A number of known estimates of the number of translates, or lattice translates, of a convex body H required to cover a convex body K are obtained as consequences of two simple results.

Let [0;a1(ξ), a2(ξ),…] denote the continued fraction expansion of ξ∈[0, 1]. The problem of estimating the fractional dimension of sets of continued fractions emerged in late twenties in papers by Jarnik [6, 7] and Besicovitch [1] and since then has been addressed by a number of authors (see [2, 4, 5, 8, 9]). In particular, Good [4] proved that the set of all ξ, for which an(ξ)→∞ as n→∞ has the Hausdorff dimension ½ For the set of continued fractions whose expansion terms tend to infinity doubly exponentially the dimension decreases even further. More precisely, let

Hirst [5] showed that dim On the other hand, Moorthy [8] showed that dim where

We present a continuum theory for smectic liquid crystals which allows variable molecular tilt and layer spacing by relating the two. This compressible theory is employed to predict the effect of ‘homeotropic’ type ordering at the plates on a planar Sm C sample. Such a boundary condition induces dilation of individual Sm C layers near the bounding plates.

In this paper we investigate the solutions in integers x, y, z, X, Y, Z of the system

where P is a real number that may be taken to be arbitrarily large, and the (fixed) integer exponent h satisfies h≥4. The system has 6P3 + O(P2) “trivial” solutions in which x, y, z are a permutation of X, Y, Z. Our result implies that the number of non-trivial solutions is at most o(P3), so that the total number of solutions is asymptotic to 6P3.

Let (Sn)n = 1,2,… be the strictly increasing sequence of those natural numbers that can be represented as the sum of three cubes of positive integers. The estimate

is easily proved as follows: Let x1 be the largest natural number with Then This procedure is iterated by choosing x2 and then x3 as the largest natural numbers satisfying and Thus Since this implies (1).

Recently, it has been shown that tight or almost tight upper bounds for the discrepancy of many geometrically denned set systems can be derived from simple combinatorial parameters of these set systems. Namely, if the primal shatter function of a set system ℛ on an n-point set X is bounded by const. md, then the discrepancy disc (ℛ) = O(n(d−1)/2d) (which is known to be tight), and if the dual shatter function is bounded by const. md, then disc We prove that for d = 2, 3, the latter bound also cannot be improved in general. We also show that bounds on the shatter functions alone do not imply the average (L1)-discrepancy to be much smaller than the maximum discrepancy: this contrasts results of Beck and Chen for certain geometric cases. In the proof we give a construction of a certain asymptotically extremal bipartite graph, which may be of independent interest.

A symposium held in Durham in July 1995 brought together an international collection of key mathematicians, theoretical physicists and experimentalists in the areas of liquid crystals and polymeric systems. Many of the participants met together for the first time, and the symposium stimulated new collaborative interactions among applied mathematicians and others who, prior to this meeting, were working separately on either liquid crystals or polymer fluids. The symposium also enhanced further interchanges of ideas between industry and academia. The flavour of this meeting is captured by the contents of this special issue which resulted from presentations given by invited speakers.

Certain convergent search algorithms can be turned into chaotic dynamic systems by renormalisation back to a standard region at each iteration. This allows the machinery of ergodic theory to be used for a new probabilistic analysis of their behaviour. Rates of convergence can be redefined in terms of various entropies and ergodic characteristics (Kolmogorov and Rényi entropies and Lyapunov exponent). A special class of line-search algorithms, which contains the Golden-Section algorithm, is studied in detail. Their associated dynamic systems exhibit a Markov partition property, from which invariant measures and ergodic characteristics can be computed. A case is made that the Rényi entropy is the most appropriate convergence criterion in this environment.