Related papers: On the Sausage Catastrophe in 4 Dimensions

The exploration of the densest sphere packings is a fundamental problem in mathematics and a wide variety of sciences including materials science. We present our exhaustive computational exploration of the densest ternary sphere packings…

The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $3$-dimensional space of constant curvature was settled by B\"or\"oczky and Florian for the hyperbolic space $\HYP$ in \cite{BF64} and by proving…

We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality.…

Recent empirical investigations suggest a universal scaling law for the spatial structure of social networks. It is found that the probability density distribution of an individual to have a friend at distance $d$ scales as $P(d)\propto…

Reduction of the $\eta$-deformed sigma model on ${\rm AdS}_5 \times {\rm S}^5$ to the two-dimensional squashed sphere $({\rm S}^2)_{\eta}$ can be viewed as a special case of the Fateev sausage model where the coupling constant $\nu$ is…

Let's have $n$ points in the space such that the maximum distance between any of them is $a$. We prove that there exists a sphere of radius $r \leq a \frac{\sqrt(6)}{4}$ that contains in its interior or on its surface all these points.…

We analyze Sz\"oll\H{o}si's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these…

BPS states of N=2, D=4 Super Yang-Mills theories with ADE flavor symmetry arise as junctions joining a D3-brane to a set of 7-branes defining the enhanced flavor algebra. We show that the familiar BPS spectrum of SU(2) theories with N_f <=…

This paper is devoted to establishing the compactness and existence results of the solutions to the prescribing fractional $Q$-curvatures problem of order $2\sigma$ on $n$-dimensional standard sphere when $ n-2\sigma=2$, $\sigma=1+m/2,$…

The smallest maximum kissing-number Voronoi polyhedron of 3d spheres is the icosahedron and the tetrahedron is the smallest volume that can show up in Delaunay tessalation. No periodic lattice is consistent with either and hence these dense…

Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $\pi/\sqrt{12}$ in any dimension.

We revisit the scaling properties of growing spheres randomly seeded in d=2,3 and 4 dimensions using a mean-field approach. We model the insertion probability without assuming a priori a functional form for the radius distribution. The…

Let $s(n)$ be the side length of the smallest square into which $n$ non-overlapping unit squares can be packed. In 2010, the author showed that $s(13)=4$ and $s(46)=7$. Together with the result $s(6)=3$ by Keaney and Shiu, these results…

In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at most $\pi/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity, this gives…

Cold dark matter (CDM) can be thought of as a 2D (or 3D) sheet of particles in 4D (or 6D) phase-space due to its negligible velocity dispersion. The large-scale structure, also called the cosmic web, is thus a result of the topology of the…

We study a scenario to derive four-dimensional modular flavor symmetric models from higher dimensional theory by assuming the compactification consistent with the modular symmetry. In our scenario, wavefunctions in extra dimensional compact…

We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…

We study the time evolution of wave packets at the mobility edge of disordered non-interacting electrons in two and three spatial dimensions. The results of numerical calculations are found to agree with the predictions of scaling theory.…

Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by…

In 1966, Kleitman determined the maximum size of a family of subsets of $[n]$ with bounded symmetric difference. Liao, Liu and Yan recently established a vector-space analogue in the cases $n=d+1$ and $n>2d$, and asked for the sharp bound…