Carlos Rafael Payares Guevara, Fabi'an Antonio Arias Amaya
. After the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the sim- ple contragredient Lie 2-algebra G ( F 4 ,a ) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G ( F 4 ,a ).
{"title":"Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4","authors":"Carlos Rafael Payares Guevara, Fabi'an Antonio Arias Amaya","doi":"10.33044/REVUMA.1555","DOIUrl":"https://doi.org/10.33044/REVUMA.1555","url":null,"abstract":". After the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the sim- ple contragredient Lie 2-algebra G ( F 4 ,a ) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G ( F 4 ,a ).","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42542577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Let R be a commutative ring with identity, and let Z ( R ) be the set of zero-divisors of R . The weakly zero-divisor graph of R is the undirected (simple) graph W Γ( R ) with vertex set Z ( R ) ∗ , and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann( x ) and s ∈ ann( y ) such that rs = 0. It follows that W Γ( R ) contains the zero-divisor graph Γ( R ) as a subgraph. In this paper, the connectedness, diameter, and girth of W Γ( R ) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of W Γ( R ) is studied.
{"title":"The weakly zero-divisor graph of a commutative ring","authors":"M. Nikmehr, A. Azadi, R. Nikandish","doi":"10.33044/REVUMA.1677","DOIUrl":"https://doi.org/10.33044/REVUMA.1677","url":null,"abstract":". Let R be a commutative ring with identity, and let Z ( R ) be the set of zero-divisors of R . The weakly zero-divisor graph of R is the undirected (simple) graph W Γ( R ) with vertex set Z ( R ) ∗ , and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann( x ) and s ∈ ann( y ) such that rs = 0. It follows that W Γ( R ) contains the zero-divisor graph Γ( R ) as a subgraph. In this paper, the connectedness, diameter, and girth of W Γ( R ) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of W Γ( R ) is studied.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69487259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We are interested in a signed graph ˙ G which admits a decomposition into a totally disconnected (i.e., without edges) star complement and a signed graph ˙ S induced by the star set. In this study we derive certain properties of ˙ G ; for example, we prove that the number of (distinct) eigenvalues of ˙ S does not exceed the number of those of ˙ G . Some particular cases are also considered.
{"title":"Signed graphs with totally disconnected star complements","authors":"Z. Stanić","doi":"10.33044/REVUMA.1480","DOIUrl":"https://doi.org/10.33044/REVUMA.1480","url":null,"abstract":". We are interested in a signed graph ˙ G which admits a decomposition into a totally disconnected (i.e., without edges) star complement and a signed graph ˙ S induced by the star set. In this study we derive certain properties of ˙ G ; for example, we prove that the number of (distinct) eigenvalues of ˙ S does not exceed the number of those of ˙ G . Some particular cases are also considered.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45764694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For every U ⊂ Diffvol(T) there is a measure of finite support contained in U which is uniformly expanding. 0. Introduction Let μ be a probability measure in Diff(M) where M is a closed manifold of dimension d := dim(M). We denote μ(1) = μ and μ(n) = μ∗μ(n−1). Note that μ(n) is the pushforward by the composition of the product measure μ in (Diff(M))n. Definition 0.1 ([8, 4]). A probability measure μ in Diff(M) is uniformly expanding if there exists N > 0 such that for every (x, v) ∈ T 1M one has that ∫ log ‖Dxfv‖ dμ(N)(f) > 2. This is a robust1 condition on μ. This notion as well as similar ones have been studied extensively recently, as it allows one to describe quite precisely the stationary measures for random walks with μ as law (see below for more discussion). Here we will make a remark (which can be related to some results, e.g. in [3, 6, 14]) that points in the direction of the abundance of uniform expansion. Theorem 0.2. For every open set U in Diffvol(T) there is a finitely supported probability measure μ whose support is contained in U and μ is uniformly expanding. As a consequence of the results of [5, 13, 6] one deduces that: Corollary 0.3. For every U ⊂ Diffvol(T) there is a probability measure μ finitely supported in U such that the orbit of every point under the random walk on T2 produced by μ equidistributes in T2. Moreover, for every μ′ close to μ in the weak-∗ 2020 Mathematics Subject Classification. 37H15. Rafael Potrie was partially supported by CSIC 618, FCE-1-2017-1-135352. This work was started while the author was a Von Neumann fellow at IAS, funded by the Minerva Research Foundation Membership Fund and NSF DMS-1638352. 1To be precise, if μ has compact support, then there is a neighborhood U of its support such that any measure μ′ which has support in U and is weak-∗-close to μ, is also uniformly expanding (see (3.1) below).
{"title":"A remark on uniform expansion","authors":"R. Potrie","doi":"10.33044/revuma.2896","DOIUrl":"https://doi.org/10.33044/revuma.2896","url":null,"abstract":"For every U ⊂ Diffvol(T) there is a measure of finite support contained in U which is uniformly expanding. 0. Introduction Let μ be a probability measure in Diff(M) where M is a closed manifold of dimension d := dim(M). We denote μ(1) = μ and μ(n) = μ∗μ(n−1). Note that μ(n) is the pushforward by the composition of the product measure μ in (Diff(M))n. Definition 0.1 ([8, 4]). A probability measure μ in Diff(M) is uniformly expanding if there exists N > 0 such that for every (x, v) ∈ T 1M one has that ∫ log ‖Dxfv‖ dμ(N)(f) > 2. This is a robust1 condition on μ. This notion as well as similar ones have been studied extensively recently, as it allows one to describe quite precisely the stationary measures for random walks with μ as law (see below for more discussion). Here we will make a remark (which can be related to some results, e.g. in [3, 6, 14]) that points in the direction of the abundance of uniform expansion. Theorem 0.2. For every open set U in Diffvol(T) there is a finitely supported probability measure μ whose support is contained in U and μ is uniformly expanding. As a consequence of the results of [5, 13, 6] one deduces that: Corollary 0.3. For every U ⊂ Diffvol(T) there is a probability measure μ finitely supported in U such that the orbit of every point under the random walk on T2 produced by μ equidistributes in T2. Moreover, for every μ′ close to μ in the weak-∗ 2020 Mathematics Subject Classification. 37H15. Rafael Potrie was partially supported by CSIC 618, FCE-1-2017-1-135352. This work was started while the author was a Von Neumann fellow at IAS, funded by the Minerva Research Foundation Membership Fund and NSF DMS-1638352. 1To be precise, if μ has compact support, then there is a neighborhood U of its support such that any measure μ′ which has support in U and is weak-∗-close to μ, is also uniformly expanding (see (3.1) below).","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47376826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.33044/REVUMA.V62N1A02
H. Mejjaoli, K. Trimeche
We define the localization operators associated with Laguerre wavelet transforms. Next, we prove the boundedness and compactness of these operators, which depend on a symbol and two admissible wavelets on Lα(K), 1 ≤ p ≤ ∞.
{"title":"Time-frequency analysis associated with the Laguerre wavelet transform","authors":"H. Mejjaoli, K. Trimeche","doi":"10.33044/REVUMA.V62N1A02","DOIUrl":"https://doi.org/10.33044/REVUMA.V62N1A02","url":null,"abstract":"We define the localization operators associated with Laguerre wavelet transforms. Next, we prove the boundedness and compactness of these operators, which depend on a symbol and two admissible wavelets on Lα(K), 1 ≤ p ≤ ∞.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49146491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.33044/REVUMA.V62N1A03
M. Carena, M. Toschi
. Given an A p -Muckenhoupt weight on a fractal obtained as the attractor of an iterated function system, we construct a sequence of approximating weights, which are simple functions belonging uniformly to the A p class on the approximating spaces.
{"title":"Uniform approximation of Muckenhoupt weights on fractals by simple functions","authors":"M. Carena, M. Toschi","doi":"10.33044/REVUMA.V62N1A03","DOIUrl":"https://doi.org/10.33044/REVUMA.V62N1A03","url":null,"abstract":". Given an A p -Muckenhoupt weight on a fractal obtained as the attractor of an iterated function system, we construct a sequence of approximating weights, which are simple functions belonging uniformly to the A p class on the approximating spaces.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44514312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-29DOI: 10.33044/revuma.v61n2a17
Amir Shehzad, M. Binyamin, H. Mahmood
The classification of right unimodal and bimodal hypersurface singularities over a field of positive characteristic was given by H. D. Nguyen. The classification is described in the style of Arnold and not in an algorithmic way. This classification was characterized by M. A. Binyamin et al. [Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 61(109) (2018), no. 3, 333–343] for the case when the corank of hypersurface singularities is ≤ 2. The aim of this article is to characterize the right unimodal and bimodal hypersurface singularities of corank 3 in an algorithmic way by means of easily computable invariants such as the multiplicity, the Milnor number of the given equation, and its blowing-up. On the basis of this characterization we implement an algorithm to compute the type of the right unimodal and bimodal hypersurface singularities without computing the normal form in the computer algebra system Singular.
H. D. Nguyen给出了正特征域上的右单峰和双峰超曲面奇异的分类。分类是用阿诺德的方式描述的，而不是用算法的方式。M. A. Binyamin等人对该分类进行了描述。数学。Soc。科学。数学。鲁曼尼(N.S.) 61(109) (2018)， no。[3,333 - 343]对于超曲面奇点的corank≤2的情况。本文的目的是利用多重性、给定方程的米尔诺数及其爆破等易计算的不变量，用算法刻画corank 3的右单峰和双峰超曲面奇异性。在此刻画的基础上，我们实现了一种在计算机代数系统奇异中不计算范式的情况下计算右单峰和双峰超曲面奇异类型的算法。
{"title":"Characterization of hypersurface singularities in positive characteristic","authors":"Amir Shehzad, M. Binyamin, H. Mahmood","doi":"10.33044/revuma.v61n2a17","DOIUrl":"https://doi.org/10.33044/revuma.v61n2a17","url":null,"abstract":"The classification of right unimodal and bimodal hypersurface singularities over a field of positive characteristic was given by H. D. Nguyen. The classification is described in the style of Arnold and not in an algorithmic way. This classification was characterized by M. A. Binyamin et al. [Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 61(109) (2018), no. 3, 333–343] for the case when the corank of hypersurface singularities is ≤ 2. The aim of this article is to characterize the right unimodal and bimodal hypersurface singularities of corank 3 in an algorithmic way by means of easily computable invariants such as the multiplicity, the Milnor number of the given equation, and its blowing-up. On the basis of this characterization we implement an algorithm to compute the type of the right unimodal and bimodal hypersurface singularities without computing the normal form in the computer algebra system Singular.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"6 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41256944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-28DOI: 10.33044/revuma.v61n2a16
J. Kratica, Dragan Matic, V. Filipović
. We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph GP ( n,k ), we prove that if k = 1 and n ≥ 4 then both the weakly convex domination number γ wcon ( GP ( n,k )) and the convex domination number γ con ( GP ( n,k )) are equal to n . For k ≥ 2 and n ≥ 13, γ wcon ( GP ( n,k )) = γ con ( GP ( n,k )) = 2 n , which is the order of GP ( n,k ). Special cases for smaller graphs are solved by the exact method. For a flower snark graph J n , where n is odd and n ≥ 5, we prove that γ wcon ( J n ) = 2 n and γ con ( J n ) = 4 n .
{"title":"Weakly convex and convex domination numbers for generalized Petersen and flower snark graphs","authors":"J. Kratica, Dragan Matic, V. Filipović","doi":"10.33044/revuma.v61n2a16","DOIUrl":"https://doi.org/10.33044/revuma.v61n2a16","url":null,"abstract":". We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph GP ( n,k ), we prove that if k = 1 and n ≥ 4 then both the weakly convex domination number γ wcon ( GP ( n,k )) and the convex domination number γ con ( GP ( n,k )) are equal to n . For k ≥ 2 and n ≥ 13, γ wcon ( GP ( n,k )) = γ con ( GP ( n,k )) = 2 n , which is the order of GP ( n,k ). Special cases for smaller graphs are solved by the exact method. For a flower snark graph J n , where n is odd and n ≥ 5, we prove that γ wcon ( J n ) = 2 n and γ con ( J n ) = 4 n .","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48691959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-22DOI: 10.33044/revuma.v61n2a13
H. Abass, C. Izuchukwu, O. Mewomo
. We introduce a viscosity iterative algorithm for approximating a common solution of a modified split generalized equilibrium problem and a fixed point problem for a quasi-pseudocontractive mapping which also solves some variational inequality problems in real Hilbert spaces. The proposed iterative algorithm is constructed in such a way that it does not require the prior knowledge of the operator norm. Furthermore, we prove a strong conver- gence theorem for approximating the common solution of the aforementioned problems. Finally, we give a numerical example of our main theorem. Our result complements and extends some related works in the literature.
{"title":"Viscosity approximation method for modified split generalized equilibrium and fixed point problems","authors":"H. Abass, C. Izuchukwu, O. Mewomo","doi":"10.33044/revuma.v61n2a13","DOIUrl":"https://doi.org/10.33044/revuma.v61n2a13","url":null,"abstract":". We introduce a viscosity iterative algorithm for approximating a common solution of a modified split generalized equilibrium problem and a fixed point problem for a quasi-pseudocontractive mapping which also solves some variational inequality problems in real Hilbert spaces. The proposed iterative algorithm is constructed in such a way that it does not require the prior knowledge of the operator norm. Furthermore, we prove a strong conver- gence theorem for approximating the common solution of the aforementioned problems. Finally, we give a numerical example of our main theorem. Our result complements and extends some related works in the literature.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47447825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-11DOI: 10.33044/revuma.v61n2a12
Catarina P. Avelino, Altino F. Santos
{"title":"Triangular spherical dihedral f-tilings: the $(pi/2, pi/3, pi/4)$ and $(2pi/3, pi/4, pi/4)$ family","authors":"Catarina P. Avelino, Altino F. Santos","doi":"10.33044/revuma.v61n2a12","DOIUrl":"https://doi.org/10.33044/revuma.v61n2a12","url":null,"abstract":"","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"9 1","pages":"367-387"},"PeriodicalIF":0.5,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78821246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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