arXiv subjects

Mauricio Duarte

Publications and source records attributed to Mauricio Duarte.

9 recordsLinked to original sources

arXiv2022

No Agreement Without Loss: Learning and Social Choice in Peer Review

In peer review systems, reviewers are often asked to evaluate various features of submissions, such as technical quality or novelty. A score is given to each of the predefined features and based on these the reviewer has to provide an overall quantitative recommendation. It may be assumed that each reviewer has her own mapping from the set of features to a recommendation, and that different reviewers have different mappings in mind. This introduces an element of arbitrariness known as commensuration bias. In this paper we discuss a framework, introduced by Noothigattu, Shah and Procaccia, and then applied by the organizers of the AAAI 2022 conference. Noothigattu, Shah and Procaccia proposed to aggregate reviewer's mapping by minimizing certain loss functions, and studied axiomatic properties of this approach, in the sense of social choice theory. We challenge several of the results and assumptions used in their work and report a number of negative results. On the one hand, we study a trade-off between some of the axioms proposed and the ability of the method to properly capture agreements of the majority of reviewers. On the other hand, we show that dropping a certain unrealistic assumption has dramatic effects, including causing the method to be discontinuous.

cs.AI↗

arXiv2022

Upper bound on the number of collisions of pinned billiard balls

We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of possible pseudo-collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of $n$ pinned balls in a $d$-dimensional space. The proof is based on analysis of foldings, i.e., mappings that formalize the idea of folding a piece of paper along a crease. We prove an upper bound for the size of an orbit of a point subjected to foldings.

math.DS↗

arXiv2021

Fermi acceleration in rotating drums

Consider hard balls in a bounded rotating drum. If there is no gravitation then there is no Fermi acceleration, i.e., the energy of the balls remains bounded forever. If there is gravitation, Fermi acceleration may arise. A number of explicit formulas for the system without gravitation are given. Some of these are based on an explicit realization, which we derive, of the well-known microcanonical ensemble measure.

math-ph↗

arXiv2020

Powers of Brownian Green Potentials

In this article we study stability properties of $g_O$, the standard Green kernel for $O$ an open regular set in $R^d$. In $d\ge 3$ we show that $g_O^\beta$ is again a Green kernel of a Markov Feller process, for any power $\beta\in [1,d/(d-2))$. In dimension $d=2$, if $O$ is an open Greenian regular set, we show the same result for $g_O^\beta$, for any $\beta\ge 1$ and for the kernel $\exp(\alpha g_O)$, when $\alpha \in (0,2\pi)$.

math.PR↗

arXiv2018

On pinned billiard balls and foldings

We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of $n$ pinned balls in a $d$-dimensional space, in terms of $n$, $d$ and the locations of ball centers. As a first step, we study foldings, i.e., mappings that formalize the idea of folding a piece of paper along a crease.

math.DS↗

arXiv2018

On the number of hard ball collisions

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than $n^{-n}$, then the total number of collisions is bounded by $n^{(5/2+\varepsilon)n}$, for any fixed $\varepsilon>0$ and large $n$. We also show that if there is a number of collisions larger than $n^{cn}$ for an appropriate $c>0$, then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.

math.DS↗

arXiv2018

A lower bound for the number of elastic collisions

We prove by example that the number of elastic collisions of $n$ balls of equal mass and equal size in $d$-dimensional space can be greater than $n^3/27$ for $n\geq 3$ and $d\geq 2$. The previously known lower bound was of order $n^2$.

math.DS↗

arXiv2015

Gravitation versus Brownian motion

We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.

math.PR↗

arXiv2015

Asymptotics for the heat kernel in multicone domains

A multi cone domain $Ω\subseteq \mathbb{R}^n$ is an open, connected set that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion killed upon exiting $Ω$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\lim_{t\to\infty} t^{1+α}p(t,x,y)$ in terms of the Martin boundary of $Ω$ at infinity, where $α>0$ depends on the geometry of $Ω$. We next derive an analogous result for $t^{κ/2}\mathbb{P}_x(T >t)$, with $κ= 1+α- n/2$, where $T$ is the exit time form $Ω$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.

math.PR↗