It is well known that every point of the closed unit interval I can be expressed as the midpoint of two points of the Cantor ternary set D. See [2, p. 549] and [3, p. 105]. Regarding J as a one dimensional compact convex set, it seems natural to try to generalize the above result to higher dimensional convex sets. We prove in section 3 that every convex polytope K in Euclidean space Rd contains a topological copy C of D such that each point of K is expressible as a midpoint of two points of C. Also, we give necessary and sufficient conditions on a planar compact convex set for it to contain a copy of D with the midpoint property above. In the final section we prove a result on minimal midpoint sets.

Gurland [1] by making use of the Cramer-Rao lower bound for the variance of an unbiased estimator obtained the following inequality

1

for real values of α and δ satisfying α+δ>0, α≠0, δ>0. He used the fact that (Γ(δ)Γ(δ+α))xα is an unbiased estimator of θα where θ is the parameter for the density function

Recently Hunsaker and Naimpally [2] have proved: The pointwise closure of an equicontinuous family of point compact relations from a compact T2-space to a locally compact uniform space is locally compact in the topology of uniform convergence. This is a generalization of the same result of Fuller [1] for single valued continuous functions.

For a range space which is locally compact normal and uniform theorem B below is an improvement on the result of Hunsaker and Naimpally quoted above [see Remark 3 at the end of this paper].

If we add the operation of composition to the polynomial ring R[X], where R is a commutative ring with identity, we get a tri-operational algebra . A full ideal or tri-operational ideal of is the kernel of a tri-operational homomorphism on . This is equivalent [4, pp. 73–74] to the following: A full ideal of is a ring ideal A of R[x] such that f°g ∈ A for every f ∈ A and g ∈ R[x].

It has long been known that the existence of a Souslin line entails (and is entailed by) the existence of a Souslin tree; indeed such a tree can be built from the open subsets of the line in a natural way. It will be shown that less onerous restrictions on a topological space than orderability allow the construction to proceed. For example, to the expected requirements-that the space satisfy the countable chain condition and not be separable, one can add the hypothesis of local connectivity, and that either first category sets be nowhere dense or that nowhere dense sets be separable.

All rings have an identity, all homomorphisms map identities to identities, all homomorphisms on algebras over fields are algebra homomorphisms. A ring R is a quotient-embeddable ring (a QE-ring) if for any proper ideal a of R there is an endomorphism of R whose kernel is the ideal a. A QE-ring U is a receptor of R if for any proper ideal a of R there is a homomorphism from R to U whose kernel is the ideal a.

Theorem. A ring R has a receptor if and only if it is a K-algebra over some field K contained in the center of R. If R is a commutative K-algebra of this type, then it has a commutative receptor.

Let A be an n × n complex matrix. The singular values of A are the non-negative square-roots of the eigenvalues of A*A. G. N. De Oliviera [4] gave a necessary condition for the existence of a matrix A with a1..., an as diagonal elements and α1,..., αn as singular values. We shall give another necessary condition which implies the above author’s condition and we show that this is also a sufficient condition for the case n =2.

Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.

An element a of a semigroup S is n-potent if there exist a1, a2,..., ak∈S such that a = a1a2...ak and If S is a group, the set of n-potent elements is a normal subgroup of S and the set of 1-potent elements is the commutator subgroup of S.

A map f: X → Y is said to be universal if for every map g:X → Y there exists an x ∈ X such that f(x) = g(x). In [2] W. Holsztynski observed that if B is a Boltyanskiĭ continuum (see [1]), then there exists a universal map f:B→ I2 such that the product map fxf:BxB→I2×I2 is not universal. Using this he showed that B can be replaced by a two-dimensional polyhedron. He did not, however, give a concrete example. We exhibit explicitly a two-dimensional polyhedron K and a universal map f:K→ I2 such that f×f:K×K→ I2×I2 is not universal.

In [2] the structure of all semiperfect rings with abelian group of units has been obtained in terms of commutative local rings. It follows easily that the structure of semiperfect rings with elementary abelian group of units is determined by commutative local rings whose unit groups are elementary abelian. In this note such local rings are completely characterized. It is shown that a local ring having an elementary abelian group of units has characteristic two, four or eight and is a homomorphic image of ZkG/E(ZkG) where G is some elementary 2-group and E(ZkG) is the ideal of ZkG generated by {1 - u2:u∈(ZkG)*}.

When I agreed to give the 1976 Jeffery-Williams Lecture to the Canadian Mathematical Congress, I did not envisage that it would be my lot to give what might now be called the first memorial lecture in this series of lectures. During the past year, Ralph Jeffery and Lloyd Williams both died, and their deaths brought forward many deserved tributes to these two wonderful men. My lecture today will forge a link with my mathematical past, a past in which both of these men played important roles.

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

The Gibbs phenomenon of the Fourier series for different summability methods has been investigated by various authors. In this note, we study the same for the [S, αn] method of summability introduced by Meir and Sharma [3]. The corresponding result for the [T, αn] method of summability due to Powell [5] can be worked out in exactly the same way.

The elements cnk of the [S, αn] matrix are defined by the relations:

1

where 0<αn<1. The [S, αn] matrix is regular if and only if

There are few known metrization theorems for manifolds (locally Euclidean, connected, Hausdorff space). It is well known that for manifolds metrizability, second countability, Lindelöf's condition, σ-compactness and paracompactness are equivalent. Although these conditions imply separability, the latter does not imply any of the former (see Example 2.2), as is often believed. A common source of metrization for a covering manifold is that lifted from the base manifold [8; p. 181].

For 2-manifolds, the presence of a complex analytic structure gives us a metrization theorem; it has been shown [1] that such manifolds are topologically characterized as those which are orientable and second countable.

We give a general method for constructing subgroups which either cover or avoid each chief factor of the finite soluble group G. A strongly pronorrnal subgroup V, a prefrattini subgroup W, an -normalizer D and intersections and products of V, W, and D axe all constructable. The constructable subgroups can be characterized by their cover-avoidance property and a permutability condition as in the results of J. D. Gillam [4] for prefrattini subgroups and -normalizers.

As observed by A. Bielecki and others ([1], [3]) the Banach contraction principle, when applied to the theory of differential equations, provides proofs of existence and uniqueness of solutions only in a local sense. S. C. Chu and J. B. Diaz ([2]) have found that the contraction principle can be applied to operator or functional equations and even partial differential equations if the metric of the underlying function space is suitably changed.

Fieldhouse observed that any finitely presented left R-module P is projective with respect to pure exact sequences, i.e.

can always be completed to a commutative diagram when the sequence is pure exact. A left R-module A is absolutely pure if it is a pure submodule of every module which contains it.

If X and Y are infinite-dimensional Banach spaces, then Y is the inductive limit of Banach spaces each isomorphic to X.