Related papers: A Breakthrough in Sphere Packing: The Search for M…
We investigate, by "a la Marcinkiewicz" techniques applied to the (asymptotic) density function, how dense systems of equal spheres of $\rb^{n}, n \geq 1,$ can be partitioned at infinity in order to allow the computation of their density as…
The immensely fruitful concept of Grothendieck topology or covering issued from the efforts of algebraic geometers to study "sheaf-like" objects defined on categories more general than the lattice of open sets on a topological space. In the…
It is shown here how prior estimates on the local shape of the universe can be used to reduce, to a small region, the full parameter space for the search of circles in the sky. This is the first step towards the development of efficient…
Policy Brief on "Latest Developments and Opportunities in Sky Survey", distilled from the corresponding panel that was part of the discussions during S20 Policy Webinar on Astroinformatics for Sustainable Development held on 6-7 July 2023.…
We highlight recent advances in neutrino astrophysics, the open issues and the interplay with neutrino properties. We emphasize the important progress in our understanding of neutrino flavor conversion in media. We discuss the case of solar…
We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all…
In this paper, we build upon the work of Gluck and Warner who showed in 1983 that the set of positively oriented fibrations of a 3-sphere by oriented great circles is in bijection with the set of distance-decreasing maps from the 2-sphere…
In this paper, we prove some differentiable sphere theorems and topological sphere theorems for submanifolds in K\"ahler manifold, especially in complex space forms.
Using a recently developed method to simulate percolation on large clusters of distributed machines [N. R. Moloney and G. Pruessner, Phys. Rev. E 67, 037701 (2003)], we have numerically calculated crossing, spanning and wrapping…
We study, via the replica method of disordered systems, the packing problem of hard-spheres with a square-well attractive potential when the space dimensionality, d, becomes infinitely large. The phase diagram of the system exhibits…
In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically…
We study the relationship between local and global density for sphere packings, and in particular the convergence of packing densities in large, compact regions to the Euclidean limit. We axiomatize key properties of sphere packing bounds…
This is an invited commentary on "Geometrically incompatible confinement of solids", B. Davidovitch, Y. Sun and G. M. Grason (PNAS, doi:10.1073/pnas.1815507116 , arxiv:1809.06919).
Revolutionary advances in both theory and technology have launched cosmology into its most exciting period of discovery yet. Unanticipated components of the universe have been identified, promising ideas for understanding the basic features…
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…
The results presented at the 28th International Cosmic Ray Conference on neutrino and muon physics are summarized. Neutrinos and muons provide a huge amount of information on particle interactions up to very high energies and on fundamental…
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of…
In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of…
If a collection of identical particles is poured into a container, different shapes will fill to different densities. But what is the shape that fills a container as close as possible to a pre-specified, desired density? We demonstrate a…
In a recent Letter, Cummer et al. give a description of material parameters for acoustic wave propagation giving rise to a 3D spherical cloak, and verify the cloaking phenomenon on the level of scattering coefficients. A similar…