Pub Date : 2019-01-01DOI: 10.15446/RECOLMA.V53N1.81045
F. Pipicano, Juan Carlos Muñoz Grajales
We establish existence of periodic standing waves for a model to describe the propagation of a light pulse inside an optical fiber taking into account the Kerr effect. To this end, we apply the Lyapunov Center Theorem taking advantage that the corresponding standing wave equations can be rewritten as a Hamiltonian system. Furthermore, some of these solutions are approximated by using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the system of standing wave equations. Our numerical simulations are found to be in accordance with our analytical results.
{"title":"Existence of periodic standing wave solutions for a system describing pulse propagation in an optical fiber","authors":"F. Pipicano, Juan Carlos Muñoz Grajales","doi":"10.15446/RECOLMA.V53N1.81045","DOIUrl":"https://doi.org/10.15446/RECOLMA.V53N1.81045","url":null,"abstract":"We establish existence of periodic standing waves for a model to describe the propagation of a light pulse inside an optical fiber taking into account the Kerr effect. To this end, we apply the Lyapunov Center Theorem taking advantage that the corresponding standing wave equations can be rewritten as a Hamiltonian system. Furthermore, some of these solutions are approximated by using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the system of standing wave equations. Our numerical simulations are found to be in accordance with our analytical results.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.15446/RECOLMA.V53N1.81045","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45376477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.15446/RECOLMA.V53N1.81044
H. Boua, M. Karmouni, A. Tajmouati
In this paper, we show a spectral inclusion of integrated semigroups for Saphar, essentially Saphar and quasi-Fredholm spectra.
在本文中，我们证明了Saphar的积分半群的谱包涵，本质上是Saphar和准fredholm谱。
{"title":"Spectral Inclusions for Quasi-Fredholm and Saphar Spectra for Integrated Semigroups","authors":"H. Boua, M. Karmouni, A. Tajmouati","doi":"10.15446/RECOLMA.V53N1.81044","DOIUrl":"https://doi.org/10.15446/RECOLMA.V53N1.81044","url":null,"abstract":"In this paper, we show a spectral inclusion of integrated semigroups for Saphar, essentially Saphar and quasi-Fredholm spectra.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.15446/RECOLMA.V53N1.81044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48040172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-07DOI: 10.15446/recolma.v53nsupl.84010
Pierre Clavier, Li Guo, S. Paycha, Bin Zhang
This is a survey on renormalisation in algebraic locality setup highlighting the role that locality morphisms can play for renormalisation purposes. After describing the general framework to build locality regularisation maps, we illustrate renormalisation by locality algebra homomorphisms on three examples, the renormalisation of conical zeta functions at poles, the definition of branched zeta functions and their evaluation at poles and finally the values of iterated integrals stemming from Kreimer's toy model.
{"title":"Renormalisation via locality morphisms","authors":"Pierre Clavier, Li Guo, S. Paycha, Bin Zhang","doi":"10.15446/recolma.v53nsupl.84010","DOIUrl":"https://doi.org/10.15446/recolma.v53nsupl.84010","url":null,"abstract":"This is a survey on renormalisation in algebraic locality setup highlighting the role that locality morphisms can play for renormalisation purposes. After describing the general framework to build locality regularisation maps, we illustrate renormalisation by locality algebra homomorphisms on three examples, the renormalisation of conical zeta functions at poles, the definition of branched zeta functions and their evaluation at poles and finally the values of iterated integrals stemming from Kreimer's toy model.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.15446/recolma.v53nsupl.84010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43251363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-01DOI: 10.15446/recolma.v53nsupl.84013
W. Santos
The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in geometric invariant theory. The concept of observable subgroup was introduced in the early 1960s with the purpose of studying extensions of representations from an afine algebraic subgroup to the whole group. The extent of its importance in representation and invariant theory in particular for Hilbert's 14th problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of strong observability was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of observable action of an afine algebraic group on an afine variety, launching a series of new applications and opening a surge of very interesting activity. In another direction around 2006, the related concept of observable adjunction was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.
{"title":"A short survey on observability","authors":"W. Santos","doi":"10.15446/recolma.v53nsupl.84013","DOIUrl":"https://doi.org/10.15446/recolma.v53nsupl.84013","url":null,"abstract":"The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in geometric invariant theory. The concept of observable subgroup was introduced in the early 1960s with the purpose of studying extensions of representations from an afine algebraic subgroup to the whole group. The extent of its importance in representation and invariant theory in particular for Hilbert's 14th problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of strong observability was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of observable action of an afine algebraic group on an afine variety, launching a series of new applications and opening a surge of very interesting activity. In another direction around 2006, the related concept of observable adjunction was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.15446/recolma.v53nsupl.84013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44167496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-07DOI: 10.15446/recolma.v53nsupl.84099
M. Wilson
In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra.
{"title":"Connes-Landi spheres are homogeneous spaces","authors":"M. Wilson","doi":"10.15446/recolma.v53nsupl.84099","DOIUrl":"https://doi.org/10.15446/recolma.v53nsupl.84099","url":null,"abstract":"In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.15446/recolma.v53nsupl.84099","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47087749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-05DOI: 10.15446/recolma.v54n1.89776
Jose Andr'es Quintero, C. Uzc'ategui
We study the concept of a premetric space introduced by F. Richman in the context of constructive mathematics, and present a method for completing them.
我们研究了F.Richman在构造数学中引入的预度量空间的概念，并提出了一种完成它们的方法。
{"title":"Completion of premetric spaces","authors":"Jose Andr'es Quintero, C. Uzc'ategui","doi":"10.15446/recolma.v54n1.89776","DOIUrl":"https://doi.org/10.15446/recolma.v54n1.89776","url":null,"abstract":"We study the concept of a premetric space introduced by F. Richman in the context of constructive mathematics, and present a method for completing them.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47210546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-30DOI: 10.15446/recolma.v54n1.89777
Christian Weiss
The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places n points on a circle, at angles of z, 2z, … nz from the starting point. The theorem was first proven in 1958 by Sós and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.
{"title":"Deducing Three Gap Theorem from Rauzy-Veech induction","authors":"Christian Weiss","doi":"10.15446/recolma.v54n1.89777","DOIUrl":"https://doi.org/10.15446/recolma.v54n1.89777","url":null,"abstract":"The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places n points on a circle, at angles of z, 2z, … nz from the starting point. The theorem was first proven in 1958 by Sós and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47840764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-18DOI: 10.15446/recolma.v53nsupl.83958
I. Angiono, Agust'in Garc'ia Iglesias
This article serves a two-fold purpose. On the one hand, it is asurvey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of Cartan type B2.
{"title":"Pointed Hopf algebras: a guided tour to the liftings","authors":"I. Angiono, Agust'in Garc'ia Iglesias","doi":"10.15446/recolma.v53nsupl.83958","DOIUrl":"https://doi.org/10.15446/recolma.v53nsupl.83958","url":null,"abstract":"This article serves a two-fold purpose. On the one hand, it is asurvey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of Cartan type B2.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.15446/recolma.v53nsupl.83958","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44827149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-01DOI: 10.15446/RECOLMA.V52N2.77153
J. C. Martínez
By using an argument based on the structure of the locally compact scattered spaces, we prove in a direct way the following result shown by Jech and Shelah: there is a family {Bα : α < ω1} of subsets of ω1 such that the following conditions are satisfied: (a) max Bα - α, (b) if α ∈ Bβ then Bα ⊆ Bβ, (c) if δ ≤ α and δ is a limit ordinal then Bα ∩ δ is not in the ideal generated by the sets Bβ, β < α, and by the bounded subsets of δ, (d) there is a partition {An : n ∈ ω} of ω1 such that for every α and every n, Bα ∩An is finite.
{"title":"A direct proof of a theorem of Jech and Shelah on PCF algebras","authors":"J. C. Martínez","doi":"10.15446/RECOLMA.V52N2.77153","DOIUrl":"https://doi.org/10.15446/RECOLMA.V52N2.77153","url":null,"abstract":"By using an argument based on the structure of the locally compact scattered spaces, we prove in a direct way the following result shown by Jech and Shelah: there is a family {Bα : α < ω1} of subsets of ω1 such that the following conditions are satisfied: (a) max Bα - α, (b) if α ∈ Bβ then Bα ⊆ Bβ, (c) if δ ≤ α and δ is a limit ordinal then Bα ∩ δ is not in the ideal generated by the sets Bβ, β < α, and by the bounded subsets of δ, (d) there is a partition {An : n ∈ ω} of ω1 such that for every α and every n, Bα ∩An is finite.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":"23 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41248487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-01DOI: 10.15446/RECOLMA.V52N2.77158
Víctor L. Fernández, Carina Murciano
In this paper we show a method to characterize logical matrices by means of a special kind of structures, called here discriminant structures for this purpose. Its definition is based on the discrimination of each truthvalue of a given (finite) matrix M = (A, D), w.r.t. its belonging to D. From this starting point, we define a whole class SM of discriminant structures. This class is characterized by a set of Boolean equations, as it is shown here. In addition, several technical results are presented, and it is emphasized the relation of the Discriminant Structures Semantics (D.S.S) with other related semantics such as Dyadic or Twist-Structure.
{"title":"Discriminant Structures Associated to Matrix Semantics","authors":"Víctor L. Fernández, Carina Murciano","doi":"10.15446/RECOLMA.V52N2.77158","DOIUrl":"https://doi.org/10.15446/RECOLMA.V52N2.77158","url":null,"abstract":"In this paper we show a method to characterize logical matrices by means of a special kind of structures, called here discriminant structures for this purpose. Its definition is based on the discrimination of each truthvalue of a given (finite) matrix M = (A, D), w.r.t. its belonging to D. From this starting point, we define a whole class SM of discriminant structures. This class is characterized by a set of Boolean equations, as it is shown here. In addition, several technical results are presented, and it is emphasized the relation of the Discriminant Structures Semantics (D.S.S) with other related semantics such as Dyadic or Twist-Structure.","PeriodicalId":38102,"journal":{"name":"Revista Colombiana de Matematicas","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46003312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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