The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

A class of mixed type functional differential equations with piecewise constant arguments is studied. The initial value problem is discussed and necessary and sufficient conditions for existence and uniqueness are obtained.

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

C. M. Petty has conjectured the minimum value for a certain affine-invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.

In this paper time-harmonic surface wave motion for progressive waves incident normally on and scattered by a partially immersed fixed vertical barrier in water of infinite depth is considered in the presence of surface tension. The problem for the velocity potential is solved, as others have been previously, by first supposing that the free-surface slopes at the barrier are prescribed and the formal solution in terms of these is obtained explicitly by complex-variable methods. To simplify the calculation the known solution corresponding to zero free-surface slopes at the barrier is subtracted out first and emphasis is placed on determining the residual potential. Finally, an appropriate dynamical edge condition is imposed on the formal solution to determine the required values of the edge-slope constants and hence fully solve the transmission problem. The problem was first examined some time ago using a complex-variable reduction procedure before the advent of this condition, although an explicit formal solution was not obtained, that earlier work forms a basis for the present investigation. It is noted in conclusion how the solution of the problem for waves generated by a partially immersed non-uniform heaving vertical plate may easily be obtained in a similar manner, since the formal solution required is just the residual potential determined in our main problem.

We determine all continuous translation invariant simple valuations on the space of convex bodies in d-dimensional Euclidean space.

The following question of V. Stakhovskii was passed to us by N. Dolbilin [4]. Take the barycentric subdivision of a triangle to obtain six triangles, then take the barycentric subdivision of each of these six triangles and so on; is it true that the resulting collection of triangles is dense (up to similarities) in the space of all triangles? We shall show that it is, but that, nevertheless, the process leads almost surely to a flat triangle (that is, a triangle whose vertices are collinear).

We present a new characterization of σ-fragmentability and illustrate its usefulness by proving some results relating analyticity and crfragmentability. We show, for instance, that a Banach space with the weak topology is σ-fragmented if, and only if, it is almost Čech-analytic and that an almost Čech-analytic topological space is σ-fragmented by a lower-semicontinuous metric if, and only if, each compact subset of the space is fragmented by the metric.

Throughout this paper we assume that k is a given positive integer. As usual, B(x, r) denotes the closed ball with centre at x∈ℝk and radius r > 0. Let μ be a Radon measure on ℝk, that is, μ is locally finite and Borel regular. For s ≥ 0, the lower and upper s–dimensional densities of μ at x are denned respectively by

and

We consider the differential equation

where w(x) = xα for α > -1, q is a real-valued member of (0, ∞) and λ is a complex number with

The distribution of squarefree binomial coefficients. For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

Let Δn = {(x, y): x, y are integers 1 ≤ x, y ≤ n} be the n x n square array of integer lattice points in the plane.

An asymptotic theory is developed for a class of fourth-order differential equations. Under a general conditions on the coefficients of the differential equation we obtained the forms of the asymptotic solutions such that the solutions have different orders of magnitude for large x.

Recently Bombieri and Sperber have jointly created a new construction for estimating exponential sums on quasiprojective varieties over finite fields. In this paper we apply their construction to estimate hybrid exponential sums on quasiprojective varieties over finite fields. In doing this we utilize a result of Aldolphson and Sperber concerning the degree of the L-function associated with a certain exponential sum.

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set

obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

This paper depends on results of Baranovskii [1], [2]. The covering radius R(L) of an n-dimensional lattice L is the radius of smallest balls with centres at points of L which cover the whole space spanned by L. R(L) is closely related to minimal vectors of classes of the quotient . The convex hull of all minimal vectors of a class Q is a Delaunay polytope P(Q) of dimension ≤, dimension of L. Let be a maximal squared radius of P(Q) of dimension n (of dimension less than n, respectively). If , then . This is the case in the well-known Barnes-Wall and Leech lattices. Otherwise, . This is a refinement of a result of Norton ([3], Ch. 22).

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.