Related papers: Cavity approach to sphere packing in Hamming space
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the…
We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we…
We describe a new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes. We believe that in some of the smaller cases, these packings are in fact optimal. When applied to the previously…
Cosmological emulators of observables such as the Cosmic Microwave Background (CMB) spectra and matter power spectra commonly use training data sampled from a Latin hypercube. This method often incurs high computational costs by covering…
In this paper we present a new algorithm for a layout optimization problem: this concerns the placement of weighted polygons inside a circular container, the two objectives being to minimize imbalance of mass and to minimize the radius of…
The recently introduced nested sampling algorithm allows the direct and efficient calculation of the partition function of atomistic systems. We demonstrate its applicability to condensed phase systems with periodic boundary conditions by…
The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbours, is studied using an analytical model that extends previous results…
Based on results from the physics and mathematics literature which suggest a series of clearly defined conjectures, we formulate three simple scenarios for the fate of hard sphere crystallization in high dimension: (A) crystallization is…
We introduce a novel quantum computing heuristic for solving the irregular strip packing problem, a significant challenge in optimizing material usage across various industries. This problem involves arranging a set of irregular polygonal…
We consider the problem of embedding a subset of $\mathbb{R}^n$ into a low-dimensional Hamming cube in an almost isometric way. We construct a simple, data-oblivious, and computationally efficient map that achieves this task with high…
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…
We use numerical simulation to investigate and analyze the way that rigid disks and spheres arrange themselves when compressed next to incommensurate substrates. For disks, a movable set is pressed into a jammed state against an ordered…
In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a…
We revisit the densest binary sphere packings (DBSP) under the periodic boundary conditions and present an updated phase diagram, including newly found 12 putative densest structures over the $x - \alpha$ plane, where $x$ is the relative…
We introduce the new problems of quantum packing, quantum covering, and quantum paving. These problems arise naturally when considering an algebra of non-commutative operators that is deeply rooted in quantum physics as well as in Gabor…
Recently, we enumerate up to isometry, all locally rigid circle packings on the unit sphere with number of circles N<12. This problem is equivalent to the enumeration of irreducible contact graphs. In this paper we show that by using the…
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…
In this paper we introduce an approximate method to solve the quantum cavity equations for transverse field Ising models. The method relies on a projective approximation of the exact cavity distributions of imaginary time trajectories…
Given a collection of $m$ sets from a universe $\mathcal{U}$, the Maximum Set Coverage problem consists of finding $k$ sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial…
Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…