If a Banach space X admits a continuously Fréchet differentiable function with bounded nonempty support, then X* admits a projectional resolution of identity and a continuous linear one-to-one map into co(Γ).

The natural, first order version of Peano's axioms (the theory T with 0, the successor function and an induction schema) is shown to possess the following nonstandard model: the natural numbers together with a collection of ‘infinite’ elements isomorphic to the integers. In fact, a complete list of the models of this theory is obtained by showing that T is equivalent to the apparently weaker theory with the induction axiom replaced by axioms stating that there are no finite cycles under the successor function and that 0 is the only non-successor.

This note exhibits a family of m – (ν, k, λ) supplementary difference sets with parameters ν = 2mf + 1, where ν = p is a prime and f is odd, k = mf and λ = m(mf−1)/2. These sets are composed of a union of cosets of 2m-th power residues of p.

The modified Hankel transform arises naturally in connection with certain semigroup operations on measures in probability theory. We give a tauberian theorem for this transform when certain higher moments exist. The probabilistic significance of our result is that it translates a regularity condition on the transform into a direct condition on the measure. This complements earlier results by Pitman and Bingham for the trigonometric and the modified Hankel transform respectively.

A solution is given for un + 1 in terms of u1 and u0, where the elements of the sequence {un} satisfy the linear difference equation

Two linearly independent solutions of the equation are written as determinants and relations are given which can be used to check the evaluation of these determinants.

A model is proposed for the population growth of a rare species after predation pressure is lifted. A geometric probability argument is used to suggest that, as population density increases, breeding encounters become more frequent and the consequent rate of increase is much steeper than the exponential. This may explain the population explosion of crown-of-thorns starfish recorded recently.

Fibonacci groups are the groups

where r is a natural number. The groups F(2, 8) and F(2, 10) are shown to he infinite, thus leaving F(2, 9) as the only Fibonacci group whose finiteness or infiniteness has not been determined.

Let α1, …, αm be distinct complex numbers and τ(1), …, τ(m) be non-negative integers. We obtain conditions under which the functions

form a perfect system, that is, for every set ρ(1), …, ρ(m) of non-negative integers, there are polynomials a1 (z), …, am (z) with respective degrees exactly ρ(1)−1, …, ρ(m)−1, such that the function

has a zero of order at least ρ(1) + … + ρ(m)−1 at the origin. Moreover, subject to the evaluation of certain determinants, we give explicit formulae for the approximating polynomials a1 (z), …, am (z).

We present the solution of a long-standing problem, namely, the determination of a set of polynomials in two independent variables which are biorthogonal over a triangular region to a set of polynomials previously introduced by Appell. Some elementary properties of our polynomials are investigated.

This paper gives convergence properties and applications of the discrete analogs of reproducing kernels for various families of harmonic functions. From these results information is obtained on the solution of interpolation problems, the convergence of the discrete Neumann's function, and the solution to problems involving the discrete biharmonic operator.

Let G be a group of collineations in a protective plane Π of order n. Suppose that one of the point orbits of G is an oval of Π, and that G acts regularly on this orbit. We prove that G fixes a non-incident point-line pair if either n is even, or n is odd and G is abelian, or n ≠ 11, 23, 59 is odd and is a pseudo-conic. It is then easy to deduce information about the lengths of the other orbits of G, and about the structure of G as an abstract group.

Two fixed point theorems for a subset C of a normed vector space X are established by using the concept of centre. These results differ from previous fixed point theorems in that X is assumed to have a topology T as well as a norm. The norm is required to be lower semi-continuous with respect to T and C is required to be convex, bounded with respect to the norm and compact with respect to T.

It is shown that if a variety V of (universal) algebras, defined by a set Σ of identities, is closed under the construction of power algebras then V can be defined by the subset Σ′ of Σ consisting of those identities ν = w of Σ such that every variable in ν = w occurs exactly once on both sides.

On a Banach space the numerical range of an unbounded operator has a certain density property in the scalar field. Consequently all hermitian and dissipative operators are bounded. For a smooth or separable or reflexive Banach space the numerical range of an unbounded operator is dense in the scalar field.

Kaplansky's observation, namely, a commutative ring R is (von Neumann) regular if and only if each simple R-module is injective, is generalized to projective modules over a commutative ring.

It is shown how torsionfree modules can be used to characterize certain important classes of orders over Dedekind rings. In particular, we show that an order is Gorenstein if and only if each of its lattices can be embedded as a pure sublattice of a free lattice. We also show that an order is hereditary if and only if the tensor product of any of its right lattices with any of its left lattices is torsionfree over the ground domain.

It is shown that any selfmapping (X, f) can be equivariantly and naturally embedded in the selfmapping of the form (X1, f1) ∪ (x2, f2) × (Y, g) where f1 and f2 are contractive relative to suitably chosen metrics and g is a bijection with all points periodic.

A Fritz John type sufficient optimality theorem is proved for nonlinear programming problems in finite dimensional complex space over polyhedral cones, which may include equality as well as inequality constraints.

In this note we present sufficient conditions for the continuity of the total kth variation of a function defined on a closed interval [a, b]. We also give an integral representation for total feth variation, thus obtaining an extension of the classical result

Multivariate versions of Mercer's Theorem and the usual expansions of the resolvent and Fredholm determinant are shown to hold for an n × n symmetric kernel N(x, y) with arbitrary domain in Rp under weakened continuity conditions. Further, the resolvent and determinant of N(x, y) − a(x)b(y) are given in terms of those of N(x, y).