arXiv subjects

Jaume de Dios Pont

Publications and source records attributed to Jaume de Dios Pont.

23 records · Page 2Linked to original sources

arXiv2023

A new proof of the description of the convex hull of space curves with totally positive torsion

We give new proofs of the description convex hulls of space curves $γ: [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(γ', γ'', \ldots, γ^{(d)})$ are positive. In particular, we recover parametric representation of the boundary of the convex hull, different formulas for its surface area and the volume of the convex hull, and the solution to a general moment problem corresponding to $γ$.

math.PR↗

arXiv2021

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.

math.CA↗

arXiv2021

Role Detection in Bicycle-Sharing Networks Using Multilayer Stochastic Block Models

In urban spatial networks, there is an interdependency between neighborhood roles and the transportation methods between neighborhoods. In this paper, we classify docking stations in bicycle-sharing networks to gain insight into the human mobility patterns of three major United States cities. We propose novel time-dependent stochastic block models (SBMs), with degree-heterogeneous blocks and either mixed or discrete block membership, which classify nodes based on their time-dependent activity patterns. We apply these models to (1) detect the roles of bicycle-sharing docking stations and (2) describe the traffic within and between blocks of stations over the course of a day. Our models successfully uncover work, home, and other districts; they also reveal activity patterns in these districts that are particular to each city. Our work has direct application to the design and maintenance of bicycle-sharing systems, and it can be applied more broadly to community detection in temporal and multilayer networks with heterogeneous degrees.

cs.SI↗

arXiv2021

On classical inequalities for autocorrelations and autoconvolutions

In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights, including Gaussian functions (as studied by the second author and Ramos) and characteristic function (as originally studied by Barnard and Steinerberger). Moreover, via a discretization argument and numerical analysis, we find some almost optimal approximation for the best constant allowed in this inequality. We also discuss some other related problem about autoconvolutions.

math.CA↗

arXiv2020

A geometric lemma for complex polynomial curves with applications in Fourier restriction theory

The aim of this paper is to prove a uniform Fourier restriction estimate for certain $2-$dimensional surfaces in $\mathbb R^{2n}$. These surfaces are the image of complex polynomial curves $γ(z) = (p_1(z), \dots, p_n(z))$, equipped with the complex equivalent to the affine arclength measure. This result is a complex-polynomial counterpart to a previous result by Stovall [Sto16] in the real setting. As a means to prove this theorem we provide an alternative proof of a geometric inequality by Dendrinos and Wright [DW10] that extends the result to complex polynomials.

math.CA↗