On F-Algebroids and Dubrovin's Duality

In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid.Comment: 13 pages; v2 has small changes, it has improved exposition. Revised version to appear in: "Archivum Mathematicum

Fixation and escape times in stochastic game learning

Evolutionary dynamics in finite populations is known to fixate eventually in the absence of mutation. We here show that a similar phenomenon can be found in stochastic game dynamical batch learning, and investigate fixation in learning processes in a simple 2x2 game, for two-player games with cyclic interaction, and in the context of the best-shot network game. The analogues of finite populations in evolution are here finite batches of observations between strategy updates. We study when and how such fixation can occur, and present results on the average time-to-fixation from numerical simulations. Simple cases are also amenable to analytical approaches and we provide estimates of the behaviour of so-called escape times as a function of the batch size. The differences and similarities with escape and fixation in evolutionary dynamics are discussed.Comment: 19 pages, 9 figure

The Pandemic\u27s Impact on the Efficacy of Chaplaincy at the Baldwin State Prison

Chaplains usually provide pastoral care for inmates, staff members, and counseling for offenders and offenders’ families. Because of the pandemic, they have not provided adequate pastoral care and counseling in their respective facilities. The restrictions imposed by the pandemic have negatively impacted the chaplain’s ability to adequately provide pastoral care and counseling in most federal and state correctional institutions. The Department of Georgia Correctional facilities, particularly at the Baldwin State prison, has also experienced this decrease in pastoral care and counseling due to the pandemic. In the Georgia Department of Corrections, all visitations, including lawyers and clergy, have been suspended since March 2020. Due to these restrictions, the chaplain of the Baldwin State prison in Georgia cannot coordinate the various religious services representing the different religious groups housed in the facility. Secondly, the facility has seven dormitories that house one hundred and seven inmates (107) per dormitory, and the COVID-19 virus can spread exponentially due to this overcrowded condition. The pandemic has severely hindered the Baldwin State prison\u27s chaplain\u27s ability to provide this critical pastoral care and counseling for staff members and offenders. This body of research implications is discussed: the restrictions imposed by the pandemic and the overcrowded conditions at this facility have impeded the Baldwin State chaplain\u27s ability to adequately provide the pastoral care and counseling required for this facility during the COVID-19 pandemic. This lack of pastoral care and counseling results in moral issues and increased mental health challenges throughout the facility