In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

The first example of a non-trivial group satisfying the normalizer condition but having trivial centre was obtained by Heineken and Mohamed (1). In fact, the group they constructed satisfies the stronger condition that all its proper subgroups are subnormal and nilpotent. In this note we use a construction suggested by their approach to obtain such a group G as a subgroup of the restricted wreath product Cp ≀ Cp∞. The fact that G is given as a subgroup of a well known group makes it rather easier to investigate its properties, and in particular to see that its centre Z(G) is trivial.

Constructions are given of sets F(k), (k = 6, 7) of residues mod q(k), (for a suitably chosen integer q(k)) such that F(k) – F(k) contains all residues, while

has a gap of an assigned number of consecutive residues.

Recall that in (2) we showed that it was possible to make transverse a homology manifold and PL-manifold inside a large dimensional homology manifold, subject to being able to do some general position inside the large manifold. In (1) we were able to relax the condition that one of the submanifolds be a PL-manifold to it being a homotopy manifold. The way in which the making transverse was achieved was via a system of h-cobordisms from the original situation to the transverse one. The problem we tackle here is that of making a map between homology manifolds transverse regular. Thus we ask: given a map f: M → N of homology manifolds with P a proper submanifold of N, is it possible to homotop f to a map g: M → N such that g−1(P) is a proper submanifold of M and g induces a map from the normal bundle of g−1(P) in M to the normal bundle of P in N?

Zeeman(5) once bravely made the following conjecture:

Conjecture. Suppose K is a contractible 2-complex and I is the unit interval [0, 1]. Then K × I collapses.

This conjecture is interesting since it trivially implies the Poincaré conjecture. For if C is a homotopy 3-cell, then it has a contractible 2-dimensional spine, so that C × I would be a collapsible 4-manifold. But then C × I would be a 4-cell, with C = C × 0 in its 3-sphere boundary, and the PL Schoenfliess theorem shows that C is a 3-cell.

Let M be a C∞ manifold smoothly embedded as a convex hypersurface of Euclidean n-space n. M has constant width if the distance between each pair of parallel supporting hyperplanes in n is equal. This is equivalent to the condition that any chord of M normal to M at one end-point be normal to M at both end-points.

Let (E, ) be a topological vector space with a positive cone C. Jameson (3) says that C given an open decomposition on E if V ∩ C − V ∩ C is a -neighbourhood of 0 whenever V is a -neighbourhood of 0. The concept of open decompositions plays an important rôle in the theory of ordered topological vector spaces; see (3). It is clear that C is generating if C gives an open decomposition on E; the converse is true for Banach spaces with a closed cone, by Andô's theorem (cf. (1) or (9)). Therefore the following question arises naturally:

(Q 1) Let (E, ) be a locally convex space with a positive cone C. What condition on is necessary and sufficient for the cone C to give an open decomposition on E?

Let (E, τ) be a complete, semi-metrizable topological vector space. Let p be a pseudo-norm (not to be confused with a semi-norm, cf. (8), p. 18) inducing the topology τ. For each positive real number r, let

Let f be a continuous linear function from E into a topological vector space F. The open mapping theorem of Banach may be stated as follows: If f is nearly open, that is, if the closure of each f(Vr) is a neighbourhood of O in F then whenever β > α > O; in particular, each f(Vr) is a neighbourhood of O. We note that f, identifying with its graph, is a closed linear subspace of the product space E × F. In this paper, we shall employ techniques developed by Kelley (6) and Baker (1) to extend the theorem to the case where f is taken to be a closed cone in E × F. The generalized theorem throws some light onto the duality theory of ordered spaces. In particular, the theorem of Andô–Ellis is generalized to (not assumed, a priori to be complete) normed vector spaces.

This result does not appear to be in the literature, though the method of proof is related to those used in (1) and, particularly, (2) to prove A. H. Stone's theorem that metrics have paracompact topologies.

Let X be a complex normed space and let T be a Hermitian linear operator on X (i.e. the numerical range of T is real). It was proved by Bonsall (see (4); Theorem 10.13) that then

for every u ∈ X. This inequality was improved by the author (3) to

Inequality (1) is closely related (see (3)) to some inequalities of Hadamard (6) and Kolmogorov (7) about the successive derivatives of functions in L∞(−∞, ∞). It was also shown in (3) (and was, in fact, shown already in (7)) that the constant 2 is best possible in (1). However, as we shall see, inequality (1) can be sharpened considerably if T attains its norm on u, i.e. if ‖Tu‖ = ‖T‖‖u‖.

Suppose (en) is a basis of a Banach space E, and that (e′n) is the dual sequence in E′. Then if (e′n) is a basis of E′ in the norm topology (i.e. (en) is shrinking) it follows that E′ is norm separable: it is easy to give examples of spaces E for which this is not so. Therefore there are plenty of spaces which cannot have a shrinking basis. This leads one to consider whether it might not be reasonable to replace the norm topology on E′ by one which is always separable (provided E is separable). Of course, the weak*-topology σ(E′, E) is one possibility (Köthe (17), p. 259); then it is trivial that (e′n) is a weak*-basis of E′. However, if the weak*-topology is separable, then so is the Mackey topology τ(E′, E) on E′, and so we may ask whether (e′n) is a basis of (E′,τ(E′, E)).

In 1968, Amemiya and Kōmura ((1), p. 275) gave an example of a separable and incomplete Montel space. Their example depended upon constructing a subspace of ω, the countable product of real lines, which had several properties and in particular had codimension in ω. There is an error on page 276, line 5 of the proof of the Hilfssatz concerning this construction. The constructed subspace has codimension one. A more delicate construction is needed here; in fact, an example given by Webb ((7), p. 360) is sufficient to make this correction.

In many contributions to the theory of Hausdorff measures, it has been the practice to place certain restrictions on the measure functions used. These restrictions usually ensure both the monotonicity and the continuity of the functions. The purpose of this work is to find conditions under which the restrictions of monotonicity and continuity may be relaxed. We shall be working mostly with sets in Euclidean space, although in the final theorem, we work in Hilbert space to show that some of our previous results do not generalize to this space. I am grateful to the referee for suggesting improvements in the proofs of the results.

A convolution formula is established for Bell polynomials. This is expressed in seven equivalent ways and used to derive further properties of these polynomials. The application of these results to some twenty-seven special polynomial sets is shown and illustrated in the case of binomial, Hermite, Gegenbauer and generalized Bernoulli sets.

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined by

where V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured that

where ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

The method of integral equations is the most familiar method of proving existence theorems for the Helmholtz equation of acoustics. The wave potentials are expressed as surface distributions of wave sources (for the Neumann problem) or wave dipoles (for the Dirichlet problem). By a wave source is meant the free-space wave source. The source and dipole strengths for the exterior potentials are found to be solutions of Fredholm integral equations of the second kind which are, however, singular at a certain discrete set of frequencies corresponding to eigensolutions of the interior problems. The existence of exterior solutions at the expected frequencies can still be shown, but the proof involves a detailed and complicated study of the interior solutions. It is physically evident that this difficulty arises from the method of solution and not from the nature of the problem.

A unified variational approach to a class of second-order integral in-equalities is presented. A special case recently considered in a different manner by Anderson, Arthurs and Hall (1) is recovered.

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.

Consider a queueing system M/G/s with the arrival intensity λ, the service time distribution function B(t) (B(0) < 1) having a finite mean and the waiting room size N ≤ ∞. If s < ∞ and N = ∞, we shall also assume that its relative traffic intensity is less than 1. Since the arrival process of this system is Poisson, it is immediate that in this case the distribution of the number of arrivals during an interval is infinitely divisible.

It is shown that the solutions of the field equations generated by the most general effective quadratic Lagrangian aR2 + bRklRkl which are static, singularity-free, of class C4 and asymptotically flat (of rank A3) are (i) conformally flat when 3a + b = 0, (ii) flat when 3a + b ≠ 0 (b ≠ 0), and (iii) have zero scalar curvature when b = 0.