Irreducible morphisms in the bounded derived category

AbstractWe study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ, denoted Db(Λmod), by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in Cb(Λproj) either fj is split monomorphism for all j∈Z or split epimorphism, for all j∈Z. Moreover, we prove that all the non-trivial components of the Auslander–Reiten quiver of Cb(Λproj) are of the form ZA∞

Graphs with isolation number equal to one third of the order

A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is known that $\iota (G)\le n/3$, if $G$ is a connected graph of order $n$, $n\ge 3$, distinct from $C_5$. The main result of this work is the characterisation of unicyclic and block graphs of order $n$ with isolating number equal to $n/3$. Moreover, we provide a family of general graphs attaining this upper bound on the isolation number.Comment: 15 pages, 12 figure

As dificultades de incorporación das mulleres matemáticas á universidade

[RESUMO] Votamos una ollada as primeiras mulleres matemáticas no eido universitario en Europa dende finais do século XIX, analizamos tamén a situación actual en datos porcentuais facendo especial fincapé nas mulleres españolas e nas galegas. Destacamos as singularidades dalgunhas mulleres que nun ámbito desfavorable sobresaíron polos seus logros académicos e/ou científicos

On the Degree in Categories of Complexes of Fixed Size

This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s10485-019-09557-x.[Abstract]: We consider Λ an artin algebra and n ≥ 2. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander-Reiten component of Cn(proj Λ) with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in Cn(proj Λ) belong to such a category. For a finite dimensional hereditary algebra H over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which Cn(proj H) is of finite type.The first and second authors thankfully acknowledge partial support from CONICET and EXA558/14 from Universidad Nacional de Mar del Plata, Argentina. The third author thankfully acknowledge support from Ministerio Español de Economía y Competitividad and FEDER (FFI2014-51978-C2-2-R). The first author is a researcher from CONICET.Argentina. Universidad Nacional de Mar del Plata; EXA558/14Cn(proj Λ) H n ≥ 2

Isolation Number versus Domination Number of Trees

[Abstract] If = (Vɢ,Eɢ) is a graph of order n, we call ⊆ Vɢ an isolating set if the graph induced by Vɢ − Nɢ[] contains no edges. The minimum cardinality of an isolating set of is called the isolation number of , and it is denoted by (). It is known that () ≤ ⁿ⁄₃ and the bound is sharp. A subset ⊆ Vɢ is called dominating in if Nɢ[] = Vɢ. The minimum cardinality of a dominating set of is the domination number, and it is denoted by (). In this paper, we analyze a family of trees where () = (), and we prove that (T) = ⁿ⁄₃ implies () = (). Moreover, we give different equivalent characterizations of such graphs and we propose simple algorithms to build these trees from the connections of stars.CITIC, as Research Center accredited by Galician University System, is funded by "Consellería de Cultura, Educación e Universidade from Xunta de Galicia", supported in an 80% through ERDF Funds, ERDF Operational Programme Galicia 2014-2020, and the remaining 20% by "Secretaría Xeral de Universidades (Grant ED431G 2019/01). This research was also funded by Agencia Estatal de Investigación of Spain (PID2019-104958RB-C42 and TIN2017-85160-C2-1-R) and ERDF funds of the EU (AEI/FEDER, UE).Xunta de Galicia; ED431G 2019/0

Resolving Sets Tolerant to Failures in Three-Dimensional Grids

[Abstract] An ordered set S of vertices of a graph G is a resolving set for G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any k vertices from the set. This is equivalent to finding (k+1)-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differs in at least k+1 coordinates. This problem is also related with the study of the (k+1)-metric dimension of a graph, defined as the minimum cardinality of a (k+1)-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of k≥1 for which there exists a (k+1)-resolving set and construct such a resolving set of minimum cardinality in almost all cases.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. M. Mora is supported by projects H2020-MSCA-RISE-2016-734922 CONNECT, PID2019-104129GB-I00/MCIN/AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation and Gen.Cat. DGR2017SGR1336; M. J. Souto-Salorio is supported by project PID2020-113230RB-C21 of the Spanish Ministry of Science and Innovation. Open Access funding provided thanks to the CRUECSIC agreement with Springer NatureGeneralitat de Catalunya; DGR2017SGR133

Resolving Sets Tolerant to Failures in Three-Dimensional Grids

[Abstract] An ordered set S of vertices of a graph G is a resolving set for G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any k vertices from the set. This is equivalent to finding (k+1)-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differs in at least k+1 coordinates. This problem is also related with the study of the (k+1)-metric dimension of a graph, defined as the minimum cardinality of a (k+1)-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of k≥1 for which there exists a (k+1)-resolving set and construct such a resolving set of minimum cardinality in almost all cases.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. M. Mora is supported by projects H2020-MSCA-RISE-2016-734922 CONNECT, PID2019-104129GB-I00/MCIN/AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation and Gen.Cat. DGR2017SGR1336; M. J. Souto-Salorio is supported by project PID2020-113230RB-C21 of the Spanish Ministry of Science and Innovation. Open Access funding provided thanks to the CRUECSIC agreement with Springer NatureGeneralitat de Catalunya; DGR2017SGR133

The Auslander-Reiten quiver of the category of m-periodic complexes

Let $\mathcal{A}$ be an additive $k-$category and $\mathbf{C}_{\equiv m}(\mathcal{A})$ be the category of $m-$periodic objects. For any integer $m>1$, we study conditions under which the compression functor ${\mathcal F}_m :\mathbf{C}^{b}(\mathcal{A}) \rightarrow \mathbf{C}_{\equiv m}(\mathcal{A})$ preserves or reflects irreducible morphisms. Moreover, we find sufficient conditions for the functor ${\mathcal F}_m$ to be a Galois $G$-covering in the sense of \cite{BL}. If in addition $\mathcal{A}$ is a dualizing category and \mbox{mod}\, \mathcal{A} has finite global dimension then $\mathbf{C}_{\equiv m}(\mathcal{A})$ has almost split sequences. In particular, for a finite dimensional algebra $A$ with finite strong global dimension we determine how to build the Auslander-Reiten quiver of the category \mathbf{C}_{\equiv m}(\mbox{proj}\, A). Furthermore, we study the behavior of sectional paths in \mathbf{C}_{\equiv m}(\mbox{proj}\, A), whenever $A$ is any finite dimensional $k-$algebra over a field $k$.Comment: 24 page

Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/. This version of the article Brozos-Vázquez, M., Pereira-Sáez, M. J., Souto-Salorio, M. J., & Tarrío-Tobar, A. D. (2019). “Classification of the relative positions between a small ellipsoid and an elliptic paraboloid” has been accepted for publication in Computer Aided Geometric Design, 72, 34–48. The Version of Record is available online at https://doi.org/10.1016/j.cagd.2019.05.002.[Abstract]: We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid (small meaning that the two surfaces cannot be tangent at two points simultaneously when one is moved with respect to the other). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.The authors wish to thank the referees for extremely valuable comments and suggestions, which were essential to improve the final version of the paper. Supported by Projects ED431F 2017/03, TIN2017-85160-C2-1-R, MTM2016-75897-P and MTM2016-78647-P (AEI/FEDER, UE).Xunta de Galicia; ED431F 2017/0