Related papers: Combinatorial cube packings in cube and torus
Let $\Delta$ be the optimal packing density of $\mathbb R^n$ by unit balls. We show the optimal packing density using two sizes of balls approaches $\Delta + (1 - \Delta) \Delta$ as the ratio of the radii tends to infinity. More generally,…
The random packing fraction of binary particles in D-dimensional Euclidean space R^D is studied using a geometric approach. First, the binary packing fraction of assemblies with small size difference are studied, using the excluded volume…
We show that every ridge unfolding of an $n$-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. Moreover, the geometry of the…
In 2019, Andrews investigated integer partitions in which all parts of a given parity are smaller than those of the opposite parity and introduced eight partition functions based on the parity of the smaller parts and parts of a given…
We study the d-dimensional hypercube knapsack problem where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping…
Dense random packings of hard particles are useful models of granular media and are closely related to the structure of nonequilibrium low-temperature amorphous phases of matter. Most work has been done for random jammed packings of…
Packing problems have been of great interest in many diverse contexts for many centuries. The optimal packing of identical objects has been often invoked to understand the nature of low temperature phases of matter. In celebrated work,…
There exists a significant conjecture in the local Langlands correspondence that A-packets are ABV-packets. For the case $G=GL_n$, the conjecture reduces to ABV-packets for orbits of Arthur type in $GL_n$ being singletons, which is a…
In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…
Porous and heterogeneous materials are found in many applications from composites, membranes, chemical reactors, and other engineered materials to biological matter and natural subsurface structures. In this work we propose an integrated…
We report a numerical calculation of the total number of disordered jammed configurations $\Omega$ of $N$ repulsive, three-dimensional spheres in a fixed volume $V$. To make these calculations tractable, we increase the computational…
In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the…
A Costas array is a permutation array for which the vectors joining pairs of $1$s are all distinct. We propose a new three-dimensional combinatorial object related to Costas arrays: an order $n$ Costas cube is an array $(d_{i,j,k})$ of size…
In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts increasing the coverage are accepted. A finite system eventually gets congested, and we study the statistics of congested…
We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard so that we cannot expect to find…
In this paper, we use techniques of enumerative combinatorics to study the following problem: we count the number of ways to split $n$ balls into nonempty, ordered bins so that the most crowded bin has exactly $k$ balls. We find closed…
This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial…
This is the first in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
We introduce notions of combinatorial blowups, building sets, and nested sets for arbitrary meet-semilattices. This gives a common abstract framework for the incidence combinatorics occurring in the context of De Concini-Procesi models of…