MD - Matemàtica Discreta
http://hdl.handle.net/2117/3546
2017-03-29T11:16:30ZThe conjugacy problem in extensions of Thompson's group F
http://hdl.handle.net/2117/102759
The conjugacy problem in extensions of Thompson's group F
Burillo Puig, José; Matucci, Francesco; Ventura Capell, Enric
We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Gonçalves in [4] showing that F has property R8, and which can be extended to show that Thompson’s group T also has property R8.
The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-016-1403-9
2017-03-21T17:43:46ZBurillo Puig, JoséMatucci, FrancescoVentura Capell, EnricWe solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Gonçalves in [4] showing that F has property R8, and which can be extended to show that Thompson’s group T also has property R8.Extreme weights in Steinhaus triangles
http://hdl.handle.net/2117/102012
Extreme weights in Steinhaus triangles
Brunat Blay, Josep M.; Maureso Sánchez, Montserrat
Let {0=w0<w1<w2<…<wm0=w0<w1<w2<…<wm} be the set of weights of binary Steinhaus triangles of size n , and let Wibe the set of sequences in F2n that generate triangles of weight wi. In this paper we obtain the values of wi and the corresponding sets Wi for i¿{2,3,m}i¿{2,3,m}, and partial results for i=m-1i=m-1.
2017-03-07T09:28:43ZBrunat Blay, Josep M.Maureso Sánchez, MontserratLet {0=w0<w1<w2<…<wm0=w0<w1<w2<…<wm} be the set of weights of binary Steinhaus triangles of size n , and let Wibe the set of sequences in F2n that generate triangles of weight wi. In this paper we obtain the values of wi and the corresponding sets Wi for i¿{2,3,m}i¿{2,3,m}, and partial results for i=m-1i=m-1.Degree of commutativity of infinite groups
http://hdl.handle.net/2117/101873
Degree of commutativity of infinite groups
Antolin, Yago; Martino, Armando; Ventura Capell, Enric
We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.
First published in Proceedings of the American Mathematical Society in volum 145, number 2, 2016, published by the American Mathematical Society
2017-03-02T15:58:36ZAntolin, YagoMartino, ArmandoVentura Capell, EnricWe prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.Perspectivas en combinatoria
http://hdl.handle.net/2117/101761
Perspectivas en combinatoria
Noy Serrano, Marcos; Serra Albó, Oriol
2017-03-01T09:06:19ZNoy Serrano, MarcosSerra Albó, OriolComputing the canonical representation of constructible sets
http://hdl.handle.net/2117/100350
Computing the canonical representation of constructible sets
Brunat Blay, Josep Maria; Montes Lozano, Antonio
Constructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.
2017-01-31T09:20:18ZBrunat Blay, Josep MariaMontes Lozano, AntonioConstructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.Bounding the gap between a free group (outer) automorphism and its inverse
http://hdl.handle.net/2117/98389
Bounding the gap between a free group (outer) automorphism and its inverse
Ladra, Manuel; Silva, Pedro V.; Ventura Capell, Enric
For any finitely generated group GG , two complexity functions aGaG and ßGßG are defined to measure the maximal possible gap between the norm of an automorphism (respectively, outer automorphism) of GG and the norm of its inverse. Restricting attention to free groups FrFr , the exact asymptotic behaviour of a2a2 and ß2ß2 is computed. For rank r¿3r¿3 , polynomial lower bounds are provided for arar and ßrßr , and the existence of a polynomial upper bound is proved for ßrßr .
The final publication is available at Springer via http://dx.doi.org/10.1007/s13348-015-0133-3.
2016-12-15T18:01:37ZLadra, ManuelSilva, Pedro V.Ventura Capell, EnricFor any finitely generated group GG , two complexity functions aGaG and ßGßG are defined to measure the maximal possible gap between the norm of an automorphism (respectively, outer automorphism) of GG and the norm of its inverse. Restricting attention to free groups FrFr , the exact asymptotic behaviour of a2a2 and ß2ß2 is computed. For rank r¿3r¿3 , polynomial lower bounds are provided for arar and ßrßr , and the existence of a polynomial upper bound is proved for ßrßr .Algorithmic recognition of infinite cyclic extensions
http://hdl.handle.net/2117/96690
Algorithmic recognition of infinite cyclic extensions
Cavallo, Bren; Delgado Rodríguez, Jordi; Kahrobaei, Delaram; Ventura Capell, Enric
We prove that one cannot algorithmically decide whether a finitely presented Z-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for deranged outer automorphisms.
2016-11-15T16:32:01ZCavallo, BrenDelgado Rodríguez, JordiKahrobaei, DelaramVentura Capell, EnricWe prove that one cannot algorithmically decide whether a finitely presented Z-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique Z-extensions, and the semi-conjugacy problem for deranged outer automorphisms.Put three and three together: Triangle-driven community detection
http://hdl.handle.net/2117/89696
Put three and three together: Triangle-driven community detection
Prat Pérez, Arnau; Domínguez Sal, David; Brunat Blay, Josep Maria; Larriba Pey, Josep
Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its applications in many fields such as biology, social networks, or network traffic analysis. Although the existing metrics used to quantify the quality of a community work well in general, under some circumstances, they fail at correctly capturing such notion. The main reason is that these metrics consider the internal community edges as a set, but ignore how these actually connect the vertices of the community. We propose the Weighted Community Clustering (WCC), which is a new community metric that takes the triangle instead of the edge as the minimal structural motif indicating the presence of a strong relation in a graph. We theoretically analyse WCC in depth and formally prove, by means of a set of properties, that the maximization of WCC guarantees communities with cohesion and structure. In addition, we propose Scalable Community Detection (SCD), a community detection algorithm based on WCC, which is designed to be fast and scalable on SMP machines, showing experimentally that WCC correctly captures the concept of community in social networks using real datasets. Finally, using ground-truth data, we show that SCD provides better quality than the best disjoint community detection algorithms of the state of the art while performing faster.
2016-09-08T08:14:33ZPrat Pérez, ArnauDomínguez Sal, DavidBrunat Blay, Josep MariaLarriba Pey, JosepCommunity detection has arisen as one of the most relevant topics in the field of graph data mining due to its applications in many fields such as biology, social networks, or network traffic analysis. Although the existing metrics used to quantify the quality of a community work well in general, under some circumstances, they fail at correctly capturing such notion. The main reason is that these metrics consider the internal community edges as a set, but ignore how these actually connect the vertices of the community. We propose the Weighted Community Clustering (WCC), which is a new community metric that takes the triangle instead of the edge as the minimal structural motif indicating the presence of a strong relation in a graph. We theoretically analyse WCC in depth and formally prove, by means of a set of properties, that the maximization of WCC guarantees communities with cohesion and structure. In addition, we propose Scalable Community Detection (SCD), a community detection algorithm based on WCC, which is designed to be fast and scalable on SMP machines, showing experimentally that WCC correctly captures the concept of community in social networks using real datasets. Finally, using ground-truth data, we show that SCD provides better quality than the best disjoint community detection algorithms of the state of the art while performing faster.Polygons as sections of higher-dimensional polytopes
http://hdl.handle.net/2117/86389
Polygons as sections of higher-dimensional polytopes
Padrol Sureda, Arnau; Pfeifle, Julián
We show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n >= 7 can be obtained as a section of a (2 + [n/7])-dimensional polytope with at most [6n/7] vertices; and provides a geometric proof of the fact that every nonnegative n x rn matrix of rank 3 has nonnegative rank not larger than [6min(n,m)/7]. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).
2016-04-28T14:55:10ZPadrol Sureda, ArnauPfeifle, JuliánWe show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n >= 7 can be obtained as a section of a (2 + [n/7])-dimensional polytope with at most [6n/7] vertices; and provides a geometric proof of the fact that every nonnegative n x rn matrix of rank 3 has nonnegative rank not larger than [6min(n,m)/7]. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).Lower bounds on the maximum number of non-crossing acyclic graphs
http://hdl.handle.net/2117/86044
Lower bounds on the maximum number of non-crossing acyclic graphs
Huemer, Clemens; Mier Vinué, Anna de
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega (12.52(N)) non-crossing spanning trees and Omega (13.61(N)) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12(N)) for the number of non-crossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.
2016-04-21T10:27:51ZHuemer, ClemensMier Vinué, Anna deThis paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega (12.52(N)) non-crossing spanning trees and Omega (13.61(N)) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12(N)) for the number of non-crossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.