Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,theta)$, we study the existence of a contact structure conformal to $theta$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green's function of the $P'$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. Then we exhibit an Aubin type result guaranteeing the existence of a minimizer for the total mass which yields the classical LHLS inequality.
{"title":"Logarithmic Hardy-Littlewood-Sobolev inequality on pseudo-Einstein 3-manifolds and the logarithmic Robin mass","authors":"Ali Maalaoui","doi":"10.5565/publmat6722302","DOIUrl":"https://doi.org/10.5565/publmat6722302","url":null,"abstract":"Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,theta)$, we study the existence of a contact structure conformal to $theta$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green's function of the $P'$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. Then we exhibit an Aubin type result guaranteeing the existence of a minimizer for the total mass which yields the classical LHLS inequality.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47547179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space.
{"title":"Coercivity for travelling waves in the Gross-Pitaevskii equation in $mathbb{R}^2$ for small speed","authors":"D. Chiron, Eliot Pacherie","doi":"10.5565/PUBLMAT6712307","DOIUrl":"https://doi.org/10.5565/PUBLMAT6712307","url":null,"abstract":"In a previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42935441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.
{"title":"A small closed convex projective 4-manifold via Dehn filling","authors":"Gye-Seon Lee, Ludovic Marquis, Stefano Riolo","doi":"10.5565/PUBLMAT6612215","DOIUrl":"https://doi.org/10.5565/PUBLMAT6612215","url":null,"abstract":"In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42112004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.
{"title":"Induced Hopf Galois structures and their local Hopf Galois modules","authors":"Daniel Gil-Muñoz, A. Rio","doi":"10.5565/PUBLMAT6612204","DOIUrl":"https://doi.org/10.5565/PUBLMAT6612204","url":null,"abstract":"The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42925188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively.
{"title":"Stallings automata for free-times-abelian groups: intersections and index","authors":"Jordi Delgado, E. Ventura","doi":"10.5565/publmat6622209","DOIUrl":"https://doi.org/10.5565/publmat6622209","url":null,"abstract":"We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. \u0000This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44209732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds.
{"title":"Iteration of functions and contractibility of acyclic 2-complexes","authors":"I. Leary","doi":"10.5565/publmat6512110","DOIUrl":"https://doi.org/10.5565/publmat6512110","url":null,"abstract":"We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42365727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Laura Cladek, Polona Durcik, B. Krause, Jos'e Madrid
We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.
{"title":"Directional maximal function along the primes","authors":"Laura Cladek, Polona Durcik, B. Krause, Jos'e Madrid","doi":"10.5565/PUBLMAT6522113","DOIUrl":"https://doi.org/10.5565/PUBLMAT6522113","url":null,"abstract":"We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46627777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, some equivalences and implications between which we prove. Considering the rings of endomorphisms of modules as topological rings in the finite topology, we establish a close connection between the conjectural concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. In particular, we show that a module $Sigma$-coperfect over its endomorphism ring has a perfect decomposition provided that the endomorphism ring is commutative, and that all countably indexed local direct summands are direct summands in any countably generated endo-$Sigma$-coperfect module.
{"title":"Topologically semisimple and topologically perfect topological rings","authors":"L. Positselski, J. Šťovíček","doi":"10.5565/PUBLMAT6622202","DOIUrl":"https://doi.org/10.5565/PUBLMAT6622202","url":null,"abstract":"Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, some equivalences and implications between which we prove. Considering the rings of endomorphisms of modules as topological rings in the finite topology, we establish a close connection between the conjectural concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. In particular, we show that a module $Sigma$-coperfect over its endomorphism ring has a perfect decomposition provided that the endomorphism ring is commutative, and that all countably indexed local direct summands are direct summands in any countably generated endo-$Sigma$-coperfect module.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42074313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $mathbb{R}^6$ (resp. $mathbb R^5$) with regular (resp. singular corank 1) surfaces in $mathbb R^5$ (resp. $mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $mathbb R^6$ and singular corank 1 surfaces in $mathbb R^4$.
{"title":"Relating second order geometry of manifolds through projections and normal sections","authors":"P. B. Riul, R. O. Sinha","doi":"10.5565/publmat6512114","DOIUrl":"https://doi.org/10.5565/publmat6512114","url":null,"abstract":"We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $mathbb{R}^6$ (resp. $mathbb R^5$) with regular (resp. singular corank 1) surfaces in $mathbb R^5$ (resp. $mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $mathbb R^6$ and singular corank 1 surfaces in $mathbb R^4$.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46876389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressable in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality $1le x^n$ is locally countable if and only if $n=1$.
{"title":"Locally countable pseudovarieties","authors":"J. Almeida, O. Kl'ima","doi":"10.5565/PUBLMAT6712303","DOIUrl":"https://doi.org/10.5565/PUBLMAT6712303","url":null,"abstract":"The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressable in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality $1le x^n$ is locally countable if and only if $n=1$.","PeriodicalId":54531,"journal":{"name":"Publicacions Matematiques","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44319357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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