We prove that Dirichlet stereohedra for cubic crystallographic groups cannot have more than 105 f... more We prove that Dirichlet stereohedra for cubic crystallographic groups cannot have more than 105 facets. This improves a previous bound of 162 [3].
We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-d... more We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. In two previous papers, D. Bochiş and the second author studied the problem for noncubic groups. This paper deals with "full" cubic groups, while "quarter" cubic groups are left for a subsequent paper. Here, "full" and "quarter" refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston. This paper's main result is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups.
El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más int... more El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más interés está despertando últimamente, ya que, a lo largo de los años, se le han atribuido muchos beneficios en relación a la motivación de los alumnos y la mejora de las, ahora llamadas, competencias clave. No obstante, fuera de trabajos de investigación, ha sido poco estudiada.
Sociedad Matemática de Profesores de Cantabria, 2016
El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más int... more El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más interés está despertando últimamente, ya que, a lo largo de los años, se le han atribuido muchos beneficios en relación a la motivación de los alumnos y la mejora de las, ahora llamadas, competencias clave. No obstante, fuera de trabajos de investigación, ha sido poco estudiada.
AbstractGiven a family P ¼fP 1 ;...;P m g of m polygonal regions (possibly intersecting) in the p... more AbstractGiven a family P ¼fP 1 ;...;P m g of m polygonal regions (possibly intersecting) in the plane, we consider the problem oflocating a straight line ‘ intersecting the convex hull of P and such that min k d(P k ,‘) is maximal. We give an algorithm thatsolves the problem in time O((m 2 + nlogm)logn) using O(m 2 + n) space, where n is the total number of vertices ofP 1 ,...,P m . The previous best running time for this problem was O(n 2 ) [J. Janardan, F.P. Preparata, Widest-corridor prob-lems, Nordic Journal of Computing 1 (1994) 231–245]. We also improve the known complexity for several variants of thisproblem which include a three dimensional version – the maximin plane problem –, the weighted problem and consideringmeasuring distance different to the Euclidean one. 2006 Elsevier B.V. All rights reserved. Keywords: Location; Computational geometry; Linear facility; Duality 1. IntroductionThe advances in computational geometry havegiven rise to the development of efficient algorit...
Resumen Un diagrama de m´õnima distancia (abreviadamente DMD) para un digrafo circulante de multi... more Resumen Un diagrama de m´õnima distancia (abreviadamente DMD) para un digrafo circulante de multiple lazo es una herramienta muy utilizada para calcular el diametro y la distancia media de un digrafo circulante. Los digrafos circulantes de doble lazo han sido muy estudiados y se sabe que cada uno tiene a lo mas dos DMD's asociados, que tienen "forma de L". No obstante, en contra de lo que cabr´õa esperar, nosotros en este trabajo demostramos que existen digrafos circulantes de triple lazo con un numero arbitrario de DMD's asociados. Nuestros metodos se basan en las relaciones existentes entre DMD's e ideales monomiales, introducidos por Gomez y otros.
ABSTRACT Resumen. Probamos que los estereoedros de Dirichlet para los grupos cristalog africos tr... more ABSTRACT Resumen. Probamos que los estereoedros de Dirichlet para los grupos cristalog africos tridimensionales cúbicos "llenos" no pueden tener más de 25 caras. Experimentalmente, encontramos estereoedros con 17 caras. Palabras clave. Estereoedro, grupo cristalográfico, teselaciones monoedrales.
Given a family P = {P 1 , ..., P m } of m polygonal regions (possibly intersecting) in the plane,... more Given a family P = {P 1 , ..., P m } of m polygonal regions (possibly intersecting) in the plane, we consider the problem of locating a straight line intersecting the convex hull of P and such that min k d(P k ,) is maximal. We give an algorithm that solves the problem in time O((m 2 + n log m) log n)) using O(m 2 + n) space, where n is the total number of vertices of P 1 , ..., P m. The previous best running time for this problem was O(n 2) [21]. We also improve the known complexity for several variants of this problem which include a three dimensional version-the maximin plane problem-, the weighted problem and considering measuring distance different to the Euclidean one.
Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop ... more Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form ("L-shape"). In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.
Avances En Matematica Discreta En Andalucia V Encuentro Andaluz De Matematica Discreta La Linea De La Concepcion Julio De 2007 2007 Isbn 978 84 9828 133 0 Pags 223 230, 2007
We prove that Dirichlet stereohedra for cubic crystallographic groups cannot have more than 105 f... more We prove that Dirichlet stereohedra for cubic crystallographic groups cannot have more than 105 facets. This improves a previous bound of 162 [3].
We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-d... more We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. In two previous papers, D. Bochiş and the second author studied the problem for noncubic groups. This paper deals with "full" cubic groups, while "quarter" cubic groups are left for a subsequent paper. Here, "full" and "quarter" refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston. This paper's main result is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups.
El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más int... more El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más interés está despertando últimamente, ya que, a lo largo de los años, se le han atribuido muchos beneficios en relación a la motivación de los alumnos y la mejora de las, ahora llamadas, competencias clave. No obstante, fuera de trabajos de investigación, ha sido poco estudiada.
Sociedad Matemática de Profesores de Cantabria, 2016
El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más int... more El trabajo por proyectos es una de las herramientas para el desarrollo del currículum que más interés está despertando últimamente, ya que, a lo largo de los años, se le han atribuido muchos beneficios en relación a la motivación de los alumnos y la mejora de las, ahora llamadas, competencias clave. No obstante, fuera de trabajos de investigación, ha sido poco estudiada.
AbstractGiven a family P ¼fP 1 ;...;P m g of m polygonal regions (possibly intersecting) in the p... more AbstractGiven a family P ¼fP 1 ;...;P m g of m polygonal regions (possibly intersecting) in the plane, we consider the problem oflocating a straight line ‘ intersecting the convex hull of P and such that min k d(P k ,‘) is maximal. We give an algorithm thatsolves the problem in time O((m 2 + nlogm)logn) using O(m 2 + n) space, where n is the total number of vertices ofP 1 ,...,P m . The previous best running time for this problem was O(n 2 ) [J. Janardan, F.P. Preparata, Widest-corridor prob-lems, Nordic Journal of Computing 1 (1994) 231–245]. We also improve the known complexity for several variants of thisproblem which include a three dimensional version – the maximin plane problem –, the weighted problem and consideringmeasuring distance different to the Euclidean one. 2006 Elsevier B.V. All rights reserved. Keywords: Location; Computational geometry; Linear facility; Duality 1. IntroductionThe advances in computational geometry havegiven rise to the development of efficient algorit...
Resumen Un diagrama de m´õnima distancia (abreviadamente DMD) para un digrafo circulante de multi... more Resumen Un diagrama de m´õnima distancia (abreviadamente DMD) para un digrafo circulante de multiple lazo es una herramienta muy utilizada para calcular el diametro y la distancia media de un digrafo circulante. Los digrafos circulantes de doble lazo han sido muy estudiados y se sabe que cada uno tiene a lo mas dos DMD's asociados, que tienen "forma de L". No obstante, en contra de lo que cabr´õa esperar, nosotros en este trabajo demostramos que existen digrafos circulantes de triple lazo con un numero arbitrario de DMD's asociados. Nuestros metodos se basan en las relaciones existentes entre DMD's e ideales monomiales, introducidos por Gomez y otros.
ABSTRACT Resumen. Probamos que los estereoedros de Dirichlet para los grupos cristalog africos tr... more ABSTRACT Resumen. Probamos que los estereoedros de Dirichlet para los grupos cristalog africos tridimensionales cúbicos "llenos" no pueden tener más de 25 caras. Experimentalmente, encontramos estereoedros con 17 caras. Palabras clave. Estereoedro, grupo cristalográfico, teselaciones monoedrales.
Given a family P = {P 1 , ..., P m } of m polygonal regions (possibly intersecting) in the plane,... more Given a family P = {P 1 , ..., P m } of m polygonal regions (possibly intersecting) in the plane, we consider the problem of locating a straight line intersecting the convex hull of P and such that min k d(P k ,) is maximal. We give an algorithm that solves the problem in time O((m 2 + n log m) log n)) using O(m 2 + n) space, where n is the total number of vertices of P 1 , ..., P m. The previous best running time for this problem was O(n 2) [21]. We also improve the known complexity for several variants of this problem which include a three dimensional version-the maximin plane problem-, the weighted problem and considering measuring distance different to the Euclidean one.
Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop ... more Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form ("L-shape"). In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.
Avances En Matematica Discreta En Andalucia V Encuentro Andaluz De Matematica Discreta La Linea De La Concepcion Julio De 2007 2007 Isbn 978 84 9828 133 0 Pags 223 230, 2007
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