted average of the partial sums of a series provides a quick and moderately powerful sum for any series in which the ratio of successive terms varies slowly along the series and this ratio is not close to +1. Some properties of the sum are examined.

A characterization of the maximal normal p-nilpotent subgroup of a finite p-solvable group is obtained for primes p ≠ 2 or 3.

The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.

If {Ln} is a sequence defined by Ln = min {Ln-a +Ln-b, Ln-c +Ln-d}, with a, b, c, d positive integers, then one can ask if necessarily Ln = Ln-b + Ln-b, for all sufficiently large n.

The answer is yes if a and b are relatively prime, Ln > 0 initially, and λ < μ, where λ-a + λ-b = 1, μ-c + μ-d = 1. The answer is no if instead a and b have greatest common divisor k ≥2, with c ≡ 0 (mod k), d ≢ 0 (mod k).

A theorem proved by de Leeuw for Γ = Rn and later generalized by Lohoué and Saeki states that if Γ is an LCA group, Γ0 a closed subgroup thereof, π the canonical mapping from Γ onto Γ/Γ0 and Φ a Fourier multiplier of type (p, p) on Γ/Γ0, then Φ ० π is a Fourier multiplier of type (p, p) on Γ. We show here that if 1 ≤ p < q ≤ ∞, Γ and Φ is a Fourier multiplier of type (p, q) on Γ/Γ, then Φ ० π is a Fourier multiplier of type (p, q) on Γ and if Γ0 is a non-compact subgroup of Γ and Φ ० π is a Fourier multiplier of type (p, q) on Γ for some p and q satisfying 1 ≤ p ≤ q ≤ ∞, then Φ is zero. We prove also that if Φ is a Fourier multiplier of type (p, q) and Γ/Γ0, whee 1 ≤ q < p ≤ ∞ and Γ is discrete, then Φ ० π is a Fourier multiplier of type (p, q) on Γ.

A subset S of an additive group G is called a maximal sum-free set in G if (S+S) nS = Φ and |S| ≥ |T| for every sum-free set T in G. In this note, we prove a conjecture of Yap concerning the structure of maximal sum-free sets in finite abelian groups of order divisible by 3 but not divisible by any prime congruent to 2 modulo 3.

This work is concerned with the forced convection of heat in a circular tube. The fluid flow is assumed to be laminar Poiseuille flow, and the physical parameters; viscosity, density, conductivity; are assumed to be independent of temperature changes. Viscous dissipation terms are also ignored, and there are no heat sources in the fluid. The problem is treated for the case of a step change in the wall temperature, and the eigenvalues have been obtained as an expansion in powers of the Péclet number for the smaller values, and in an asymptotic form for the larger values. The temperature distribution in the fluid in the neighbourhood of the temperature jump has been calculated for two values of the Péclet number, as have the Nusselt numbers.

It has been shown that the existence of a (left) approximate identity in the tensor product A ⊗ B of Banach algebras A and B, where α is an admissible algebra norm on A ⊗ B, implies the existence of approximate identities in A and B. The question has been raised as to whether the boundedness of the approximate identity in A ⊗αB implies the boundedness of the approximate identities in A and B. This paper answers the question affirmatively with a being the greatest cross-norm.

In this paper, the problem of heat transfer to laminar Poiseuille flow in a circular tube is discussed for the case of an insulated tube with a ring source of heat on the boundary. The solution is developed analytically for low values of the Péclet number, and formulae for calculating the eigenvalues and coefficients have been obtained. The temperature distributions in the neighbourhood of the source have been calculated for two values of the Péclet number. The extension to the case of arbitrary wall flux has also been discussed.

Let G be a group with a normal nilpotent subgroup N such that G/N is periodic and nilpotent. If G(p)/N is the Sylow p-subgroup of G/N and Q(p) is the maximal p-radicable subgroup of N, it is shown that G has a bound on the subnormal indices of its subnormal subgroups if and only if there is a positive integer c such that G(p)/Q(p) is nilpotent of class at most c, for all primes p. It is also shown that if G is a periodic metanilpotent group and Q is its maximal radicable abelian normal subgroup then G has a bound on its subnormal indices if and only if there is a positive integer c such that for all primes p the Sylow p-subgroups of G/Q are nilpotent of class at most c.

Led on by a paper of Brown and Dancer with the same title, this paper gives two further theorems on asymptotic expansions of composite functions.

Let X be a locally convex linear topological space. A point z in an ultralimit enlargement of X is pre-near-standard if and only it is finite and for every equicontinuous subset S′ of the dual space X′, a point z′ belongs to *S′ ∩ μσ(X′, X) (0) only if z′ (z) is infinitesimal.

We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).

Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α1 ≥ … ≥ αa, β1 ≥ … ≥ βb, γ1 ≥ … ≥ γn, respectively. In this paper four classes of inequalities are proved comparing the αi. and βj with the γk. The simplest of these is: if the subscripts is, js satisfy 1 ≤ i1 < … < im ≤ α, 1 ≤ j1 < … < jm ≤ b.