Related papers: Hyperplane Equipartitions Plus Constraints
Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an $\mathbb{F}_q$-subspace $U$ of $V$ $h$-scattered if $U$ meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most…
Most research into similarity search in metric spaces relies upon the triangle inequality property. This property allows the space to be arranged according to relative distances to avoid searching some subspaces. We show that many common…
In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation $-\Delta u - (k^2+ {\rm i} \varepsilon)u = f$, with absorption…
The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as…
Teichm\"uller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given…
We study the fundamental problem of clustering $n$ points into $K$ groups drawn from a mixture of isotropic Gaussians in $\mathbb{R}^d$. Specifically, we investigate the requisite minimal distance $\Delta$ between mean vectors to partially…
The purpose of this paper is to expose and investigate natural phase-space formulations of two longstanding problems in the restriction theory of the Fourier transform. These problems, often referred to as the Stein and Mizohata--Takeuchi…
Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate…
The spline space $C_k^r(\Delta)$ attached to a subdivided domain $\Delta$ of $\R^{d} $ is the vector space of functions of class $C^{r}$ which are polynomials of degree $\le k$ on each piece of this subdivision. Classical splines on planar…
The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamics formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE…
We investigate two related problems concerning the dimension of joint eigenspaces of the Laplace--Beltrami operator and a finite set of Hecke operators on $\mathbb{X}=\mathrm{PGL}_2(\mathbb{Z})\backslash \mathbb{H}$. First, we consider…
We consider DPG methods with optimal test functions and broken test spaces based on ultra-weak formulations of general second order elliptic problems. Under some assumptions on the regularity of solutions of the model problem and its…
This manuscript develops a new framework to analyze and design iterative optimization algorithms built on the notion of Integral Quadratic Constraints (IQC) from robust control theory. IQCs provide sufficient conditions for the stability of…
Huang, McKinnon, and Satriano conjectured that if $v \in \mathbb{R}^n$ has distinct coordinates and $n \geq 3$, then a hyperplane through the origin other than $\sum_i x_i = 0$ contains at most $2\lfloor n/2 \rfloor (n-2)!$ of the vectors…
We study distributed protocols for finding all pairs of similar vectors in a large dataset. Our results pertain to a variety of discrete metrics, and we give concrete instantiations for Hamming distance. In particular, we give improved…
We investigate a first-order mean field planning problem of the form \begin{equation} \left\lbrace\begin{aligned} -\partial_t u + H(x,Du) &= f(x,m) &&\text{in } (0,T)\times \mathbb{R}^d, \\ \partial_t m - \nabla\cdot (m\,H_p(x,Du)) &= 0…
Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and…
Optimal statistical decisions should transcend the language used to describe them. Yet, how do we guarantee that the choice of coordinates - the parameterisation of an optimisation problem - does not subtly dictate the solution? This paper…
Given $mp$ different $p$-planes in general position in $(m+p)$-dimensional space, a classical problem is to ask how many $p$-planes intersect all of them. For example when $m = p = 2$, this is precisely the question of "lines meeting four…
In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients…