MD  Matemàtica Discreta
http://hdl.handle.net/2117/3546
Fri, 29 Jul 2016 19:59:05 GMT
20160729T19:59:05Z

Polygons as sections of higherdimensional polytopes
http://hdl.handle.net/2117/86389
Polygons as sections of higherdimensional polytopes
Padrol Sureda, Arnau; Pfeifle, Julián
We show that every heptagon is a section of a 3polytope with 6 vertices. This implies that every ngon 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).
Thu, 28 Apr 2016 14:55:10 GMT
http://hdl.handle.net/2117/86389
20160428T14:55:10Z
Padrol Sureda, Arnau
Pfeifle, Julián
We show that every heptagon is a section of a 3polytope with 6 vertices. This implies that every ngon 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 noncrossing acyclic graphs
http://hdl.handle.net/2117/86044
Lower bounds on the maximum number of noncrossing 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 noncrossing spanning trees and forests. We show that the socalled double chain point configuration of N points has Omega (12.52(N)) noncrossing spanning trees and Omega (13.61(N)) noncrossing forests. This improves the previous lower bounds on the maximum number of noncrossing spanning trees and of noncrossing 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 noncrossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.
Thu, 21 Apr 2016 10:27:51 GMT
http://hdl.handle.net/2117/86044
20160421T10:27:51Z
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 noncrossing spanning trees and forests. We show that the socalled double chain point configuration of N points has Omega (12.52(N)) noncrossing spanning trees and Omega (13.61(N)) noncrossing forests. This improves the previous lower bounds on the maximum number of noncrossing spanning trees and of noncrossing 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 noncrossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.

Finite automata for Schreier graphs of virtually free groups
http://hdl.handle.net/2117/83851
Finite automata for Schreier graphs of virtually free groups
Silva, Pedro V.; Soler Escrivà, Xaro; Ventura Capell, Enric
The Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on BassSerre theory. Complexity issues and applications are also discussed.
Fri, 04 Mar 2016 17:02:48 GMT
http://hdl.handle.net/2117/83851
20160304T17:02:48Z
Silva, Pedro V.
Soler Escrivà, Xaro
Ventura Capell, Enric
The Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on BassSerre theory. Complexity issues and applications are also discussed.

On the probability of planarity of a random graph near the critical point
http://hdl.handle.net/2117/83733
On the probability of planarity of a random graph near the critical point
Noy Serrano, Marcos; Ravelomanana, Vlady; Rue, Juanjo
Let G(n, M) be the uniform random graph with n vertices and M edges. Erdos and Renyi (1960) conjectured that the limiting probability; lim(n >infinity) Pr{G(n, n/2) is planar}; exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture, and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each lambda, we find an exact analytic expression for; p(lambda) = lim(n >infinity) Pr {G (n, n/2 (1 + lambda(n1/3))) is planar}.; In particular, we obtain p(0) approximate to 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n/2) being seriesparallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.
Wed, 02 Mar 2016 18:49:21 GMT
http://hdl.handle.net/2117/83733
20160302T18:49:21Z
Noy Serrano, Marcos
Ravelomanana, Vlady
Rue, Juanjo
Let G(n, M) be the uniform random graph with n vertices and M edges. Erdos and Renyi (1960) conjectured that the limiting probability; lim(n >infinity) Pr{G(n, n/2) is planar}; exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture, and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each lambda, we find an exact analytic expression for; p(lambda) = lim(n >infinity) Pr {G (n, n/2 (1 + lambda(n1/3))) is planar}.; In particular, we obtain p(0) approximate to 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n/2) being seriesparallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.

The conjugacy problem for freebycyclic groups
http://hdl.handle.net/2117/79985
The conjugacy problem for freebycyclic groups
Martino, Armando; Ventura Capell, Enric
We show that the conjugacy problem is solvable in [finitely
generated free]bycyclic groups, by using a result of O. Maslakova
that one can algorithmically find generating sets for the fixed sub
groups of free group automorphisms, and one of P. Brinkmann that
one can determine whether two cyclic words in a free group are
mapped to each other by some power of a given automorphism. The
algorithm effectively computes a conjugating element, if it exists. We
also solve the power conjugacy problem and give an algorithm to rec
ognize if two given elements of a finitely generated free group are
Reidemeister equivalent with respect to a given automorphism.
Thu, 26 Nov 2015 18:18:08 GMT
http://hdl.handle.net/2117/79985
20151126T18:18:08Z
Martino, Armando
Ventura Capell, Enric
We show that the conjugacy problem is solvable in [finitely
generated free]bycyclic groups, by using a result of O. Maslakova
that one can algorithmically find generating sets for the fixed sub
groups of free group automorphisms, and one of P. Brinkmann that
one can determine whether two cyclic words in a free group are
mapped to each other by some power of a given automorphism. The
algorithm effectively computes a conjugating element, if it exists. We
also solve the power conjugacy problem and give an algorithm to rec
ognize if two given elements of a finitely generated free group are
Reidemeister equivalent with respect to a given automorphism.

The automorphism group of a freebycyclic group in rank 2
http://hdl.handle.net/2117/79983
The automorphism group of a freebycyclic group in rank 2
Bogopolski, Oleg; Martino, Armando; Ventura Capell, Enric
Let
¡
be an automorphism of a free group
F
2
of rank
2 and let
M
¡
=
F
2
o
¡
Z
be the corresponding mapping torus of
¡
.
We prove that the group
Out
(
M
¡
) is usually virtually cyclic. More
over, we classify the cases when this group is Ønite depending on the
conjugacy class of the image of
¡
in
GL
2
(
Z
)
Thu, 26 Nov 2015 17:54:44 GMT
http://hdl.handle.net/2117/79983
20151126T17:54:44Z
Bogopolski, Oleg
Martino, Armando
Ventura Capell, Enric
Let
¡
be an automorphism of a free group
F
2
of rank
2 and let
M
¡
=
F
2
o
¡
Z
be the corresponding mapping torus of
¡
.
We prove that the group
Out
(
M
¡
) is usually virtually cyclic. More
over, we classify the cases when this group is Ønite depending on the
conjugacy class of the image of
¡
in
GL
2
(
Z
)

Absolutetype shaft encoding using LFSR sequences with a prescribed length
http://hdl.handle.net/2117/79981
Absolutetype shaft encoding using LFSR sequences with a prescribed length
Fuertes Armengol, José Mª; Balle Pigem, Borja de; Ventura Capell, Enric
Maximallength binary sequences have existed for a long time. They have many interesting properties, and one of them is that, when taken in blocks of n consecutive positions, they form 2n  1 different codes in a closed circular sequence. This property can be used to measure absolute angular positions as the circle can be divided into as many parts as different codes can be retrieved. This paper describes how a closed binary sequence with an arbitrary length can be effectively designed with the minimal possible block length using linear feedback shift registers. Such sequences can be used to measure a specified exact number of angular positions using the minimal possible number of sensors that linear methods allow.
Thu, 26 Nov 2015 17:41:12 GMT
http://hdl.handle.net/2117/79981
20151126T17:41:12Z
Fuertes Armengol, José Mª
Balle Pigem, Borja de
Ventura Capell, Enric
Maximallength binary sequences have existed for a long time. They have many interesting properties, and one of them is that, when taken in blocks of n consecutive positions, they form 2n  1 different codes in a closed circular sequence. This property can be used to measure absolute angular positions as the circle can be divided into as many parts as different codes can be retrieved. This paper describes how a closed binary sequence with an arbitrary length can be effectively designed with the minimal possible block length using linear feedback shift registers. Such sequences can be used to measure a specified exact number of angular positions using the minimal possible number of sensors that linear methods allow.

Absolute type shaft encoding using LFSR sequences with prescribed length
http://hdl.handle.net/2117/79980
Absolute type shaft encoding using LFSR sequences with prescribed length
Fuertes Armengol, José Mª; Balle, Borja; Ventura Capell, Enric
Maximallength binary sequences have been known for
a long time. They have many interesting properties, one of them
is that when taken in blocks of
n
consecutive positions they form
2
n
°
1 diÆerent codes in a closed circular sequence. This property
can be used for measuring absolute angular positions as the circle
can be divided in as many parts as diÆerent codes can be retrieved.
This paper describes how can a closed binary sequence with arbitrary
length be eÆectively designed with the minimal possible blocklength,
using
linear feedback shift registers
(LFSR). Such sequences can be
used for measuring a speciØed exact number of angular positions,
using the minimal possible number of sensors that linear methods
allow
Thu, 26 Nov 2015 17:28:19 GMT
http://hdl.handle.net/2117/79980
20151126T17:28:19Z
Fuertes Armengol, José Mª
Balle, Borja
Ventura Capell, Enric
Maximallength binary sequences have been known for
a long time. They have many interesting properties, one of them
is that when taken in blocks of
n
consecutive positions they form
2
n
°
1 diÆerent codes in a closed circular sequence. This property
can be used for measuring absolute angular positions as the circle
can be divided in as many parts as diÆerent codes can be retrieved.
This paper describes how can a closed binary sequence with arbitrary
length be eÆectively designed with the minimal possible blocklength,
using
linear feedback shift registers
(LFSR). Such sequences can be
used for measuring a speciØed exact number of angular positions,
using the minimal possible number of sensors that linear methods
allow

Completion and decomposition of a clutter into representable matroids
http://hdl.handle.net/2117/78129
Completion and decomposition of a clutter into representable matroids
Martí Farré, Jaume; Mier Vinué, Anna de
This paper deals with the question of completing a monotone increasing family of subsets Gamma of a finite set Omega to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we prove that such vectorial completions of the family of subsets Gamma exist and, in addition, we show that the minimal vectorial completions of the family Gamma provide a decomposition of the clutter Lambda of the inclusionminimal elements of Gamma. The computation of such vectorial decomposition of clutters is also discussed in some cases. (C) 2015 Elsevier Inc. All rights reserved.
Thu, 22 Oct 2015 11:23:00 GMT
http://hdl.handle.net/2117/78129
20151022T11:23:00Z
Martí Farré, Jaume
Mier Vinué, Anna de
This paper deals with the question of completing a monotone increasing family of subsets Gamma of a finite set Omega to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we prove that such vectorial completions of the family of subsets Gamma exist and, in addition, we show that the minimal vectorial completions of the family Gamma provide a decomposition of the clutter Lambda of the inclusionminimal elements of Gamma. The computation of such vectorial decomposition of clutters is also discussed in some cases. (C) 2015 Elsevier Inc. All rights reserved.

Fixed subgroups are compressed in surface groups
http://hdl.handle.net/2117/77518
Fixed subgroups are compressed in surface groups
Zhang, Qiang; Ventura Capell, Enric; Wu, Jianchun
For a compact surface Sigma (orientable or not, and with boundary or not), we show that the fixed subgroup, Fix B, of any family B of endomorphisms of pi(1)(Sigma) is compressed in pi(1)(Sigma), i.e. rk(Fix B) <= rk(H) for any subgroup FixB <= H <= pi(1)(Sigma). On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, G, of finitely many free and surface groups, and give a characterization of when G satisfies that rk(Fix phi) <= rk(G) for every phi is an element of Aut(G).
Electronic version of an article published as International journal of algebra and computation, Vol. 25 (5), 2015, p. 865887
DOI: 10.1142/S0218196715500228 © [copyright World Scientific Publishing Company]
Thu, 08 Oct 2015 14:23:23 GMT
http://hdl.handle.net/2117/77518
20151008T14:23:23Z
Zhang, Qiang
Ventura Capell, Enric
Wu, Jianchun
For a compact surface Sigma (orientable or not, and with boundary or not), we show that the fixed subgroup, Fix B, of any family B of endomorphisms of pi(1)(Sigma) is compressed in pi(1)(Sigma), i.e. rk(Fix B) <= rk(H) for any subgroup FixB <= H <= pi(1)(Sigma). On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, G, of finitely many free and surface groups, and give a characterization of when G satisfies that rk(Fix phi) <= rk(G) for every phi is an element of Aut(G).