arXiv subjects

Mario Chater

Publications and source records attributed to Mario Chater.

3 recordsLinked to original sources

arXiv2018

Fixing Match-Fixing: Optimal schedules to promote competitiveness

In the last round of the FIFA World Cup group stage, games for which the outcome does not affect the selection of the qualified teams are played with little enthusiasm. Furthermore, a team that has already qualified may take into account other factors, such as the opponents it will face in the next stage of the competition so that, depending on the results in the other groups and the scheduling of the next stage, winning the game may not be in its best interest. Even more critically, there may be situations in which a simple draw will qualify both teams for the next stage of the competition. Any situation in which the two opposing teams do not play competitively is detrimental to the sport, and, above all, can lead to collusion and match-fixing opportunities. We here develop a relatively general method of evaluating competitiveness and apply it to the current format of the World Cup group stage. We then propose changes to the current format in order to increase the stakes in the last round of games of the group stage, making games more exciting to watch and, at the same time, reducing any collusion opportunities. We appeal to the same method to evaluate a "groups of 3" format which will be introduced in the 2026 World Cup edition as well as a format similar to the one of the current Euro UEFA Cup.

stat.AP↗

arXiv2017

Least Squares Shadowing method for sensitivity analysis of differential equations

For a parameterized hyperbolic system $\frac{du}{dt}=f(u,s)$ the derivative of the ergodic average $\langle J \rangle = \lim_{T \to \infty}\frac{1}{T}\int_0^T J(u(t),s)$ to the parameter $s$ can be computed via the Least Squares Shadowing algorithm (LSS). We assume that the sytem is ergodic which means that $\langle J \rangle$ depends only on $s$ (not on the initial condition of the hyperbolic system). After discretizing this continuous system using a fixed timestep, the algorithm solves a constrained least squares problem and, from the solution to this problem, computes the desired derivative $\frac{d\langle J \rangle}{ds}$. The purpose of this paper is to prove that the value given by the LSS algorithm approaches the exact derivative when the discretization timestep goes to $0$ and the timespan used to formulate the least squares problem grows to infinity.

math.DS↗

arXiv2017

Simplified Least Squares Shadowing sensitivity analysis for chaotic ODEs and PDEs

This paper develops a variant of the Least Squares Shadowing (LSS) method, which has successfully computed the derivative for several chaotic ODEs and PDEs. The development in this paper aims to simplify Least Squares Shadowing method by improving how time dilation is treated. Instead of adding an explicit time dilation term as in the original method, the new variant uses windowing, which can be simpler to implement, especially for PDEs.

math.DS↗