Manuel Cruz | Searcharxiv

arXiv2020

Rigidity of complete self-shrinkers whose tangent planes omit a nonempty set

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones. We also prove an analogous result for self-expanders.

math.DG↗

arXiv2018

A time-dependent scoring system for soccer leagues

In this paper, a new continuous scoring system for soccer is proposed, based on the proportion of time that a team is winning, losing or tied. Several simulations are made applying this technique to complete seasons of different leagues. As some preliminary fundamental analysis previews, the simulation results show that this proposal compacts the gap between several teams, increasing the competitiveness of the championships and making the matches even more challenging as every minute may count in order to get the best rank in the final standings. Based on those results, some characteristics of the model are highlighted and a discussion on some of the advantages, disadvantages and practical issues regarding a hypothetical implementation of this model is made. Finally, some ideas to tackle the less positive effects of an implementation in real leagues is also provided.

stat.AP↗