Let d be an integer greater than $1$, and let t be fixed such that $\frac {1}{d} < t < \frac {1}{d-1}$. We prove that for any $n_0$ chosen sufficiently large depending on t, the d-dimensional cubes of sidelength $n^{-t}$ for $n \geq n_0$ can perfectly pack a cube of volume $\sum _{n=n_0}^{\infty } \frac {1}{n^{dt}}$. Our work improves upon a previously known result in the three-dimensional case for when $\frac {1}{3} < t \leq \frac {4}{11} $ and $n_0 = 1$ and builds upon recent work of Terence Tao in the two-dimensional case.

Research from specialised hospital feeding programmes in the United States has shown effectiveness of a variety of treatments for packing (not swallowing food or liquid in the mouth) to increase swallowing and consumption. One potential component used in clinical practice has not been evaluated in the literature to our knowledge. This component is move-on and involves moving on to the next bite presentation rather than waiting for swallowing (i.e., clean mouth). A 5-year-old female with autism spectrum disorder and avoidant/restrictive food intake disorder participated in a home setting in Australia. We used a withdrawal/reversal single-case experimental design for a move-on component added to a treatment package. With move-on added, latency to clean mouth decreased and consumption increased to 100%. After the treatment evaluation, additional procedures (interspersal, redistribution) were needed in full plate and portion meals. Food variety was increased to 116 regular texture foods across all food groups. All (100%) of admission goals were met. Parents were trained to high procedural integrity, and the protocol was generalised to the community. Gains maintained to 1-month follow-up.

The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.

The spatial and temporal pattern of cone packing during marmoset foveal development was explored to understand the variables involved in creating a high acuity area. Retinal ages were between fetal day (Fd) 125 and 6 years. Cone density was determined in wholemounts using a new hexagonal quantification method. Wholemounts were labeled immunocytochemically with rod markers to identify reliably the foveal center. Cones were counted in small windows and density was expressed as cones × 103/mm2 (K). Two weeks before birth (Fd 125–130), cone density had a flat distribution of 20–30 K across the central retina encompassing the fovea. Density began to rise at postnatal day 1 (Pd 1) around, but not in, the foveal center and reached a parafoveal peak of 45–55 K by Pd 10. Between Pd 10 and 33, there was an inversion such that cone density at the foveal center rose rapidly, reaching 283 K by 3 months and 600 K by 5.4 months. Peak foveal density then diminished to 440 K at 6 months and older. Counts done in sections showed the same pattern of low foveal density up to Pd 1, a rapid rise from Pd 30 to 90, followed by a small decrease into adulthood. Increasing foveal cone density was accompanied by 1) a reduction in the amount of Müller cell cytoplasm surrounding each cone, 2) increased stacking of foveal cone nuclei into a mound 6–10 deep, and 3) a progressive narrowing of the rod-free zone surrounding the fovea. Retaining foveal cones in a monolayer precludes final foveal cone densities above 60 K. However, high foveal adult cone density (300 K) can be achieved by having cone nuclei stack into columns and without reducing their nuclear diameter. Marmosets reach adult peak cone density by 3–6 months postnatal, while macaques and humans take much longer. Early weaning and an arboreal environment may require rapid postnatal maturation of the marmoset fovea.

The literature on Anglo-South American trade during the first half of the 19th century has taken British exports for granted. There are no specific considerations of textile exports, which were the backbone of British trade to the continent. Accordingly, when explaining the growth of British exports, historians have paid tribute solely to economic developments in South America. Important developments taking place in Britain have long been neglected. This paper provides the first account of the impact that improvements in the packing of textiles to protect against seawater damages had on British exports to distant markets, focusing on the particular markets of Chile and the River Plate c.1810-1859.

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$. Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$. We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

This study presents an application to optimize the use of an L-cut guillotine machine. The application has two distinct parts to it; first, a number of rectangular shapes are placed on as few metal sheets as possible by using genetic algorithms. Second, the sequence for cutting these pieces has to be generated. The guillotine's numeric control then uses this sequence to make the cuts.

An effective way to build ordered materials with micrometer- or submicrometer-sized features is to pack together monodisperse (equal-sized) colloidal particles. But most monodisperse particles in this size range are spheres, and thus one problem in building new micrometer-scale ordered materials is controlling how spheres pack. In this article, we discuss how this problem can be approached by constructing and studying packings in the few-sphere limit. Confinement of particles within containers such as micropatterned holes or spherical droplets can lead to some unexpected and diverse types of polyhedra that may become building blocks for more complex materials. The packing processes that form these polyhedra may also be a source of disorder in dense bulk suspensions.

A strip of radius $r$ in the hyperbolic plane is the set of points within distance $r$ of a given geodesic. Wedefine the density of a packing of strips of radius $r$ and prove that this density cannot exceed

$$ \mathcal{S}(r)=\frac{3}{\pi}\sinh r\mathrm{arccosh}\biggl(1+\frac{1}{2\sinh^2r}\biggr). $$

This bound is sharp for every value of $r$ and provides sharp bounds on collaring theorems for simple geodesics onsurfaces.

AMS 2000 Mathematics subject classification: Primary 51M09; 52C15. Secondary 51M04

We present direct evidence for a change in protein structural specificity due to hydrophobic core packing. High resolution structural analysis of a designed core variant of ubiquitin reveals that the protein is in slow exchange between two conformations. Examination of side-chain rotamers indicates that this dynamic response and the lower stability of the protein are coupled to greater strain and mobility in the core. The results suggest that manipulating the level of side-chain strain may be one way of fine tuning the stability and specificity of proteins.