We first prove a basic theorem that if a set of polynomials satisfies an orthogonality relation with respect to integration, the set also satisfies an orthogonality relation with respect to summation. This result is then used to derive the Gaussian quadrature formula. The orthogonality relations give rise to interpolation formulas and a connection between the coefficients in these interpolation formulas is established. Finally, the analysis is used to get an estimate of the error in the Gaussian quadrature formula. Some error coefficients are evaluated in the cases where the orthogonal polynomials are those of Jacobi, Laguerre, Hermite and Bessel.

Our main purpose in this paper is to obtain more precise information about two problems which we investigated in Hartley (1971a). They are as follows: Problem 1. Let G be a countable locally finite group and π be a set of primes. Suppose that G = HK, H◃G, H∩K = 1, where H is a normal π′-subgroup of G, K is a π-group and Ck(H) = 1. If we assume that the Sylow (that is, maximal) π-subgroups of G are conjugate, what can we say about the structure of K?

This paper concerns the completions of partially ordered groups introduced by Fuchs (1965a) and the author (to appear); the p.o. groups under consideration are, generally, abelian tight Riesz groups, and so, throughout, the word “group” will refer to an abelian group.

In section 3 we meet the cornerstone of the work, the central product theorem, by means of which we can interpret the Cauchy completion of a tight Riesz group in terms of the completion of any of its o-ideals; one particularly important case arises when the group has a minimal o-ideal. Such a minimal o-ideal is o-simple, and in section 6 the completion of an isolated o-simple tight Riesz group is shown to be a tight Riesz real vector space.

Let R, C be the additive groups of the real, complex numbers respectively. Using the Axiom of Choice (A.C.), these groups may be shown to be isomorphic. We show that this cannot be proved in Zermelo-Fraenkel set theory (see e.g. Fraenkel, Bar-Hillel and Levy (1973)) without the additional assumption of A.C. This is one of the most “concrete” used of the Axiom of Choice of which I know. THEOREM 1 (assuming (A.C)). C ≅ R.

It is well-known that for any set X, px, the semigroup of all partial transformations on X, can be embedded in Jx∪a for some a ∉ X (see for example Clifford and Preston (1967) and Ljapin (1963)). Recently Magill (1967) has considered a special case of what we call ‘generalised partial transformation semigroups’. We show here that any such semigroup can always be embedded in a full transformation smigroup in which the operation is not in general equal to the usual composition of mappinas. We then examine conditions under which such a semigroup, (J x, θ), is isomorphic to the semigroup, under composition, of all transformations on the same set X.

A further development of the concept of contractors for non-linear functionals is presented by introducing two-point contractors. On this basis two-point secant methods are proposed for finding roots of non-linear functionals in Banach spaces.

Abstract A wake similarity solution for symmetric uniform shear flows merging at the trailing edge of a flat plate has associated with it an eigenfunction problem which was overlooked by Hakkinen and O'Neil (1967). An asymptotic formula for large eigenvalues is obtained and compared with another such formula related to both the Goldstein (1930) inner wake solution and Tillett's (1968) similarity solution for a jet emerging from a two-dimensional channel.

In this note we give an example of a homotopy commutative H-space X that is not dominated by any homotopy associative, homotopy commutative H-space. In particular, X is not dominated by Ω2S2X.

Products of positive measures play a very important role in analysis. The purpose of this paper is to construct a theory of products of two measures taking values in two (possibly different) Banach spaces. A Fubini theorem is obtained which generalizes the Fubinitheorem for the Bochner integral (Dunford and Schwartz (1958), Theorem 9, page 190), and hence also the classical result.

We use the theory of vector integration presented in Dinculeanu (1967). Our arguments rely upon a standard sort of application of the dominated convergence theorem (cf. Dunford and Schwartz (1958), Theorem 9, page 190), and therefore do not appear to generalize to any theory of integration where this theorem is lacking (e.g. Bartle (1956)).

Let R be an integral domain. It is well known (see Lambek (1971), Stenström (1971)), that idempotent filters of right ideals, torsion radicals and trosio theories are in one-to-one correspondence, but that different idempotent filters F of right ideals may lead to the same rings of quotiens Rf. We have always R ⊃ Rf ⊂Qmax(R). Given this situation one can ask a number of questions. For example: Describe all different idempotent filters for a given ring. Determine all different rings of quotients. When do different filters lead to the same ring of quotients? When are all rings between R and Qmax(R) of the form RF. When is every RF of the form RS−1, where S is an Ore system?

Consider a set x together with a σ-algebra B of subsets of x. Let G be a family of B-measurable transformations on x, let p(X) be the convex set of all prbability measures on B and let I be the convex set of all G-invariant probablity measures in p(X). For μµ p(X) we define Bµ = {A ∈ B: µ(g A Δ A) = 0 for all g ∈ G} and we define B0 = {A ∈B: gA = A for all g ∈ G}. Then B0 ⊆ Bµ and both are σ-subalgebras of B. G is said to act transitively on X if for x ∈ X, y ∈ X, gx = y for some g ∈ G.

In this paper the problem of constructing an arc of minimum length joining two fixed points: P1, P2, in an arbitrary subset: S, of a metric space is considered. The approach taken is a departure from the classical methods of the calculus of variations in that it is topological character, making use of the properties of sets rather than differential calculus.

In recent years, there has been considerable interest in the refining of thin plate theories. In this paper, the method of matching asymptotic expansions is used to obtain one such refinement which is believed to be an improvement on several previous results. Previous authors(Habip (1967), Widera (1969)) attempted such refinemnts within the framework of a partially nonlinear theory of elasticity whereas in the present work all terms neglected by these authors have been retained.

Let and .

In 1951 Mohanty proved the following theorem: .

In this paper a general theorem on summability |R,l (w), 1 | of Σ An(x) has been given which improves upon Mohanty's result in different ways (see Corollaries 1, 2 and 3) and it is also shown that some of the results of this note are the best possible.

A mapping κ: P(X) → P(X) is a quasi-closure operator (see Thron (1966) page 44) if (i) □κ = □, and for all A, B ∈ P(X) we have (ii) A ⊆ Aκ, and (iii) (A ⋓ B)κ = Aκ ∪ Bκ one easily deduces that such operators have the further property: (iv) if A ⊆ B ⊆ X, then Aκ if κ also satisfies: (v) Aκ2 ⊆ Aκ for all A ⊆ X, then κ is called a Kuratowski closure operator.

Hashimoto (1952; Theorems 8.3 and 8.5) proved the following theorems: Theorem A. If L, is a distributive lattice, then there exists a generalized Boolean alegbra Lr and an isomorphism from the lattice of all congruence relations of L onto the lattice of all congruence relations of Lr.

The purpose of this paper is to present three somewhat disparate results on free objects in three different classes of λ-groups. The first is that no proper ideal of a finitely generated free vector lattice can itself be a free vector lattice. Second, each free abelian lgroup is characteristically simple. The third result is that each disjoint subset of a free (non-abelian) lgroup is countable.

Výborný and I (1972) proved maximum principles for a quasilinear elliptic operator where the boundary satisfied a smoothness condition weaker than the interior sphere property. In this paper I extend these to parabolic operators of a similar form and through a simple device to weakly coupled systems of such operators. Finally, I extend all of these results to an operator similar to the “parabolic” case of an operator introduced by Redheffer (1971). His conditions on the coefficients are replaced by conditions analogous to those Dow and Výborný (1972).

Let be regular and unvivalent in the unit disk D and map D onto a region R. A point w0∈ R is called a star-center point of f(z), or of R, if tf(z)+(1−t)w0∈ R for z∈D and 0≦ t≦ 1. In this paper we consider only functions of the form (1) where w = 0 is a star-center point, i.e., those functions that are starlike with respect to the origin.

If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.