The study of general multiplier theorems (Kuhn-Tucker Conditions) for constrained optimization problems has led to extensions of the notion of a differentiable arc. Abadie [1], Varaiya [10], Guignard [5], Zlobec [11] and Massam [12] investigated the so called cone of tangent vectors to a point in a set for optimization purposes.

A simple proof is given of a result of Hmelevskiï on the solutions of the equation [x, y] = [u, υ] over a free group for any specified u, υ. To illustrate, the equation is solved explicitly for (u, υ) = (a, b), (a2, b), ([a, b], c) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commutators in the automorphism group of this free group are determined.

It has been shown previously that the Lp(μ) spaces for 1 < p ≤ 2 satisfy a weak parallelogram law, and the same methods can be used to show that the Lp(μ) spaces for 2 ≤ p <∞ satisfy a related weak parallelogram law. This paper obtains several equivalent characterizations of Banach spaces which satisfy one of these two weak parallelogram laws. One such characterization involves the conditions on the moduli of convexity and smoothness analyzed by Lindenstrauss.

This paper determines the structure of finite rings whose two sided ideals are principal as left ideals, and as right ideals. Such rings will be called principal ideal rings. Although finite rings have been studied extensively [1], [5], [12], [14] and the tools necessary for describing finite principal ideal rings have been available for thirty years, these structure theorems (which are essentially given in a more general setting in [4]) seem to have been overlooked. In particular, let or be an endomorphism of a ring V.

I show that if X is the hyperstonian space of Lebesgue measure on [0,1], then there are open sets in X×X which are not measurable for the simple product outer measure. This answers a question of M. C. Godfrey and M. Sion.

Quasigroups and groupoids with one or other of the Reidemeister or Thomsen closure conditions, the relationship among them with emphasis on their relationship to associativity viz groups, Abelian groups, have been investigated in [2], [3], [4], [5], [6], [12], and others. In [10] R- and T-groupoids, (that is, groupoids possessing one of the first two closure conditions mentioned above) which are generalizations of groups and Abelian groups were investigated. In this paper, we show that groupoids with the given identities may be described in terms of R- and T-groupoids. These results and others are used to give another proof of theorems given in [1], [7], and [5] describing the variety of all groups and Abelian groups defined by single laws.

In this note, we discuss a representation of the class of polynomials with real coefficients having all zeros in a given disk of the complex plane C, in terms of convex combinations of certain extremal polynomials of this class. The result stated in the theorem is known [1] for polynomials having n real zeros in the interval [a.b.]. In the following z will be a complex number and D[(a + b)/2, (b-a)/2] the closed disk of the complex plane centered at the real point (a + b)/2 and having radius (b-a)/2.

This expository paper surveys work on invariant subspaces and related topics which has been done in the past few years. We recommend, naturally, that the reader consult [52] for work done prior to 1973 and [54] for a discussion of some of the consequences of Lomonosov’s Lemma; (Lomonosov’s paper has now also appeared in English ([36])).

Averaging interpolation generalizes polynomials of Lagrange interpolation and also the next-to-interpolatory polynomials. This notion has recently been introduced by Motzkin, Sharma and Straus [3]. Later in [5], Saxena and Sharma have considered the convergence problem of the averaging interpolators on Tchebycheff abscissas. It appears that averaging interpolators have convergence properties similar to those of Lagrange interpolation. It is therefore reasonable to look for an extension of these operators along the lines of Hermite-Fejér interpolation. This can be done in three ways: (i) by taking assigned averages of function-values and by taking the derivatives to be zero, (ii) by taking assigned function-values and by taking the averages of derivatives to be zero, or (iii) by taking averages of function-values and by taking the averages of derivatives to be zero. The object of this note is to take the first approach. The second approach has been the subject of study by M. Botto and A. Sharma [2].

This note confirms that the existence proof for absolutely free algebras originated by Dedekind in [2] and completely developed for instance in [4] can still be carried out in a topos containing an infinite object i.e. an object N for which N ≃ N+1 if the type of the algebras considered is finite, pointed and internally projective i.e. is a finite sequence of objects, (Ij)i≤j≤k for which the functors ( )Ij preserve epimorphisms and each of which has a global section.

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.

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.

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

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.

In [2] G. Baumslag presents a list of twenty-three unsolved problems on one-relator groups. We give counterexamples to two of them.

Problem 5 asks whether a maximal locally free subgroup of a one-relator group always has finite “rank” (G has “rank” k if each finitely generated subgroup of G is contained in a k-generator subgroup of G).

If Y is a subspace of a Banach space X, either Y contains an isomorphic copy of ℓ1 or each weak Cauchy sequence in X/Y has a subsequence that is the image under the quotient mapping of a weak Cauchy sequence in X. If X is weakly sequentially complete and Y is reflexive, X/Y is weakly sequentially complete. Related structural results are given.

The object of this note is to give simpler proofs of a splitting theorem of Gaschütz [1] and a related theorem for groups with operators by using cross-homomorphisms (1-cocycles) instead of 2-cohomology.

We recall that a cross-homomorphism or 1-cocycle from a group E to an abelian normal subgroup N of E is a map f from E to N such that f(ab)=(f(a))bf(b) for all a, b ∈ E where superscript denotes conjugation. Cocycles f and h are equivalent if for some n ∈ N have h(e)=nef(e)n-1 for all e ∈ E.

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

Matrix iterative methods of solving systems of linear algebraic equations for a class of matrices which includes strictly and irreducibly diagonally dominant matrices are considered and a convergence theorem proved.