If E and F are ordered spaces, a map T: E → F is positive if

The map T is bi-positive if f ≥ 0 ⇔ Tf ≥ 0. It is a natural problem to seek information about the positive and bi-positive linear maps between pairs of ordered vector spaces. The present study is an attempt to elucidate one facet of this general problem (see, for example, (2), vol. II, § 16·7·1, p. 278).

1. It is trivial that

for all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows from

Theorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If

(ii) sn = ο(nK)as n tends to infinity

then sn = P(n), where P(x) is a polynomial of degree less than K.

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb{P}$r of dimension n and degree d ⩽ n(r − n) + 2 is Calabi–Yau, and give an example that shows this bound is also sharp.

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditions

for all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have

.

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.

Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.

A theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

A simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.

It has often been assumed in cosmology theory(1) that there exists an average density of matter in space which is everywhere greater than zero. Under this assumption the space-time M will be foliated by curves each of which represents the life history of a particle. In keeping with the postulates of general relativity theory we shall refer to these curves as geodesics. Letting X denote the space of particles one obtains a projection f: M → X which assigns to every P ∈ M the particle found at P. Conversely, given the projection f:M → X, one can recover the geodesics: they are precisely the fibres f−1(x), x∈X.

Let ∥x∥ denote distance to a nearest integer. Heilbronn's theorem in Diophantine approximation [8] may be stated as follows.

A simple formula is derived for transforming the slowly convergent series involving Bessel functions which arise in axially symmetric potential problems into more rapidly convergent series. Using it, sums of series previously found by elaborate special methods may be obtained by direct substitution. The necessary numerical coefficients are given.

1. Pólya's walk in the Euclidean plane is the motion of a particle, which takes successive independent random steps of unit length parallel to the coordinate axes, the probabilities of each of the four possible directions at each step being ¼. We imagine the particle starts at the origin, and that, in taking the ith step (i = 1,2,…) from (xi-1, yi-1) to (xi, yi), it leaves a straight track of unit length between these two points. Let Tn denote the track thus marked out in the first n steps; so that Tn is a connected chain of n straight segments each of unit length. Define the interior of Tn to be the set of points, which neither lie on Tn nor can be connected to the point at infinity without crossing Tn at least once; and let An denote the area of the interior of Tn. Associated with each walk ω, there is a sequence of random numbers {An(ω)}; what can be said about the behaviour of this sequence, especially for large n? We shall proveTheorem 1. For all ω,0 ≤ An(ω) ≤ 1/16 n2.

In this paper we consider complex projective surfaces V, defined by an equation of the form fn–1 (x, y, z) w + fn (x, y, z) = 0, where fi is homogeneous of degree i, and relate the geometry of the intersections of the piane projective curves fn–1 = 0 and fn = 0 with the singularities of V. The results we obtain clarify and generalize some of those presented by Bruce and Wall (3).

We study the prime ideal structure of the Iwasawa algebra $\Lambda_G$ of an almost simple compact $p$-adic Lie group $G$. When the Lie algebra of $G$ contains a copy of the two-dimensional non-abelian Lie algebra, we show that the prime ideal structure of $\Lambda_G$ is somewhat restricted. We also provide a potential example of a prime c-ideal of $\Lambda_G$ in the case when the Lie algebra of $G$ is ${\frak s}{\frak l}_2(\mathbb{Q}p)$.

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

In a paper which is to be published shortly formulae have been obtained for the virtual characters of the system of surfaces cut out on a primal of S4, having ordinary singularities, by adjoint primals of any order; an account of the principal results has been given elsewhere. As a first application of these formulae we propose in the present paper to investigate the canonical systems belonging to various threefolds and thence to construct the canonical models which correspond to them.

When the inner integrals in a pair of repeated integrals are absolutely convergent, inversion of the order of integration is a not too difficult problem and there are a number of general theorems which, under comparatively unrestricted appropriate conditions, can be applied to give a solution. When, on the other hand, the inner integrals are only conditionally convergent, the problem is of a very different nature and the solution, if it exists at all, is usually only to be obtained by special devices.

We continue below the note ‘Some cyclic and other inequalities’ (3).