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.

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 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].

Let K be a convex body in Ed and let skelsK denote the s-skeleton of K. Let ηs(K) denote the Hausdorff s-measure of skelsK and

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

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.

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.

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.

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

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].

An exact solution for the velocity field induced by the slow rolling motion of a sphere in contact with a permeable surface is derived. The porous solid and the viscous fluid each occupies a semi-infinite space.

It is shown that, by the use of conformal mapping, the problem is reduced to a fourth order ordinary linear boundary value problem. As the permeability of the porous body diminishes the important functionals, e.g. the force and torque resisting the rolling motion of the sphere, are divergent. The nature of this divergence is found and the dependence on the permeability of the porous solid is explicitly evaluated.

It is shown that a convex body K tiles Ed by translation if, and only if, K is a centrally symmetric d-polytope with centrally symmetric facets, such that every belt of K (consisting of those of its facets which contain a translate of a given (d – 2)-face) has four or six facets. One consequence of the proof of this result is that, if K tiles Ed by translation, then K admits a face-to-face, and hence a lattice tiling.

§1. Introduction and notation. In [1] and [2], Besicovitch demonstrated that there exist plane sets of measure zero containing line segments (and indeed entire lines) in all directions in the plane. It is natural to ask about the existence of analogous sets in Euclidean spaces of higher dimensions, and in [3] we defined an (n, k)-Besicovitch set to be a subset A of Rn, of n-dimensional Lebesgue measure zero, such that for each k-dimensional subspace Π of Rn, some translate of Π intersects A in a set of positive k-dimensional measure. (Thus Besicovitch's original constructions were for (2,1)-Besicovitch sets.) Recently, Marstrand [5] has shown (by approximating to sets by unions of cubes) that no (3, 2)-Besicovitch sets exist, and simultaneously the author [3] proved using Fourier transform methods that (n, k)- Bsicovitch sets cannot exist if k > ½n.

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.

When K is a convex body in d-dimensional euclidean space E and 0 ≤ s ≤ d, the s-skeleton of K, denoted skel(s)K, consists of those points of K which are not centres of (s + l)-dimensional balls contained in K. The s-skeleton is thus the union of the extreme faces of K having dimension at most s. The s-skeleton is a -set [see 6] and it is therefore measurable with respect to the s-dimensional Hausdorff measure ℋ(s) [see 7]; here we normalize ℋ(s) so that it assigns unit measure to the s-dimensional unit cube.

The first problem in diophantine approximation is for a given real number ξ, and positive number x to find a fraction s/t with t ≤ x which is close to ξ. This problem can be rephrased in geometric terms. Given a vector v in ℝ2, find a vector a = (a1, a2) with integer coordinates and 1 1 ≤ al ≥ x such that the vectors a and v are nearly parallel. Simultaneous approximation of d − 1 real numbers can be recast in terms of approximation of the angle between two vectors in d-dimensional Euclidean space.

Unfortunately the contours used in the proof of Lemma 3 of my recent paper «this volume pp. 62–71» have not all been specified in the appropriate way. The contours used at the foot of page 66 should be

and middle contour used near the middle of page 67 should be

Let X be a compact metric space. By an invariant measure on X we shall mean a finitely additive non-negative real-valued function μ. on the Borel σ-algebra in X, such that μ(X) = 1 and Borel subsets of X have equal measure if there exists an isometry from one onto the other (not necessarily extendable to the whole of X). We note in passing that an isometric copy of a Borel set is necessarily itself Borel, by a well-known theorem of Souslin (see [3], §39. V). In Problem 2 of the Scottish Book (17. VII. 1935; see [2]), Banach and Ulam asked whether every non-empty compact metric space admits an invariant measure. This problem remains open; we give a positive answer in a veryspecial case.

The temporal and spatial linear instability of Poiseuille flow through pipes of arbitrary cross-section is discussed for large Reynolds numbers (R). For a pipe whose aspect ratio is finite, neutral stability (lower branch) is found to be governed by disturbance modes of large axial wavelength (of order hR, where h is a characteristic cross-sectional dimension). By contrast, spatial instability for finite aspect ratios is governed by length scales between O(h) and O(hR). When the aspect ratio is increased to O(R1/7), however, these two characteristic length scales both become O(R1/7h) and a match with plane channel flow instability is achieved. Thus the general cross-section produces temporal and spatial instability if the aspect ratio is O(R1/7). Further, in the flow in a rectangular pipe neutral stability (lower branch) exists for some finite aspect ratios, while for the flow in any non-circular elliptical pipe spatial instability is possible. It is suggested that both temporal and spatial instability occur for a wide range of pipe cross-sections of finite aspect ratio. Part 2 (Smith 1979a), which studies the upper branch neutral stability, confirms the importance of the O(hR) scale modes in neutral stability for finite aspect ratios.