Related papers: Compact Packings are not always the Densest
In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In section 1 we prove that Loeb spaces are compact under…
Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure…
We prove that the intersection of a Hirsch polytope and a cube may be a non-Hirsch polytope.
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a…
The note shows an easy way to improve E.H. Smith's packing density bound in $\mathbb{R}^3$ from $0.53835...$ to $0.54755...$ .
We propose a conjecture on the density of arithmetic points in the deformation space of representations of the \'etale fundamental group in positive characteristic. This? conjecture has applications to \'etale cohomology theory, for example…
Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis,…
We characterize those derivations from the convolution algebra $\ell^1({\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial,…
In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order…
We collect various facts related loosely to random Gaussian quadrilaterals in the plane. For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.
It is well known that the lattice packing density and the lattice covering density of a triangle are $\frac{2}{3}$ and $\frac{3}{2}$ respectively. We also know that the lattices that attain these densities both are unique. Let…
In this brief note, we provide an example of non complete locally convex space $E$ with a $\sigma(E, E^*)$ closed bounded subset $C\subset E$, which is not $\sigma(E, E^*)$-compact, even if every $\varphi\in E^*$ attains its sup over $C$.
We consider four problems. Rogers proved that for any convex body $K$, we can cover ${\mathbb R}^d$ by translates of $K$ of density very roughly $d\ln d$. First, we extend this result by showing that, if we are given a family of positive…
This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof…
We provide a counterexample to a conjecture of Hildebrand which states that if $\S$ has positive lower density and is stable i.e. for all $d$, $n$ is in $\mathcal{S}$ if and only if $dn$ is in $\mathcal{S}$ except on a set of density $0$…
In this paper, we introduce the notion of circular orderability for quandles. We show that the set all right (respectively left) circular orderings of a quandle is a compact topological space. We also show that the space of right…
We consider a spherically symmetric distribution of dust and show that it is possible, under general physically reasonable conditions, for an overdensity to evolve to an underdensity (and vice versa). We find the conditions under which this…
We consider the optimal containment of polygonal regions within convex containers with the special property of 'orientedness' - an oriented region enables us to choose a preferred direction on the plane (this direction is not necessarily an…
We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. We also provide examples of maps satisfying optimality conditions for special classes of data.
We show that big bang cosmology implies a high degree of entanglement of particles in the universe. In fact, a typical particle is entangled with many particles far outside our horizon. However, the entanglement is spread nearly uniformly…