Auslander-Buchweitz approximation theory for triangulated categories

We introduce and develop an analogous of the Auslander-Buchweitz approximation theory (see \cite{AB}) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category \T, which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of \T. The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category \T equipped with a pair (\X,\omega), where \X is closed under extensions and $\omega$ is a weak-cogenerator in \X, usually under additional conditions. This reduces, among other things, to the existence of distinguished triangles enjoying special properties, and the behavior of (suitably defined) (co)resolutions, projective or injective dimension of objects of \T and the formation of orthogonal subcategories. Finally, some relationships with the Rouquier's dimension in triangulated categories is discussed.Comment: To appear at: Appl. Categor. Struct. (2011); 22 page

Cycle-finite module categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited

Coverings of Laura Algebras: the Standard Case

In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, it is not quasi-tilted of canonical type and its connecting components are standard), then this algebra has nice Galois coverings associated to the coverings of the connecting component. As a consequence, we show that the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings.Comment: The main result on the non-standard case was reformulated due to an inaccuracy in the previous version. Lemma 6.1 was removed due to a simplification. The last section on the special biserial case was removed. Typos corrected and bibliography updated. Final version to appear in Journal of Algebr

Finite Intersection Property and Dynamical Compactness

[EN] Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in Huang et al. (J Differ Equ 260(9):6800-6827, 2016). In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the -limit and the -limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.Wen Huang and Sergii Kolyada acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai. Sergii Kolyada also acknowledges the hospitality of the Max-Planck-Institute fur Mathematik (MPIM) in Bonn, the Departament de Matematica Aplicada of the Universitat Politecnica de Valencia, the partial support of Project MTM2013-47093-P, and the Department of Mathematics of the Chinese University of Hong Kong. We thank the referees for careful reading and constructive comments that have resulted in substantial improvements to this paper. Wen Huang was supported by NNSF of China (11225105, 11431012); Alfred Peris was supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P, and by GVA, Project PROMETEOII/2013/013; and Guohua Zhang was supported by NNSF of China (11671094).Huang, W.; Khilko, D.; Kolyada, S.; Peris Manguillot, A.; Zhang, G. (2018). 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Care for patients with severe mental illness: the general practitioner's role perspective

BACKGROUND: Patients with severe mental illness (SMI) experience distress and disabilities in several aspects of life, and they have a higher risk of somatic co-morbidity. Both patients and their family members need the support of an easily accessible primary care system. The willingness of general practitioners and the impeding factors for them to participate in providing care for patients with severe mental illness in the acute and the chronic or residual phase were explored. METHODS: A questionnaire survey of a sample of Dutch general practitioners spread over the Netherlands was carried out. This comprised 20 questions on the GP's 'Opinion and Task Perspective', 19 questions on 'Treatment and Experiences', and 27 questions on 'Characteristics of the General Practitioner and the Practice Organisation'. RESULTS: 186 general practitioners distributed over urban areas (49%), urbanised rural areas (38%) and rural areas (15%) of the Netherlands participated. The findings were as follows: GPs currently considered themselves as the first contact in the acute psychotic phase. In the chronic or residual phase GPs saw their core task as to diagnose and treat somatic co-morbidity. A majority would be willing to monitor the general health of these patients as well. It appeared that GP trainers and GPs with a smaller practice setting made follow-up appointments and were willing to monitor the self-care of patients with SMI more often than GPs with larger practices.GPs also saw their role as giving support and information to the patient's family.However, they felt a need for recognition of their competencies when working with mental health care specialists. CONCLUSION: GPs were willing to participate in providing care for patients with SMI. They considered themselves responsible for psychotic emergency cases, for monitoring physical health in the chronic phase, and for supporting the relatives of psychotic patients