It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.

The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).

It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.

Let G be a locally finite group, k a field of characteristic p ≥ 0 and V a right kG-module. We say that V is an -module over kG, if each p′-subgroup H of G contains a finite subgroup F with the same fixed points as H in V. (By convention, 0′ is taken as the set of all primes.) Such modules arise as elementary abelian section of -groups, a class of locally finite groups similar in many ways to the class of finite soluble groups.

The main theorem is that if V is an -module over kG with trivial Frattini submodule, and G is almost abelian, then every composition factor of V is complemented. This is a crucial ingredient in Tomkinson's theory of prefrattini subgroups in a certain subclass of . An example is given to show that the theorem breaks down for metabelian G. This leads to an example of a -group in which there are no analogues of prefrattini subgroups - the first situation where one of the standard conjugacy classes of subgroups of finite soluble groups has no decent analogue in the whole class

A question of John S. Wilson concerning indecomposable representations of metabelian groups satisfying the minimal condition for normal subgroups is answered negatively, by means of an example. It is shown that such representations need not be irreducible, even when the group being represented is an extension of an elementary abelian p–group by a quasicyclic q–group of the type first described by V.S. Čarin, and the characteristic of the field is a prime distinct from both p and q. This implies that certain techniques used in the study of metabelian groups satisfying the minimal condition for normal subgroups are not available for the corresponding class of soluble groups of derived length 3.

Using homological methods (and in particular the 5-term sequence of Hochschild-Serre in the homology of groups) a generalization of the theorem of John G. Thompson (J. Austral. Math. Soc. 16 (1973), 16–17) referred to in the title is proved.

A method of construction of designs with parameters v1 = r2 = p2, r1 = v2 = p + 1, b = p(p+1), k = p which may be used for the two-way elimination of heterogeneity is discussed. These designs were first studied in connection with estimating tobacco mosaic virus. Our designs have the advantage that every treatment occurs at most once in a row or column. We give the designs explicitly for p = 3, 4, 5.

Given a site T, that is, a category equipped with a fixed Grothendieck topology, we provide a definition of fibration for morphisms of the presheaves on T. We verify that the notion is well-behaved with respect to composition, base change, and exponentiation, and is trivial on the topos of sheaves. We compare our definition to that of Kan fibration in the semi-simplicial setting. Also we show how we can obtain a notion of fibration on our ground site T and investigate the resulting notion in certain ring-theoretic situations.

In the Appendix to a recent paper by J.J. Koliha and A.P. Leung (Math. Ann.216 (1975), 273–284), a functional calculus for continuous affine operators was constructed on the basis of the Taylor-Dunford calculus. This calculus applied only to functions defined and analytic in an open set containing the spectrum of an operator and the point λ = 1. In the present paper I examine the affine resolvent, and develop independently a more general calculus applicable to functions which are analytic in any open neighbourhood of the spectrum of an affine operator.

Let G1 and G2 be graphs and h1, h2 be hamiltonian paths (h-paths) in G1 and G2 respectively. The hamiltonian product (G1, h2)*(G2, h2) was defined by Holton. If a hamiltonian cycle exists in G2, it can give rise to 2nh-paths. Peckham conjectured that (G1, h1)*(G2, h2)≡(G1, h1)*(G2, h3) where h2 and h3 are any two of these 2nh-paths of G2. He has proved the validity of this conjecture for those h2, h3 where h3 is obtainable from h2 by a rotation along the h-cycle of G2. Here we disprove this conjecture for those h2, h3 where one is obtained from the other by a reflection of the h-cycle.

Suppose T is the natural transformation

where the variables are abelian groups. We find conditions on A, B, and C that guarantee that T is injective.

The complex nonlinear complementarity problem considered here is the following: find z such that

where S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. We study the extent to which the existence of a z ∈ S with g(z) ∈ S* (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on positive semi-definite matrices.

A recent improvement by H. Guggenheimer of the lower bound for the product of the volumes of a pair of polar reciprocal convex bodies is used to replace one of the results of Kurt Mahler, “Polar analogues of two theorems by Minkowski”, Bull. Austral. Math. Soc. 11 (1974), 121–129 by a best possible one. A similar improvement can be made for the other theorems in my note.

There is a group defined by fourth powers which did not yield to attempts to determine its order by coset enumerations. This group has now been shown to be infinite with the aid of a computer. An outline of the method is given as well as a simple direct proof inspired by the results of further computer calculations.

Recently Kurt Mahler asked: for which natural numbers N is the least common multiple of all the binomial coefficients the product of the primes less than or equal to N? We obtain a formula for the least common multiple of all the binomial coefficients of any natural number N and hence show that 2,11, and 23 are the only soultions to Mahler's problem.

The semigroup of all onto mappings of a set to itself and the semigroup of all one-to-one mappings of a set to itself are shown to have the property that every automorphism is inner.

In this note, Lp estimators for the parameters in the linear model y = Xβ are considered. In particular, it is shown that these estimators are invariant under scale transformations on the dependent variable; that is, if is an Lp estimator for β, then for any nonzero scalar a. It is shown that this result does not extend to more general transformations on y, and the invariance property does not hold for general nonlinear models.

A powerful tool in the construction of orthogonal designs has been amicable orthogonal designs. Recent results in the construction of Hadamard matrices has led to the need to find amicable orthogonal designs A, B in order n and of types (u1, U2, …, u6) and (ν1, ν2, …, νr) respectively satisfying At = -A, Bt = B, and ABt = BAt with

For simplicity, we say A, B are amicable orthogonal designs of type (u1, u2, …, us; v1, v2, …, vr).

We completely answer the question in order 8 by showing (1, 2, 2, 2; 8), (1, 2, 4; 2, 2, 4), (2, 2, 3; 2, 6), (7, 1, 7) and those designs derived from the above are the only possible.

We use our results to obtain new orthogonal designs in order 32.