Han's conjecture and Hochschild homology for null-square projective algebras

Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $\Lambda$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at the idempotents belong to $\mathcal H$, then $\Lambda$ is in $\mathcal H$. Secondly we consider a $2\times 2$ matrix algebra, with two algebras on the diagonal, two projective bimodules in the corners, and zero corner products. They are not triangular with respect to the system of the two diagonal idempotents. However, the analogous result holds, namely if both algebras on the diagonal belong to $\mathcal H$, then the algebra itself is in $\mathcal H$.Comment: To appear in Indiana University Mathematics Journal 28 page

Hochschild cohomology: some methods for computations

We present some results on computing Hochschild cohomology groups. We describe the lower cohomology groups and provide several examples. In the particular case of hereditary algebras, radical square zero algebras and incidence algebras, we construct convenient projective resolutions that allow us to compute their cohomology groups

Connected gradings and fundamental group

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k$-algebra $A$, we consider the category of all connected gradings of $A$ by a group $G$ and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of $A$, which allows its computation using complete lists of gradings.Comment: Final version to appear in Algebra & Number Theory. 26 page

Maurer-Cartan equation for gentle algebras

Let $A= \Bbbk Q/I$ be a finite-dimensional gentle algebra. In this article, under some hypothesis on the quiver $Q$, we give conditions for nilpotency of the $L_\infty$-structure on the shifted Bardzell's complex $B(A)[1]$. For nilpotent cases, we describe Maurer-Cartan elements.Comment: 26 page

Adolescent victims of physical dating violence: why do they stay in abusive relationships?

According to recent international studies, most of the adolescent victims of physical dating violence remain satisfied and committed toward their abusive relationship, giving way to long-term relationships in which the abuse tends to persist and increase in frequency and severity. The objective of this study is to contribute to the understanding of the stay/leave decision of the adolescent victims of physical dating violence. A structural equation model was estimated to explain the direct and indirect contribution of the level of satisfaction, commitment, justification of the aggression, relationship duration, psychological coercion toward commitment, and the consequences of the abuse on the victims’ decision to continue in the abusive relationship. The sample was 456 Latinx adolescents aged between 12 and 18 years. The results corroborate that the decision to leave a physically abusive dating relationship is a complex phenomenon related to subjective variables such as (a) the level of satisfaction, (b) cognitive evaluation of the aggression (especially in those couples who have been together for a long time), and (c) the exposure to psychological pressures toward commitment. Future preventive strategies must incorporate actions to help the youngest to evaluate in a more objective and adequate way, the real quality of their first relationships, and aim to modify the justification of the aggression, the recognition of the potential harm, and to foster an adequate balance between the benefits and harm of staying in the abusive relationshi

Uterus transplantation: An open door to genetic motherhood

La infertilidad de origen uterino afecta a una de cada 500 mujeres en edad reproductiva impidiéndoles ser madres biológicas. Es la única causa de esterilidad para la que no se disponía de tratamiento hasta el desarrollo del trasplante de útero, que todavía se encuentra en fase experimental. Los resultados del grupo de investigación del Dr. Brännström de Suecia están siendo prometedores, habiendo nacido cinco bebés sanos gracias a esta técnica. Se prevé una inminente demanda y polémica social en torno a los aspectos éticos y legales que la rodean. Las alternativas con las que cuentan las pacientes con este tipo de esterilidad son la adopción y la maternidad subrogada o vientre de alquiler. Ambas plantean también un reto ético-legal significativo que revisamos en el artículo. En el contexto de la legislación española, el uso de vientres de alquiler está prohibido, por lo que el trasplante de útero puede ser una solución alternativa para conseguir la maternidad en mujeres con esterilidad por causa uterina absolutaUterine factor infertility affects one in 500 women on reproductive age preventing them from being biological mothers. It is the only cause of sterility for which treatment was not available until uterine transplantation was developed, although it remains an experimental technique. The results of Dr. Brännström's research group in Sweden are promising. Five healthy babies have been born after uterus transplantation. We expect an imminent demand and social controversy around the ethical and legal aspects that surround uterus transplantation. The alternatives for patients with this kind of sterility are adoption and surrogacy. Both raise significant ethical and legal issues that we review in this article. In the context of the Spanish legislation, surrogacy is prohibited; therefore, uterine transplantation could be an alternative to achieve motherhood in patients with absolute uterine factor infertilit

Covering functors without groups

Coverings in the representation theory of algebras were introduced for the Auslander-Reiten quiver of a representation finite algebra by Riedtmann and later for finite dimensional algebras by Bongartz and Gabriel, R. Martinez-Villa and de la Pe\~na. The best understood class covering functors is that of Galois covering functors F: A -> B determined by the action of a group of automorphisms of A. In this work we introduce the balanced covering functors which include the Galois class and for which classical Galois covering-type results still hold. For instance, if F:A -> B is a balanced covering functor, where A and B are linear categories over an algebraically closed field, and B is tame, then A is tame.Comment: Some improvements have been made; in particular, the proof of Theorem 2 has been restructured and clarifie