The goal of this paper is to study global dynamics of $C^infty$-smooth slow-fast systems on the $2$-torus of class $C^infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $minmathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.
{"title":"Slow-fast torus knots","authors":"Renato Huzak, Hildeberto Jard'on-Kojakhmetov","doi":"10.36045/j.bbms.220208","DOIUrl":"https://doi.org/10.36045/j.bbms.220208","url":null,"abstract":"The goal of this paper is to study global dynamics of $C^infty$-smooth slow-fast systems on the $2$-torus of class $C^infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $minmathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89169795","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 F be the free group functor, left adjoint to the forgetful functor between the category of groups GRP and the category of sets SET. Let f from A to B, and h from A to C be two functions in SET and let Ker(F(f)) and Ker(F(h)) be the kernels of the induced morphisms between free groups. Provided that the kernel pairs Eq(f) and Eq(h) of f and h permute (such as it is the case when the pushout of f and h is a double extension in SET), this short article describes a method to rewrite a general element in the intersection of Ker(F(f)) and Ker(F(g)) as a product of generators in A which is (f,h)-symmetric in the sense of the higher covering theory of racks and quandles.
{"title":"Rewriting the elements in the intersection of the kernels of two morphisms between free groups","authors":"Franccois Renaud","doi":"10.36045/j.bbms.210310","DOIUrl":"https://doi.org/10.36045/j.bbms.210310","url":null,"abstract":"Let F be the free group functor, left adjoint to the forgetful functor between the category of groups GRP and the category of sets SET. Let f from A to B, and h from A to C be two functions in SET and let Ker(F(f)) and Ker(F(h)) be the kernels of the induced morphisms between free groups. Provided that the kernel pairs Eq(f) and Eq(h) of f and h permute (such as it is the case when the pushout of f and h is a double extension in SET), this short article describes a method to rewrite a general element in the intersection of Ker(F(f)) and Ker(F(g)) as a product of generators in A which is (f,h)-symmetric in the sense of the higher covering theory of racks and quandles.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82048629","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}
The aim of this article is to provide characterizations for subadditivity-like growth conditions for the so-called associated weight functions in terms of the defning weight sequence. Such growth requirements arise frequently in the literature and are standard when dealing with ultradifferentiable function classes defned by Braun-Meise-Taylor weight functions since they imply or even characterize important and desired consequences for the underlying function spaces, e.g. closedness under composition.
{"title":"On subadditivity-like conditions for associated weight functions","authors":"G. Schindl","doi":"10.36045/j.bbms.210127","DOIUrl":"https://doi.org/10.36045/j.bbms.210127","url":null,"abstract":"The aim of this article is to provide characterizations for subadditivity-like growth conditions for the so-called associated weight functions in terms of the defning weight sequence. Such growth requirements arise frequently in the literature and are standard when dealing with ultradifferentiable function classes defned by Braun-Meise-Taylor weight functions since they imply or even characterize important and desired consequences for the underlying function spaces, e.g. closedness under composition.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75046194","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}
A regular parallelism of real projective 3-space PG(3,R) is an equivalence relation on the line space such that every class is equivalent to the set of 1-dimensional complex subspaces of a 2-dimensional complex vector space. We shall assume that the set of classes is compact, and characterize those regular parallelisms that admit an action of a 2-dimensional torus group. We prove that there is a one-dimensional subtorus fixing every parallel class. From this property alone we deduce that the parallelism is a 2- or 3-dimensional regular parallelism in the sense of Betten and Riesinger. If a 2-torus acts, then the parallelism can be described using a so-called generalized line star which admits a 1-torus action. We also study examples of such parallelisms by constructing generalized line stars. In particular, we prove a claim which was presented by Betten and Riesinger with an incorrect proof. The present article continues a series of papers by the first author on parallelisms with large groups.
{"title":"Regular parallelisms on $mathrm{PG}(3,mathbb R)$ admitting a 2-torus action","authors":"Rainer Lowen, Gunter F. Steinke","doi":"10.36045/j.bbms.210114","DOIUrl":"https://doi.org/10.36045/j.bbms.210114","url":null,"abstract":"A regular parallelism of real projective 3-space PG(3,R) is an equivalence relation on the line space such that every class is equivalent to the set of 1-dimensional complex subspaces of a 2-dimensional complex vector space. We shall assume that the set of classes is compact, and characterize those regular parallelisms that admit an action of a 2-dimensional torus group. We prove that there is a one-dimensional subtorus fixing every parallel class. From this property alone we deduce that the parallelism is a 2- or 3-dimensional regular parallelism in the sense of Betten and Riesinger. If a 2-torus acts, then the parallelism can be described using a so-called generalized line star which admits a 1-torus action. We also study examples of such parallelisms by constructing generalized line stars. In particular, we prove a claim which was presented by Betten and Riesinger with an incorrect proof. The present article continues a series of papers by the first author on parallelisms with large groups.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84985803","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}
In this paper, we deal with the singularly perturbed problem for a linear second order differential difference equation with delay as well as advance. In order to solve the problem numerically, we construct a new difference scheme by the method of integral identities with the use interpolating quadrature rules with remainder terms in integral form. Using an appropriately non-uniform mesh of Shishkin type, we find that the method is almost first order convergent in the discrete maximum norm with respect to the perturbation parameter. Furthermore, we present the numerical experiments that their results support of the theory.
{"title":"Uniformly convergent numerical method for a singularly perturbed differential difference equation with mixed type","authors":"Erkan Çimen","doi":"10.36045/j.bbms.200128","DOIUrl":"https://doi.org/10.36045/j.bbms.200128","url":null,"abstract":"In this paper, we deal with the singularly perturbed problem for a linear second order differential difference equation with delay as well as advance. In order to solve the problem numerically, we construct a new difference scheme by the method of integral identities with the use interpolating quadrature rules with remainder terms in integral form. Using an appropriately non-uniform mesh of Shishkin type, we find that the method is almost first order convergent in the discrete maximum norm with respect to the perturbation parameter. Furthermore, we present the numerical experiments that their results support of the theory.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84573132","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}
J. Khodabandehlou, S. Maghsoudi, J. Seoane-Sepúlveda
{"title":"Lineability and algebrability within $p-$adic function spaces","authors":"J. Khodabandehlou, S. Maghsoudi, J. Seoane-Sepúlveda","doi":"10.36045/j.bbms.200416","DOIUrl":"https://doi.org/10.36045/j.bbms.200416","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86789116","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}
Jogli G. Araújo, H. Lima, Wallace F. Gomes, M. Velásquez
We study $n$-dimensional complete submanifolds immersed in a weighted warped product of the type $Itimes_fM^{n+p}_{varphi}$, whose warping function $f$ has convex logarithm and weight function $varphi$ does not depend on the real parameter $tin I$. Assuming the constancy of an appropriate support function involving the $varphi$-mean curvature vector field of such a submanifold $Sigma^n$ jointly with suitable constraints on the Bakry-Emery-Ricci tensor of $Sigma^n$, we prove that it must be contained in a slice of the ambient space. As applications, we obtain codimension reductions and Bernstein-type results for complete $varphi$-minimal bounded multi graphs constructed over the $n$-dimensional Gaussian space. Our approach is based on the weak Omori-Yau's generalized maximum principle and Liouville-type results for the drift Laplacian.
{"title":"Submanifolds immersed in a warped product with density","authors":"Jogli G. Araújo, H. Lima, Wallace F. Gomes, M. Velásquez","doi":"10.36045/j.bbms.200126","DOIUrl":"https://doi.org/10.36045/j.bbms.200126","url":null,"abstract":"We study $n$-dimensional complete submanifolds immersed in a weighted warped product of the type $Itimes_fM^{n+p}_{varphi}$, whose warping function $f$ has convex logarithm and weight function $varphi$ does not depend on the real parameter $tin I$. Assuming the constancy of an appropriate support function involving the $varphi$-mean curvature vector field of such a submanifold $Sigma^n$ jointly with suitable constraints on the Bakry-Emery-Ricci tensor of $Sigma^n$, we prove that it must be contained in a slice of the ambient space. As applications, we obtain codimension reductions and Bernstein-type results for complete $varphi$-minimal bounded multi graphs constructed over the $n$-dimensional Gaussian space. Our approach is based on the weak Omori-Yau's generalized maximum principle and Liouville-type results for the drift Laplacian.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91135889","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}
A weak stability bound for the $varepsilon$-isometry $f$ form the positive cone of a reflexive, strictly convex and Gateaux smooth Banach lattice $X$ to a Banach space $Y$ is presented. This result is used to prove the stability theorem for the $varepsilon$-isometry $f:(mathbb{R}^n)^+rightarrow Y$, where $mathbb{R}^n$ is the $n$-dimensional space equipped with a $1$-unconditional norm and $Y$ is a n-dimensional, strictly convex and Gateaux smooth space.
{"title":"Stability of $varepsilon$-isometries on the positive cones of finite-dimensional Banach spaces","authors":"Longfa Sun, Ya-jing Ma","doi":"10.36045/j.bbms.200413","DOIUrl":"https://doi.org/10.36045/j.bbms.200413","url":null,"abstract":"A weak stability bound for the $varepsilon$-isometry $f$ form the positive cone of a reflexive, strictly convex and Gateaux smooth Banach lattice $X$ to a Banach space $Y$ is presented. This result is used to prove the stability theorem for the $varepsilon$-isometry $f:(mathbb{R}^n)^+rightarrow Y$, where $mathbb{R}^n$ is the $n$-dimensional space equipped with a $1$-unconditional norm and $Y$ is a n-dimensional, strictly convex and Gateaux smooth space.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76008926","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}
The Suzuki algebra $A_{Nn}^{mu lambda}$ was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N,2n}^{mulambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1times A_1$, $A_2$, $A_2times A_2$, Super type ${bf A}_{2}(q;I_2)$ and the Nichols algebra ufo(8). There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N,2n}^{mu lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{mu lambda}$. By using a result of Masuoka, we prove that $dimmathfrak{B}(V_{abe})=infty$ under the condition $b^2=(ae)^{-1}$, $binBbb{G}_{m}$ for $mgeq 5$.
{"title":"Finite-dimensional Nichols algebras over the Suzuki algebras I: simple Yetter-Drinfeld modules of $A_{N,2n}^{mulambda}$","authors":"Yuxing Shi","doi":"10.36045/j.bbms.211101","DOIUrl":"https://doi.org/10.36045/j.bbms.211101","url":null,"abstract":"The Suzuki algebra $A_{Nn}^{mu lambda}$ was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N,2n}^{mulambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1times A_1$, $A_2$, $A_2times A_2$, Super type ${bf A}_{2}(q;I_2)$ and the Nichols algebra ufo(8). There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N,2n}^{mu lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{mu lambda}$. By using a result of Masuoka, we prove that $dimmathfrak{B}(V_{abe})=infty$ under the condition $b^2=(ae)^{-1}$, $binBbb{G}_{m}$ for $mgeq 5$.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75418723","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}
{"title":"Some aspects of multiorthomorphisms on Riesz spaces","authors":"Abderraouf Dorai, Elmiloud Chil","doi":"10.36045/j.bbms.191104","DOIUrl":"https://doi.org/10.36045/j.bbms.191104","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79624291","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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