Quivers with potentials associated to triangulated surfaces

We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky. To each ideal triangulation of a bordered surface with marked points we associate a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the quivers with potentials associated to its triangulations are rigid and hence non-degenerate.Comment: v3: 44 pages, 57 figures. Prop 29 of v2 generalized to Thm 36, some changes to References. In response to referee's comments: some examples added, more cases verified in proof of Thm 30 (formerly Thm 23). Submitted to Proc. London Math. So

Older men learning in the community : European snapshots

The full book can be found here: https://www.um.edu.mt/library/oar//handle/123456789/1780The learning partnership on older men learning was primarily an opportunity for researchers from different countries to establish a path for a future common understanding of the interface between masculinities and older adult learning, taking as our point of departure our determination to focus on the role of community and informal learning settings in the lives of older men. Project leaders decided not to take a rigid approach to or engage in a unified coordination of the cases being studied. Basically, each partner decided freely which approaches were best suited to each context. Sometimes driven by curiosity and sometimes by our desire to learn more about the relatively unknown locations of leaning for older men learning in our own countries, we considered diversity to be of outmost importance in our exploratory attitude towards a field of inquiry as of yet almost completely unexplored.peer-reviewe

Strongly primitive species with potentials I: Mutations

Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.Comment: 51 page

The Mathematics of Chinese Checkers

Our goal for this project was to expand and improve upon the findings of George I. Bell and Nicholas Fonseca, who have both written papers on optimization in Chinese Checkers. While their work focuses mainly on cooperative games between one, two, and three players, we have considered games for six players. While doing this, we have redefined the playing board in a more intuitive manner, while developing and proving its associated distance formula. As well, we have found the shortest game for six players, and are working to generalize a formula for the number of moves required to finish a six player game as fast as possible. This could further incite research to generalize a lower bound for any number of players

Editorial: open issue

In this open issue we present five papers covering different adult education and learning contexts across different geographical spaces, ranging from Spain, Germany, Eastern Europe to Canada. The topics of research range from young people to retired people, adult educators and men and women reading self-help literature

Estimation of Cost Allocation Coefficients at the Farm Level Using an Entropy Approach

This paper aims to estimate the farm cost allocation coefficients from whole farm input costs. An entropy approach was developed under a Tobit formulation and was applied to a sample of farms from the 2004 FADN data base for Alentejo region, Southern Portugal. A Generalized Maximum Entropy model and Cross Generalized Entropy model were developed to the sample conditions and were tested. Model results were assessed in terms of their precision and estimation power and were compared with observed data. The entropy approach showed to be a flexible and valid tool to estimate incomplete information, namely regarding farm costs.Generalized maximum entropy; costs; estimation; Alentejo, FADN.