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I = lim inf v n-a* log n The purpose of this paper is to combine the methods used in two earlier papers' in order to prove the following theorem. THEOREM. (1) 1 5 c(1 + 4e)/5, where c 1 -c, so that (1) is an improvement on (2) only if 0 is not too close to unity, in fact, if 0

In a recent paper on a divisor problem the author showed incidentally that there is a certain regularity in the distribution of the roots of the congruencefor variable k, where D is a fixed integer that is not a perfect square. In fact, to be more precise, it was shown that the ratios v/k, when arranged in the obvious way, are uniformly distributed in the sense of Weyl. In this paper we shall prove that a similar result is true when the special quadratic congruence above is replaced by the general polynomial congruencewhere f(u) is any irreducible primitive polynomial of degree greater than one. An entirely different procedure is adopted, since the method used in the former paper is only applicable to quadratic congruences.

IN (11) J. H. C. Whitehead introduced the theory of regular neighbourhoods, which has become a basic tool in combinatorial topology. We extend the theory in three ways. First we relativize the concept, and introduce the regular neighbourhood N of X mod Y in M, where X and Y are two compact polyhedra in the manifold M, satisfying a certain condition called link-collapsibility. We prove existence and uniqueness theorems. The idea is that N should be a neighbourhood of X— Y, but should avoid Y as much as possible. The notion is extremely useful in practice, and is illustrated by the following examples. We assume M to be closed for the examples. (i) If Y = 0 then N is a, regular neighbourhood of X. Therefore the relative theory is a generalization of the absolute theory. (ii) If X is a manifold with boundary Y, then the interior of X lies in the interior of N and the boundary of X lies in the boundary of N; in other words X is properly embedded in N. (iii) Let Ibe a cone and suppose that X n Y is contained in the base of the cone. Then N is a ball containing X— Y in its interior, In Y in its boundary, and Y — X in its exterior. The last example was used in ((12) Lemma 6), and was one of the examples which suggested the need for a relative theory. Other illustrations of the use are to be found in the proofs of Theorem 2, Corollary 8, and Lemmas 7, 8, and 9 below, in the proof of Theorem 3 of (4), and in forthcoming papers by us on isotopy. Secondly, Whitehead proved a uniqueness theorem that said that any two regular neighbourhoods were (piecewise linearly) homeomorphic. We strengthen this result by showing them to be isotopic, keeping a smaller regular neighbourhood fixed (Theorem 2). In fact they are ambient isotopic provided that they meet the boundary regularly (Theorem 3), which is always the case if M is unbounded. Thirdly, Whitehead wrote the theory in the combinatorial category, and we rewrite it in the polyhedral category. The difference is that the combinatorial category consists of simplicial complexes and piecewiselinear maps, whereas the polyhedral category consists of polyhedra and piecewise-linear maps. In this paper by a polyhedron we mean a topological

In this paper we shall set out the generalization, for n-dimensional space S n , of some recent results about complete quadrics and complete collineations in S 2 , S 3 and S 4 . For the results about complete conies in S 2 , originally introduced by Study [1], we refer the reader to papers by Severi ([2], [3]), van der Waerden [4], Semple [5]; for those about complete quadrics in S 3 , to Semple ([6], [7]); for the extension to S 4 to Alguneid [8]; for the general concept of complete collineations in S n , and for results in S 2 and S 3 , to Semple [9].

M. D. ATKINSONIn a series of papers [3, 4 and 5] on insoluble (transitive) permutation groupsof degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from asmall number of exceptions, such a group must be at least quadruply transitive.One of the results which he uses is that an insoluble 2q grou +1 p of degree p =which is not doubly primitive must be isomorphi (3, 2)c wit to PSh p =L 7. Thisresult is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extendthis proof to give the following result: a doubly transitive group of degree 2q + l,where q is prime, which is not doubly primitive, is either sharply doubly transitiveor a group of automorphisms of a bloc A = 1 ank desigd k = 3n wit. Ouh rnotation for the parameters of a block design, v, b, X, k, i r,s standard; see [9].In this paper we shall prove two results about doubly transitive but not doublyprimitive groups which resemble the two results mentioned above.

Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results isprovided m ≥ log n. Related questions and results are also discussed.

In this paper is proved a formula for the number of neighbourly d-polytopes with d + 3 vertices, when d is odd.

In this paper is proved a formula for the number of simplicial neighbourly d-polytopes with d + 3 vertices, when d is odd.

In this paper we show that, for the Apollonian or osculatory packing C0 of a curvilinear triangle T, the dimension d(C0, T) of the residual set is equal to the exponent of the packing e(Co, T) = S. Since we have [5, 6] exhibited constructible sequences λ(K) and μ(K) such that λ(K) < S < μ(K), and μ(K)–λ(K) → 0 as κ → 0, we have thus effectively determined d(C0, T). In practical terms it is thus now known that 1·300197 < d(C0, T) < 1·314534.

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ 1 …, λ r be any real numbers. Then there exists a real number α such that for all integers u > 0 and for q = 1,…,r , where C = C(r) > 0. Thus all the numbers α+ λ 1 , …, α+ λ r are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ 1 , …, λ r and μ 1 , …, μ r there exist α, β such that for all integers u > 0 and for q=1,…, r , where again C = C(r) > 0 . Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.

In a recent paper [1] I made the followingDefinition. The function f: ℤ+ ∪ {0} → ℤ is a pseudo-polynomial iffor all integers n ≥ 0, k ≥ 1.

Let K(x1, x2, , xn) be a graph without loops or multiple edges, the complement of which consists of n disjoint complete graphs of x1, x2, , xn vertices. In this paper a class of mappings of K(x1, , xn) onto the Euclidean plane is described. The minimum number of intersection points of edges for these mappings is determined. This number also involves an upper bound for the so-called crossing number cr(x1, , xn), being the minimum number of intersection points of edges for all mappings of K(x1, , xn) onto the Euclidean plane (see (28)). Equality in (28) is conjectured.

In this note I settle a question which arose out of my first paper under the above title ( cf . [1]), where I considered the classgroup C (Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C( ) of the maximal order of the rational groupring Q (Γ), and C ( ) is the product of the ideal classgroups of the algebraic number fields which occur as components of Q (Γ) and is thus in a sense known. One is then interested in the kernel D (Z(Γ)) of C (Z(Γ)) → C ( ) and in its order k (Γ). In [1] I proved that, for Γ a p -group, k (Γ) is a power of p . I also computed k (Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q ( n ) with n 4 = 1 or n 6 = 1. The numerical results obtained led me to the question whether in fact k (Γ) tends to infinity with the order of Γ.

In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible number of j-faces of a d-polytope with v vertices is achieved by a cyclic polytope C(v, d).

In a short and little known paper, Jacobi [1] gives conditions for the cubic residuacity of small primes q = 2, 3, ..., 37 to a prime p in terms of the quadratic partitionin the form L ≡ ±µM (mod q), and LM ≡ 0 (mod q).

If K is a convex body in n -dimensional space, let SK denote the closed n -dimensional sphere with centre at the origin and with volume equal to that of K . If H and K are two such convex bodies let C ( H , K ) denote the least convex cover of the union of H and K , and let V*(H, K) denote the maximum, taken over all points x for which the intersection is not empty, of the volume of the set . The object of this paper is to discuss some of the more interesting consequences of the following general theorem.

Denote by |E| the cardinal of a set E . The purpose of the present paper is to prove the following result, constituting the solution of an unpublished problem of Erdos, Hajnal and Milner.

Let be a topological property. We say that the locally finite sum theorem holds for the property if the following is true:“If {Fα : α ∈ Λ} be a locally finite closed covering of X such that each Fα possesses the property , then X possesses .” The above property is known as the locally finite sum theorem (referred to as LFST in the present note). The LFST has been of interest to many people for it holds for many interesting properties such as metrizability, paracompactness, normality, collectionwise normality, local compactness, stratifiability, the property of being a normal M-space etc. etc. In [10], a large number of properties for which the LFST holds have been noted. In the same paper, a general method for proving this has been obtained. It has been shown that if a property is such that it is preserved under finite-to-one, closed continuous maps and also preserved under disjoint topological sums, then the LFST holds for and the same has been used to establish the LFST for a large number of properties. In [9,10], several interesting consequences of the LFST have been obtained. In the present note, some more interesting consequences of it have been obtained in regular, normal, collectionwise normal and countably paracompact spaces. Also, the LFST has been established for some other topological properties

If a group G is presented in terms of generators and relations, then the classical Reidemeister-Schreier Theorem [1] gives a presentation for any subgroup of G . If G is a free product of groups G α each of which is presented in terms of generators and relations, then the main result of this paper is a presentation for any subgroup H of G , which shows the nature of H as a free product of certain subgroups of G . This result is a generalization of the celebrated Kuros Theorem [2]. It also includes the Reidemeister–Schreier Theorem and the Schreier Theorem [1] which states that any subgroup of a free group is free.

In a paper [1] published in 1956, David Gale introduced the idea of representing a convex polytope by a diagram (now called a Gale diagram). Later work by Gale, T. S. Motzkin, and more recently by M. A. Perles and B. Grunbaum, has shown the importance of this idea. Using it, a large number of results which were formerly inaccessible have been proved. For an account of Gale diagrams and their applications, the reader is referred to Grunbaum's recent book [2; §5.4 and §6.3] and we shall largely follow the notations used there.

In this paper X ( t ) denotes Brownian motion on the line 0 ≤ t E is a compact subset of (0, ∞) and F a compact subset of ( -∞, ∞). Then δ( E , F ) is the supremum of the numbers c such that In [4, 5] some lower and upper bounds for δ were found in terms of dim E and dim F, and it seemed possible that δ( E, F ) could be determined entirely by these constants; this much is false, as the examples in our last paragraph will demonstrate. Here we shall show that δ( E, F ) depends on a certain metric character η( E × F ). However, η is not calculated relative to the Euclidean metric: the set F must be compressed to compensate for the oscillations of most paths X ( t ). Fortunately, η( E × F ) can be calculated for a large enough class of sets E and F , by means of sequences of integers, to test any conjectures (and disprove most of them). In passing from η to δ we present a slight variation of Frostman's theory [3II].

1. The purpose of this paper is to give simple proofs for some recent versions of Linnik's large sieve, and some applications.The first theme of the large sieve is that an arbitrary set of Z integers in an interval of length N must be well distributed among most of the residue classes modulo p, for most small primes p, unless Z is small compared with N. Following improvements on Linnik's original result [1] by Renyi [2] and by Roth [3], Bombieri [4] recently proved the following inequality: Denote by Z(a, p) the number of integers in the set which are congruent to a modulo p.

A graph consists of a set of vertices some pairs of which are joined by a single edge. A tree is a graph with the property that each pair of vertices is connected by precisely one path, i.e. , a sequence of distinct vertices joined consecutively by edges. The complexity c of a graph G(n, k) with n vertices and k edges is the number of trees with n vertices which are subgraphs of G(n, k) . The distribution of c over the class of all graphs G(n, k) is of physical interest because it throws light on the classical many-body problem. (See, e.g. [9].) Ford and Uhlenbeck [3] gave numerical data which suggested that the distribution of c tends to normality for increasing n if k is near No moments higher than the first were known in general and they remarked in [4] that even “the second would be worth knowing”. The main object in this paper is to derive a formula for the second moment of c .

1. If w (mod 2ω 1 , 2ω 2 ) is an elliptic parameter for points of a normal elliptic curve C = 1 C n [ n − 1], then it is well known that the sets of n points in which C is met by primes have a constant parameter sum k (mod 2ω l , 2ω 2 ), and we may express this for convenience by saying that k is the prime parameter sum for the parametrisation of C by w . If we take the origin of w (the point for which w ≡ 0) to be one of the points of hyperosculation of C , then k ≡ 0, and we may say that w is a normal parameter for C . In the same way, if Γ is the Grassmannian image curve of the generators of a normal elliptic scroll 1 R 2 n [ n − 1], then a normal parametrisation of Γ defines a normal parameter w for the generators of 1 R 2 n , such that n of the generators have parameter sum zero if and only if they belong to a linear line-complex not containing all the generators of 1 R 2 n ; or, in particular, if they all meet a space [ n – 3] that is not met by every generator of the scroll. In this paper we are concerned in the first instance with the type of normal elliptic scroll 1 R 2 2 m +1 [2 m ] whose points can be represented by the unordered pairs ( u 1 ; u 2 ) of values of an elliptic parameter u (mod 2ω 1 , 2ω 2 ); and we establish a significant connection between any normal parametrisation of the generators of 1 R 2 m +1 and an associated parametric representation ( u 1 u 2 ) of its points. We also add a brief note to indicate the lines along which this kind of connection can be extended to apply to a general normal elliptic scrollar variety 1 R k mk +1 [ mk ] whose points can be represented by the unordered sets ( u 1 , …, u k ) of values of an elliptic parameter u .

In a recent paper I. J. Schoenberg [1] considered relationswhere the av are rational integers and the ζv are roots of unity. We may in (1) replace all negative coefficients av by −av replacing at the same time ζv by −ζv so that we may, if it is convenient, assume that all av are positive. If we do this and arrange the ζr so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all av are non-negative. We shall call a polygon (1) k-sided if all av are positive. The polygon is called degenerate if two of the ζv are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p-gons where p is a prime.

Throughout this paper, E will denote a finite-dimensional vector space over an ordered field . The real number field will be denoted by ℜ and its rational subfield by . Many of the basic notions in the theory of convexity (convex set, extreme point, hyperplane, etc.) can be defined in the general case just as they are when , but their behaviour may be different from that in the real case. By way of example, we consider the following theorem (due essentially to Minkowski), which is of fundamental importance both for geometric investigations and for the applications of convexity in analysis:(1) Suppose . If K is a convex subset of E which is linearly closed and linearly bounded, then. K = con ex K; that is, K is the convex hull of its set of extreme points.

Let Q(x1, …, xn) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequalityis soluble for every e > 0 in integers x1, …, xn, not all 0. The first progress towards proving this conjecture was made by Davenport in two recent papers; the result obtained involved, however, a condition on the type of the form as well as on n. We say that a non-singular Q is of type (r, n—r) if, when Q is expressed as a sum of squares of n real linear forms with positive and negative signs, there are r positive signs and n—r negative signs. It was proved that (1) is always soluble provided that

00 It x = S Qn x - If» m addition, the projections P n = £ Q,- are equicontinuous, then n = 1 «= 1 (£n)^°=1 is said to be an equi-Schauder decomposition of E. It is obvious that a Schauder basis is equivalent to a Schauder decomposition in which each subspace is one-dimensional, and that it is equi-Schauder if and only if the corresponding decomposition is equi-Schauder. For more information on Schauder decompositions see, for example [2 and 3]. In this paper, it will be shown that if E is locally convex and possesses an equiSchauder decomposition, the properties of sequential completeness, quasicompleteness or completeness of E may be related very simply to the properties of the decomposition; and that if £ possesses an equi-Schauder basis, these three types of completeness are equivalent. If (£„)*=! is a Schauder decomposition of E, the sequences ( polar topology on E. Suppose (En)™=1 is an equi-Schauder decomposition for (E, T) and let (xa)a eAbe a x-Cauchy net on E such that for each n (Qn xa)a e A converges. Then: (i) (lim Pn xa)"-i ' s a ?-Cauchy sequence. a

A one parameter solution of th equation was given by Ryley in 1825. Others were found by Richmond and myself. In the Landau memorial volume [1] recently published, there appears a joint paper, with the above title, of Davenport and Landau dating from 1935. They prove that if n is a positive integer, then positive rational solutions of (1) exist with denominators of order of magnitude O ( n 2 ). Their proof depended upon a two parameter solution of (1) due to Richmond and is very complicated.