For fixed integers k and m, with k ≥m ≥ 2, there are only finitely many runs of m consecutive integers with no prime factor exceeding k. We obatin lower bounds for the last such run. Let g(k, m) be its smallest member. For 2 ≤ m ≤ 5 it is shown that g(k, m) > kc logloglogk holds for all sufficiently large k, where c is a constant depending only on m. We also obtain a number of lower bounds with explict ranges of validity. A typical result of this type g(k, 3) > k3 holds just if k ≥ 41.

The main results of the paper give necessary and sufficient conditions as well as sufficient conditions that continuous bijections of a manifold onto itself be homeomorphisms. Such conditions include the embedding of manifolds, preserving ends, preserving closed half-rays and restrictions on boundary components. A number of counterexamples are given to likely conjectures.

Correcting some “proofs” given in an earlier paper of the same title, we prove here, among other things, that, if S is a subgroup of a topological group that is complete in a left invariant metric or locally compact, then every weakly almost periodic function on S is (left and right) uniformly continuous. We also prove a theorem related to results of R. B. Burckel and of W. W. Comfort and K. A. Ross: a topological group is pseudocompact if and only if WAP(G) = C(G).

Many transformation semigroups S over a set X have the inner-automorphism property: their automorphisms are precisely those mappings of the form g-1 · g: α → g-1αg(α∈S) where g is a fixed permutation of S. For example, the full transformation semigroup, any two sided ideal of this semigroup, and, with some exceptions for small cardinals, the alternating group and the symmetric group itself. See Malcev (1952) and Scott (1965, Chapter 11). In this paper we determine all transformation semigroups over finite X with this property. In fact, we do more: we characterize all transformation semigroups invariant under the mappings g-1 · g. We refer to these as -normal transformation semigroups.

In earlier work Borrego, Cohen and DeVun (1971a, b) topological semigroups which were uniquely representable in terms of two subsets were investigated. In this note we extend the definition and prove similar results for a larger class of semigroups.

By employing a precise form of the Hermite interpolation formula we obtain a best possible bound for the number of zeros of p-adic exponential polynomials. As companion to this quantitative result we give a best possible bound on the coefficients, if the exponential polynomial is small at sufficiently many points.

A class (called c-radicals) is defined such that, given a c-radical P, there is in any class M′ a certain internal criterion that its upper radical UM′ = P, For P a non-c-radical (called a q-radical) there exists no smallest class M such that UM = P, and P is a q-radical if and only if for some M with P = UM there exists 0 ≠ R ∈ M such that when an image ¯ of R has a non-zero image in M there exists an infinite chain of epimorphisms ¯ → R1 → R2 → … with all R1 ∈ M and no Ri, the image of any Rj, with j > i. Several examples of such rings are constructed including a ring all of whose images are primitive. Thus all radicals contained in the Jacobson radical are q-radicals.

We are concerned with the problem of the existence and uniqueness of regularly varying (in Karamata's sense) solutions ϕ of the linear functional equation in a right neighbourhood of x = 0. Under suitable conditions on the given functions f and h, the uniqueness of solutions depends essentially on whether the series Σh ∘ f1 converges or diverges; here fi denotes the i-th functional iterate of f. The existence of solutions may be proved under further assumptions.

The case of the more general linear functional equation may be reduced to that of equation (*).

In this paper we use a result from graph theory on the characterization of the line graphs of the complete bigraphs to show that if n is any integer ≥ 2 then any finite linear space having p = n2 − n or p = n2 − n + 1 points, of which at least n2 − n have degree n + 1, and q ≤ n2 + n − 1 lines is embeddable in an FPP of order n unless n = 4. If n = 4 there is only one possible exception for each of the two values of p, and for p = n2 − n, this exception can be embedded in the FPP of order 5.

The aim of this paper is to enumerate all equivalence classes of perfect septenary forms with Δ4 = 2. This is an important section of the complete enumeration of perfect septenary forms by the method which was outlined in Stacey (1975). There are nine equivalence classes of these forms, four of which were announced for the first time in Stacey (1975). This work forms part of the author's D. Phil, thesis and was done at Oxford under the supervision of Dr. B. J. Birch.

Generalized Ramsey theory for graphs was formulated and developed in the previous papers in this series. We extend the area here by introducing generalized Ramsey numbers for higher dimensional simplicial complexes. In particular we calculate explicitly the Ramsey numbers for several small “pure 2-complexes”, or more briefly plexes, in which each edge is contained in some 2–call.

The theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

In this paper we consider the (non)oscillation properties of two general nonhomogeneous nonlinear delay differential equations of order 2n using as background and motivation the techniques previously applied to the associated homogeneous delay differential equations H+ and H−. The equations N+ and N− are each reduced to homogeneous form by the introduction of transformations u(t) = y(t) – R(t) and v(t) = R(t) — y(t), where R(t) is a solution of the associated nonhomogeneous differential equation (N). We first extend certain results for the equation H+ and then develop a classification of the positive solutions of equation H−. Using this classification and the one developed by Terry (1974) for H+ we develop a natural classification of the positive solutions of N+ and N− according to the sign properties of the derivatives of u(t) and v(t). For each choice of R(t), it is seen that there are 2n + 1 types of positive solutions of N+or N–. An intermediate Riccati transformation is employed to obtain integral criteriafor the nonexistence of some of these solutions. Analysis of the Taylor remainder results in sufficient conditions for the nonexistence of other such solutions.

The Hamiltonian system Y′ = BY + CZ, Z′ = – AY – B*Z is considered where the coefficients are continuous on I = [a, ∞, C = C* ≧ 0, and A = A* ≦ 0. A solution (Y, Z) satisfying Y*Z = Z*Y is defined to be principal (coprincipal) provided that (i) Y−1 exists on I (Z−1 exists on I) and (ii) as t→∞ ( as t → ∞). Three conditions are given which are separtely equivalent to the condition that a solution is principal iff it is coprincipal. For a self-adjoint scalar operator L of order 2n, this problem is related to the deficiency index problem and to a problem of Anderson and Lazer (1970) which concerns the number of lnearly independent solutions of L (y) =0 satisfying y(k) ∈ (a, ∞) (k = 0, …, n).

Sum-free sets may be used to colour the edges of a complete graph in such a way as to avoid monochromatic triangles. We discuss the automorphism groups of such graphs. Embedding of colourings is considered. Finally we illustrate a way of constructing colourings using block designs.

The objective of the paper Aczél and Ostrowski (1973) is to show in an elementary way that any real-valued function f, defined on (0,1) and satisfying he inquality , where n ≧ 3 is fixed, is necessary of the form .

A fundamental region for the reduction of positive quaternary quadratic forms is exhibited. It is a convex polyhedral cone with twelve edges in the 10-dimensional space of quaternary quadratic forms.

Let G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

We prove here that the multiplicator of certain simply described metabelian groups is not finitely generated

A general metric is considered. Some solutions correspoding to the field equations Rik λgik = −8πEik are obtained as particular cases. Details of these solutions are also discussed. An Einstein space corresponding to the field equations Rik = λ gik is also constructed as a particular case of the general metric.