Related papers: Optimally dense packings of hyperbolic space
Define the superball with radius $r$ and center ${\boldsymbol 0}$ in $\mathbb{R}^n$ to be the set $$ \left\{{\boldsymbol x}\in\mathbb{R}^n:\sum_{j=1}^{m}\left(x_{k_j+1}^2+x_{k_j+2}^2+\cdots+x_{k_{j+1}}^2\right)^{p/2}\leq…
We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an…
In this paper we define the notion of $(p,\delta)$--Gromov hyperbolic space where we relax Gromov's {\it slimness} condition to allow that not all but a positive fraction of all triangles are $\delta$--slim. Furthermore, we study maximum…
Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This…
Optimal transport has become part of the standard quantitative economics toolbox. It is the framework of choice to describe models of matching with transfers, but beyond that, it allows to: extend quantile regression; identify discrete…
Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only…
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…
In this article we consider the ergodic risk-sensitive control problem for a large class of multidimensional controlled diffusions on the whole space. We study the minimization and maximization problems under either a blanket stability…
We develop a min-max theory for certain complete minimal hypersurfaces in hyperbolic space. In particular, we show that given two strictly stable minimal hypersurfaces that are both asymptotic to the same ideal boundary, there is a new one…
We consider the problem of dynamic optimal transport with a density constraint. We derive variational limits in terms of $\Gamma$-convergence for two singular phenomena. First, for densities constrained near a hyperplane we recover the…
Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the famous Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim…
After having investigated several types of geodesic ball packings in $\mathbf{S}^2 \times \mathbf{R}$ space, in this paper we study the locally optimal geodesic of simply and multiply transitive ball packings with equal balls to the space…
We investigate a real-life air cargo loading problem which is a variant of the three-dimensional Variable Size Bin Packing Problem with special bin forms of cuboid and non-cuboid unit load devices (ULDs). Packing is constrained by…
Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak…
The theory of stochastic representations of solutions to elliptic and parabolic PDE has been extensive. However, the theory for hyperbolic PDE is notably lacking. In this short note we give a stochastic representation for solutions of…
This paper is concerned with a shape optimization problem governed by a non-smooth PDE, i.e., the nonlinearity in the state equation is not necessarily differentiable. We follow the functional variational approach of [40] where the set of…
It is known that hyperbolic dynamical systems admit a unique invariant probability measure with maximal entropy. We prove an effective version of this statement and use it to estimate an upper bound for Hausdorff dimension of exceptional…
We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of…
We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).
Negatively-curved, maximally symmetric hyperbolic spaces enjoy a number of remarkable properties that can be traced back to Riemannian geometry, group theory and algebraic geometry. In this note we recall some such properties and find $H_n$…