In this paper we obtain matroid extensions of two important results in graph theory, namely the 4-colour theorem of Appel and Haken [1] and the 8-flow theorem of Jaeger [4]. As a corollary we prove that any bridgeless graph with no subgraph contractible to K3,3 has a nowhere zero 4-flow. These results depend heavily on a remarkable theory of splitters developed recently by Seymour [8], [9].

The celebrated theorem of Bombieri and A. I. Vinogradov states that

for any ε > 0 and A ε 0, the implied constant in the symbol «ε depending at most on ε and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applications of sieve methods. Having this in mind distinct generalizations have been investigated (see for example [15] and [2]). Y. Motohashi established a general theorem which, roughly speaking, says that if (1) holds for two arithmetic functions then it also holds for their Dirichlet convolution; for precise assumptions and statement see [11]. So far, all methods depend on the large sieve inequality (see [10])

which sets the limit x1/2 for the modulus q in (1) and in its generalizations.

It has been conjectured that, if p ≡ 1 (mod 4) is prime, and if d < 0 is a square-free discriminant with then

Where belongs to the field is the fundamental unit of Q(√k), depending on whether there are an even number or an odd number of classes per genus in Q(√d), and Ω is the genus field of Q(√d). Here the summation being over a complete set of inequivalent forms in the genus G, and

In this paper it will be shown that this conjecture is true when d is the product of two odd discriminants. An example when d is the product of three prime discriminants is discussed.

Let U0 = [(0, 1); and U1 = (0,1)]. Suppose we have a distribution of N points in , where, for k ≥ 1, is the unit cube consisting of the points y = (y1, … , yk+1) with 0 ≤ yi < 1 (i = 1 , … , k + 1). For X = (x1 ,…, xk + 1) in , let B(x) denote the box consisting of all y such that 0 ≤ yi < xi (i = 1 ,…, k + 1), and let denote the number of points of which lie in B(x).

One of the problems of solid state physics is to explain why the atoms of certain elements (such as iron) arrange themselves in a body-centred cubic lattice, rather than in the much denser face-centred cubic lattice packing [8]. Recently, we discovered the geometric significance of these body-centred lattice packings: they are fragile in the sense that, assuming we have a sphere of fixed radius at each lattice point, any perturbation which does not alter the set of nearest neighbours of any sphere results in a denser configuration [3]. In other words, in-these packings it is the density of the interstitial void between the spheres that is being maximized. This fact might seem only to deepen the mystery of why such arrangements ever appear in nature at all. Our purpose here is to demonstrate that these lattice packings are in fact quite “stable”, provided one seeks a more subtle form of stability, one prompted by physical rather than purely geometrical considerations. More precisely, we will show for those lattices which possess a high degree of symmetry–and for the bodycentred cubic lattice in particular–that the EPSTEIN ZETA FUNCTION of the lattice, ∑ |r|-2s (where |r| is the distance of the r-th lattice point from the origin and s is any fixed number greater than n/2), is locally minimized, in the sense that, any lattice, obtained by a perturbation which does not alter the set of nearest neighbours of any lattice point, has an Epstein zeta function of larger value, for any s > n/2.

A positive integer n is called a square-full integer, if p2 divides n, whenever p is a prime divisor of n. It is clear that each square-full integer can be written in the form a2b3, where a and b are positive integers; moreover, this representation is unique if we stipulate that b is square-free.

In an earlier paper [11] we have discussed the diophantine equation

and the more general equation

when l is an odd prime > 3, c a natural number (or more generally a rational number) satisfying several conditions, n also a natural number and x, y, z, non-zero rational integers. Other investigators have obtained quite a number of results concerning particularly the case n = 1. Obviously, without an essential restriction, we may assume that the natural number c contains no divisor a1n (since such a divisor can be absorbed in the power of z) and furthermore that x, y, z in (1) are relatively prime in pairs.

Let

be a real quadratic form in n variables x1,…, xn and let ‖θ‖ denote the distance from θ to the nearest integer. Danicic [3] proved that if N > 1 and ξ > 0 then there exist integers x1,…, xn, not all zero, satisfying

where C depends on n and ξ only.

The present investigation was suggested by a theorem of A. Schinzel, H. P. Schlickewei and W. M. Schmidt [6]: let Q(x) = Q(x1,…, xs) be a quadratic form with integer coefficients. Then for each natural number m there are integers x1,…, xs satisfying

and having

The result of my paper with the title given above (Mathematika, 26 (1979), 72–75) is not new; it was proved by Delone, see [2] and [3]. I have not been able to refer to either of these papers, but the result is given in [1], A similar problem was considered by Kitaoka in [5]. Professor Kneser of Göttingen tells me that he solved the problem about ten years ago in an unpublished manuscript; also that Schering's result [4] is less general and weaker than mine. I am obliged to Professor Peters of Münster for a copy of [1], giving the references [2] to [4], and also for the reference [5].

There are at least two indices used to measure the size of bounded sets of ℝn of zero measure—Hausdorff dimension (see [4] for a definition), and the density index [7].

Consider a uniform body whose behaviour is described by the linear field equations of quasi-static, coupled thermoelasticity†:

This paper concerns conditions under which the classifying space functor transforms a fibre square of topological monoids into a fibre square of spaces. Before stating our results precisely, we first develop some relevant points concerning the continuity of functors.

Let P be a (convex) d-polytope in the Euclidean space Ed and p a point of Ed not contained in P or in a supporting hyperplane of a facet of P (we use the terminology of Grunbaum [2]). The part of the boundary of P which is “visible” from p, i.e. the union of those facets whose supporting hyperplanes separate p and P, form a (d – l)-ball, whose boundary is a (d – 2)-sphere S (all balls, spheres and manifolds to be considered are piecewise-linear). The boundary-complex ℬ(P) induces a subdivision of S, which we call a (sharp) shadow-boundary ofP. is combinatorially isomorphic to the boundary complex of a poly tope which is a central projection of P from the point p on a hyperplane.

The self-complement index s(G) of a graph G is the maximum order of an induced subgraph of G whose complement is also induced in G. This new graphical invariant provides a measure of how close a given graph is to being selfcomplementary. We establish the existence of graphs G of order p having s(G) = n for all positive integers n < p. We determine s(G) for several families of graphs and find in particular that when G is a tree, s(G) = 4 unless G is a star for which s(G) = 2.

Problems about partitions of cartesian products are common in mathematics. For example, in finite and infinite combinatorics, they keep emerging in Ramsay theory, where one seeks to show that, if a product is partitioned into finitely many parts, one part at least must contain a subset of a certain specified kind. In the transition from finite to infinite products, one usually imposes restrictions of a topological nature on the partition, in order to obtain theorems analogous to those which are valid in the finite case. (See, e.g., [G-P], [Si], [E].)

In this paper we prove

Theorem 1. If N is a natural number and I is an interval of N consecutive integers, then there is a 1–1 correspondencef: {1, 2,…, N} → I such that (i, f(i)) = 1 for 1 ≤ i ≤ N.

The concept of uniform regularity of a measure on a compact space—a property which allows uniform approximation of the measure on all compact sets—was introduced and discussed in [1], [2] and [3]. Further some extensions of the notion of uniform regularity are given in [4] and [6].

For each natural number m, let N(m) denote the number of integers n with ø(n) = m, where ø denotes Euler's function. There are many interesting problems connected with the function N(m), such as the conjecture of Carmichael that N(m) is never 1 (see [9], for example) and the study of the distribution of the m for which N(m) > 0 (see Erdős and Hall [5]). In this note we shall be concerned with the maximal order of N(m).

In recent work [2] we have investigated Borel isomorphisms at the first level, i.e. mappings that together with their inverses map Fσ-sets to Fσ-sets. We are grateful to Dr. F. Topsøe for writing to point out errors in the proofs of our Theorems 2.1, 2.2 and 2.3. The errors escaped our attention because we only wrote out the first step of a complicated inductive argument and carelessly and falsely claimed that the general step of the induction was similar to the first. To be more specific, in the proof of Theorem 2.1, although we do have

we do not, in general, have

Fortunately the proofs can be corrected.