Structure Results for Multiple Tilings in 3D

We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda $ in such a way that each point of ${\mathbb {R}}^d$ gets covered exactly $k$ times, except perhaps the translated copies of the boundary of $P$ . It is known that all possible multiple tilers in ${\mathbb {R}}^3$ are zonotopes. In ${\mathbb {R}}^2$ it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537–553, 2000) that, unless $P$ is a parallelogram, the multiset of translation vectors $\Lambda $ must be a finite union of translated lattices (also known as quasi periodic sets). In that work (Kolountzakis, Discrete Comput Geom 23(4):537–553, 2000) the author asked whether the same quasi-periodic structure on the translation vectors would be true in ${\mathbb {R}}^3$ . Here we prove that this conclusion is indeed true for ${\mathbb {R}}^3$ . Namely, we show that if $P$ is a convex multiple tiler in ${\mathbb {R}}^3$ , with a discrete multiset $\Lambda $ of translation vectors, then $\Lambda $ has to be a finite union of translated lattices, unless $P$ belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes $P$ , defined by the Minkowski sum of two 2-dimensional symmetric polygons in ${\mathbb {R}}^3$ , one of which may degenerate into a single line segment. It turns out that rational two-flat zonotopes admit a multiple tiling with an aperiodic (nonquasi-periodic) set of translation vectors $\Lambda $ . We note that it may be quite difficult to offer a visualization of these 3-dimensional non-quasi-periodic tilings, and that we discovered them by using Fourier methods.     

Collective correlations,dynamics, and behavioural inconsistencies of the cryptocurrency market over time

Nonlinear Dynamics - This paper introduces new methods to study behaviours among the 52 largest cryptocurrencies between 01-01-2019 and 30-06-2021. First, we explore evolutionary correlation...     

Divisors in residue classes, constructively

Let be integers satisfying , , , and let . Lenstra showed that the number of integer divisors of equivalent to is upper bounded by . We re-examine this problem, showing how to explicitly construct all such divisors, and incidentally improve this bound to .

     

QSAR modeling using automatically updating correction libraries: application to a human plasma protein binding model

It is assumed that compounds occupying the same region of model space will be subject to similar errors in prediction, and hence, where these errors are known, they can be applied to predictions. Thus, any available measured data can be used to refine predictions of query compounds. This study describes the application of a correction library to a human plasma protein binding model. Compounds that have been measured since the model was built are entered into the library to improve predictions of current compounds. Time-series simulations were conducted to measure the time dependence of the correction library. This study demonstrates significant improvements in predictions where a library is applied, compared with both a static model and an updating model that includes recently measured data.     

Metalloporphyrin Dendrimers with Folding Arms

Rigid and flexible linkers are combined in the new dendrimer shown schematically in the picture, which contains nine metalloporphyrin units (Porph). The construction is such that the four “arms” of the dendrimer can fold in a cooperative and predetermined manner in response to added 1,4-diazabicyclo[2.2.2]octane (DABCO).     

Synthesis and characterisation of low density porous polymers by reversible addition-fragmentation chain transfer (RAFT)

PolyHIPE foams with densities of 0.05–0.1 g cm^?3 have been prepared by the polymerisation of the continuous phase of high internal phase emulsions (HIPEs). The internal aqueous phase in HIPE occupies more than 74 % of the total volume, which leads to highly porous and open-cell morphologies. In this paper a method of preparing polyHIPE foams by using reversible addition-fragmentation chain transfer (RAFT) polymerisation has been investigated. Polystyrene-co-polymethyl methacrylate (PS-co-PMMA) has been studied and by using a variety of characterisation methods, it was possible to compare the polyHIPEs prepared by the conventional free radical polymerisation (FRP) to those by RAFT polymerisation. Scanning electron microscopy images have confirmed the presence of a cellular polyHIPE structure. PS-co-PMMA polyHIPEs made by RAFT have significantly narrower molecular weight distribution with values for the polydispersity index (PDI) for PS-co-PMMA between 1.46 and 2.08 compared to 4.68 observed by FRP. The effects of different concentrations of the RAFT agent on structure, glass transition temperature (T _g) and PDI of PS-co-PMMA polyHIPE foams are presented.