Abstract:
In the category of finitely generated modules over an artinian ring, we classify all the abelian exact subcategories closed under predecessors or, equivalently, all the split torsion pairs with torsion-free class closed under quotients.

Abstract:
Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type \tilde{\mathbb{A}}.

Abstract:
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.

Abstract:
Introduction: Premature Ovarian Failure (POF) is cessation of ovarian functions before the age of 40 years old with consequent cessation of menstruation. Objective of study: The aim of this study was to evaluate the association between Premature Ovarian Failure and sexual dysfunctions and outcome of management with tibolone. Patients and Methods: Thirty-one women with Premature Ovarian Failure seen at the outpatient clinic of Maternity Hospital were enrolled into the study with 31 healthy women as control group. The instrument of data collection included two types of questionnaires to assess the effect of Premature Ovarian Failure on sexuality. All the women with POF had oral tibolone 2.5 mg for at least one year and the second questionnaire and the profiles were repeated. Results: Of the 31 women with POF that presented with sexual dysfunction (SD), 27 (87.1%) complained of one or more SD domains such as reduced frequency of coitus, dyspareunia, vaginal dryness, reduced libido and general sexual satisfaction (P < 0.01), amenorrhea (P < 0.01) and hot flashes compared to 5 (16.1%) control women (P < 0.01). Administration of tibolone was associated with significant increase in frequency of coitus, reduced dyspareunia and vaginal dryness, increase libido and general satisfaction and happiness. Reduction of sexual dysfunction was predicated on the estrogenic, progestogenic and androgenic metabolite of tibolone through the reduction of serum level of FSH and LH and increased levels of estrogen and testosterone (P < 0.01). Tibolone had no adverse effect on serum lipid profile. Conclusion: Premature Ovarian Failure is associated with sexual dysfunction. Tibolone provides an effective means of treating sexual dysfunction caused by Premature Ovarian Failure.

Abstract:
We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.

Abstract:
We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbuesch.

Abstract:
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.

Abstract:
Each acyclic graph, and more generally, each acyclic orientation of the graph associated to a Cartan matrix, allows to define a so-called frise; this is a collection of sequences over the positive natural numbers, one for each vertex of the graph. We prove that if these sequences satisfy a linear recurrence, then the Cartan matrix is of Dynkin type (if the sequences are bounded) or of Euclidean type (if the sequences are unbounded). We prove the converse in all cases, except for the exceptional Euclidean Cartan matrices; we show even that the sequences are rational over the positive natural numbers. We generalize these results by considering frises with variables; as a byproduct we obtain, for the Dynkin and Euclidean type A cases, explicit formulas for the cluster variables, over the semiring of Laurent polynomials over the positive natural numbers generated by the initial variables (which explains simultaneously positivity and the Laurent phenomenon). The general tool are the so-called SL_2-tilings of the plane; these are fillings of the whole discrete plane by elements of a ring, in such a way that each 2 by 2 connected submatrix is of determinant 1.

Abstract:
In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster automorphisms in detail for acyclic cluster algebras and cluster algebras from surfaces, and we compute this group explicitly for the Dynkin types and the Euclidean types.

Abstract:
We prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three, and give an explicit construction of the Auslander generator of its module category.