Related papers: Dense packing of space with various convex solids
This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including…
We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this…
We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant $1$.…
The isoperimetric problem is one of the oldest in geometry and it consists of finding a surface of minimum area that encloses a given volume $V$. It is particularly important in physics because of its strong relation with stability, and…
In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…
In the framework of shape constrained estimation, we review methods and works done in convex set estimation. These methods mostly build on stochastic and convex geometry, empirical process theory, functional analysis, linear programming,…
This paper presents a novel geometrical approach to investigate the convexity of a density-based cluster. Our approach is grid-based and we are about to calibrate the value space of the cluster. However, the cluster objects are coming from…
Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials.…
This short note describes the tentative form of a finite-dimensional optimization problem that may be of use in a second-generation proof of the Kepler conjecture. In the original 1998 proof of the Kepler conjecture, the form of the…
Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries $R$ such that $\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density. These are…
Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of…
Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice…
We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to…
Density level sets can be estimated using plug-in methods, excess mass algorithms or a hybrid of the two previous methodologies. The plug-in algorithms are based on replacing the unknown density by some nonparametric estimator, usually the…
It can be difficult to tell whether a trained generative model has learned to generate novel examples or has simply memorized a specific set of outputs. In published work, it is common to attempt to address this visually, for example by…
Given a convex disk $K$ and a positive integer $k$, let $\delta_T^k(K)$ and $\delta_L^k(K)$ denote the $k$-fold translative packing density and the $k$-fold lattice packing density of $K$, respectively. Let $T$ be a triangle. In a very…
We advocate an optimization procedure for variable density sampling in the context of compressed sensing. In this perspective, we introduce a minimization problem for the coherence between the sparsity and sensing bases, whose solution…
We propose a unified framework that synthesizes advances in high-dimensional lattice theory with novel computational algorithms for the shortest vector problem (SVP) to model pure root lattices and compute sphere packing densities. Building…
In many areas of research it is interesting how lattices can be filled with particles that have no nearest neighbors, or they are in limited quantities. Examples may be found in statistical physics, chemistry, materials science, discrete…
A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were…