A duality theorem of Wolfe for non-linear differentiable programming is now extended to minimization of a non-differentiable, convex, objective function defined on a general locally convex topological linear space with a non-differentiable operatorial constraint, which is regularly subdifferentiable. The gradients are replaced by subgradients. This extended duality theorem is then applied to a programming problem where the objective function is the sum of a positively homogeneous, lower semi continuous, convex function and a subdifferentiable, convex function. We obtain another duality theorem which generalizes a result of Schechter.

The relative Schoenflies theorem says that a locally flat embedding e: Sn−1 → Rn for which e−1(Rk) = Sk−1 extends to homeomorphism of the pair (Rn, Rk) provided the local collars respect Rk. In this note it is shown that the proviso is essential, at least when k = 3.

An associative locally finite algebra which is also an integral domain over an algebraically closed field is isomorphic to the ground field.

Iséki's BCK-algebras form a quasivariety of groupoids and a finite BCK-algebra must satisfy the identity (En): xyn = xyn+1, for a suitable positive integer n. The class of BCK-algebras which satisfy (En) is a variety which has strongly equationally definable principal congruences, congruence-3-distributivity, and congruence-3-permutability. Thus, a finite BCK-algebra generates a 3-based variety of BCK-algebras. The variety of bounded commutative BCK-algebras which satisfy (En) is generated by n finite algebras, each of which is semiprimal.

In the present paper we give a criterion for a function of Wiener's class to belong to the class of generalized absolute continuity, in terms of Fourier-Young coefficients {Ck}. More precisely, we prove the following theorem.

THEOREM. Let Λ = (λn,k) be a normal almost periodic matrix of real numbers such that λn,k ≥ λn,k+1 for all n and k. Then for any function f of Wiener's class Vν (1 < ν < 2) to be of class of generalized absolute continuity Ap (1 < p < ∞) it is necessary and sufficient that {|Ck|2} is summable Λ to zero.

It is shown that a simple trick involving Legendre polynomials readily yields the irrationality of ea, , π2, and of the zeros of Bessel functions of integer order. Generalisation of this idea yields the irrationality of ζ(3).

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords bisected by 0, it is shown that A(K) ≤ 4. Also, if K has three such chords, A(K) ≤ 4.5. The results are generalised to any lattice in the plane.

Let k be a finite field of characteristic 2, and let G be the three dimensional special linear group over k. The principal indecomposable modules of G over k are constructed from tensor products of the irreducible modules, and formulae for their dimensions are given.

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords through 0 of one of the following types, it is shown that A(K) ≤ 4, while if K has three of one type, A(K) ≤ 4.5. The types of chords considered are chords which partition K into two regions of equal area, chords which lie midway between parallel supporting lines of K, and chords such that K is invariant under reflection in them. The results are generalised to any lattice in the plane.

The Cauchy problem x′ = f(t, x), x(0) = x0, is considered in a non-reflexive Banach space E, where f is weakly continuous. A local existence theorem is proved using the measure of weak noncompactness.

It is proved that certain rings satisfying generalized-commutator constraints of the form [xm, yn, yn, …, yn] = 0 must have nil commutator ideal.

Let M be a closed subset of a Banach space E such that the norms of both E and E* are Fréchet differentiable. It is shown that the distance function d(·, M) is Fréchet differentiable at a point x of E ∼ M if and only if the metric projection onto M exists and is continuous at X. If the norm of E is, moreover, uniformly Gateaux differentiable, then the metric projection is continuous at x provided the distance function is Gateaux differentiable with norm-one derivative. As a corollary, the set M is convex provided the distance function is differentiable at each point of E ∼ M. Examples are presented to show that some of our hypotheses are needed.