Journal•ISSN: 1982-6907

# The São Paulo Journal of Mathematical Sciences

Springer Nature

About: The São Paulo Journal of Mathematical Sciences is an academic journal published by Springer Nature. The journal publishes majorly in the area(s): Mathematics & Combinatorics. It has an ISSN identifier of 1982-6907. Over the lifetime, 460 publications have been published receiving 1601 citations.

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TL;DR: Auslander as mentioned in this paper showed that an algebra is of finite representation type, that is, it admits only finitely many indecomposable modules up to isomorphism, if and only if its representation dimension is at most 2.

Abstract: Auslander has shown that an algebra is of finite representation type, that is, it admits only finitely many indecomposable modules up to isomorphism, if and only if its representation dimension is at most 2. We will give a proof of this fact in Section 1 as Corollary 1.9. This led Auslander to the expectation, “that this notion gives a reasonable way of measuring how far an artin algebra is from being of finite representation type.” [1, III.5, lines 2, 3]

204 citations

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TL;DR: In this article, the authors introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category and describe the shapes of its semi-stable components.

Abstract: We rst introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This uni es the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then de ne the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those for an artin algebra and specializes to an arbitrary triangulated categories, in particular to the derived category of bounded complexes of nitely generated modules over an artin algebra of nite global dimension.

71 citations

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TL;DR: In this article, the authors give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids.

Abstract: We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto $$C^*$$
-algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

41 citations

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TL;DR: A survey of principal configurations on surfaces, their structural stability and further generalizations can be found in this article, where a number of recent results developed after the assimilation into the subject of concepts and problems from the qualitative theory of differential equations (QTDE) and Dynamical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions.

Abstract: This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations (QTDE), founded by Poincar´e in 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynamical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections.

29 citations

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TL;DR: In this article, a large-scale computational experiment was conducted to investigate the structure of the number of real solutions to Osculating instances of Schubert problems, and a family of Grassmannian-based problems with upper bounds, lower bounds, gaps, and congruence modulo four was investigated.

Abstract: We describe a large-scale computational experiment study- ing structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems whose computed numbers of real solutions variously exhibit nontriv- ial upper bounds, lower bounds, gaps, and a congruence modulo four. We present a family of Schubert problems, one in each Grassmannian, and prove that their real osculating instances have the observed lower bounds and gaps.

24 citations