Let xik (i, k=1, 2, …, n) be n2 independent variables, aik be n2 constants, and .

Let x=(xik) be the determinant formed from the n2 elements xik, and the determinantal operator formed from the elements .

The important relation in the Theory of Bending between the curves of Bending Moment (B.M.), Shearing Force (S.F.), and Load, or between those of Deflection, Slope, and Bending Moment, viz., that the tangents to the first of either set intersect in a vertical line through the centroid of the corresponding area of the last, under the usual convention of drawing, is usually not proved in Engineering Treatises, or else is established in simple cases by the polygon of loads.

This bibliographical note was drawn up to accompany Mr Collignon's memoir Recherches sur l'Enveloppe des Pédales des divers points d'une Circonférence par rapport à un triangle inscrit, printed in this volume, p. 2–34; and if I had remembered (as I ought to have done) the very full bibliography given in L'Intermédiaire des Mathématiciens (Vol. 3, p. 166–168, 1896) by Mr Brocard and others, I should not have commenced it. The result, however, has been that several articles on this particular curve, not noted in the Intermédiaire, have been discovered, and I have thought it worth while to print the information thus gained.

In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

In this paper, radial basis functions that are compactly supported and give rise to positive definite interpolation matrices for scattered data are discussed. They are related to the well-known thin plate spline radial functions which are highly useful in applications for gridfree approximation methods. Also, encouraging approximation results for the compactly supported radial functions are shown.

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.

In their recently published book Bonsall and Duncan ask the following question ((1), p. 65): if A is a Banach star algebra with an identity, is it true that

The set G[x] of polynomials over a group (G, + )—as well as the polynomial functions P(G) on (G, +) form near-rings with respect to addition and composition (substitution). See [1] for polynomials and [2] for near-rings. A number of results on G[x] can be deduced from [2].

Due to [1], the polynomials in G[x] can uniquely be represented in the following “normal form”:

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if

(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

Let the Nodal cubic have the equation

or,

Three points t1, t2, t3 form an apolar triad to the curve when they satisfy an equation of the form

Kirchhoff's kinematic hypothesis that leads to an approximate two-dimensional theory of bending of elastic plates consists in assuming that the displacements have the form [1]

In general, the Dirichlet and Neumann problems for the equilibrium equations obtained on the basis of (1.1) cannot be solved by the boundary integral equation method both inside and outside a bounded domain because the corresponding matrix of fundamental solutions does not vanish at infinity [2]. However, as we show in this paper, the method is still applicable if the asymptotic behaviour of the solution is suitably restricted.

The Abel and Cesàro summabilities of two alternating gap series are investigated. We prove that the series is summable at x = 1 (in both senses), but that is not. In 1907, Hardy obtained essentially the same result for the latter series; our proof is shorter and more elementary: we use the Poisson summation formula to derive an explicit estimate for the size of the oscillations as x → 1_. This represents an example of a general method for determining the Abel summability of similar series.

In this note we consider the Dirichlet problem Δu + f(x, u)=0 in Ω, u = 0 on ∂Ω here Ω is a bounded domain in ℝn(n≧3), with smooth boundary ∂Ω. We prove the existence of strong solutions to the previous problem, which are positive if f satisfies a suitable condition. As a consequence we find that the problem with , may have positive solutions even if g is not a lower-order perturbation of Next We examine the case .

A class of finite semigroups is called a genus if it is closed under homomorphic images, subsemigroups and finite direct products. During a talk at the Symposium on Semigroups held at the University of St Andrews, in 1976, M. P. Schützenberger posed the problem of characterising the smallest genus which contains finite groups and finite semigroups, all of whose subgroups are trivial.

Binet shewed that the function

can be expanded as an inverse factorial series. This note furnishes a new and much simpler proof of his result, based on a formula which is an analogue of the Binomial Theorem for factorials.

This formula is that, if we denote by [x]n the ratio

then

where denotes the coefficient of xr in the expansion of (1 + x)m.

In this paper we consider the questions of existence and uniqueness of solutions to a singular, nonlinear boundary value problem arising from a model problem in isothermal autocatalytical chemical kinetics. The boundary value problem occurs in the construction of a small time asymptotic solution to an initial-boundary value problem (King and Needham [14]), and existence and uniqueness for the boundary value problem are required for consistency of this formal asymptotic solution.