The chain covering number Cov(P) of a poset P is the least number of chains needed to cover P. For an uncountable cardinal ν, we give a list of posets of cardinality and covering number ν such that for every poset P with no infinite antichain, Cov(P)≥ν if and only if P embeds a member of the list. This list has two elements if ν is a successor cardinal, namely [ν] 2 and its dual, and four elements if ν is a limit cardinal with cf(ν) weakly compact. For ν=ℵ 1 , a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal ν.
{"title":"The chain covering number of a poset with no infinite antichains","authors":"Uri Abraham, Maurice Pouzet","doi":"10.5802/crmath.511","DOIUrl":"https://doi.org/10.5802/crmath.511","url":null,"abstract":"The chain covering number Cov(P) of a poset P is the least number of chains needed to cover P. For an uncountable cardinal ν, we give a list of posets of cardinality and covering number ν such that for every poset P with no infinite antichain, Cov(P)≥ν if and only if P embeds a member of the list. This list has two elements if ν is a successor cardinal, namely [ν] 2 and its dual, and four elements if ν is a limit cardinal with cf(ν) weakly compact. For ν=ℵ 1 , a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal ν.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869711","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":"A simple construction of the Rumin algebra","authors":"Jeffrey S. Case","doi":"10.5802/crmath.510","DOIUrl":"https://doi.org/10.5802/crmath.510","url":null,"abstract":"","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135872826","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 study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part to questions concerning the motivic nearby cycles of restriction functions and the integral identity conjecture in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.
{"title":"Geometry of nondegenerate polynomials: Motivic nearby cycles and Cohomology of contact loci","authors":"Quy Thuong Lê, Tat Thang Nguyen","doi":"10.5802/crmath.492","DOIUrl":"https://doi.org/10.5802/crmath.492","url":null,"abstract":"We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part to questions concerning the motivic nearby cycles of restriction functions and the integral identity conjecture in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"41 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765675","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}
This article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a space of initial conditions) to zero, in particular in the case when spectral condensation occurs. The purpose of the present article is to push forward the analysis to deal with any admissible control operator. The considered setting leads to applications to one dimensional parabolic-type equations or coupled systems of such equations.
{"title":"Analysis of non scalar control problems for parabolic systems by the block moment method","authors":"Franck Boyer, Morgan Morancey","doi":"10.5802/crmath.487","DOIUrl":"https://doi.org/10.5802/crmath.487","url":null,"abstract":"This article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a space of initial conditions) to zero, in particular in the case when spectral condensation occurs. The purpose of the present article is to push forward the analysis to deal with any admissible control operator. The considered setting leads to applications to one dimensional parabolic-type equations or coupled systems of such equations.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"29 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765828","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}
Vanderléa R. Bazao, César R. de Oliveira, Pablo A. Diaz
It is shown that if X is a unitary operator so that a singular subspace of U is unitarily equivalent to a singular subspace of UX (or XU), for each unitary operator U, then X is the identity operator. In other words, there is no nontrivial generalization of Birman–Krein Theorem that includes the preservation of a singular spectral subspace in this context.
{"title":"On the Birman–Krein Theorem","authors":"Vanderléa R. Bazao, César R. de Oliveira, Pablo A. Diaz","doi":"10.5802/crmath.473","DOIUrl":"https://doi.org/10.5802/crmath.473","url":null,"abstract":"It is shown that if X is a unitary operator so that a singular subspace of U is unitarily equivalent to a singular subspace of UX (or XU), for each unitary operator U, then X is the identity operator. In other words, there is no nontrivial generalization of Birman–Krein Theorem that includes the preservation of a singular spectral subspace in this context.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"21 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220099","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}
where γ s =2 -s π -n 2 Γ(n-s 2) Γ(s 2). We also consider the behavior of the best constant 𝒞 n,s of weak type estimate for Riesz potentials, and we prove 𝒞 n,s =O(γ s s) as s→0.
其中γ s =2 -s π -n 2 Γ(n-s 2) Γ(s 2)。我们还考虑了Riesz势的弱类型估计的最佳常数 n,s的行为，并证明了当s→0时， n,s =O(γ s s)。
{"title":"Optimal weak estimates for Riesz potentials","authors":"Liang Huang, Hanli Tang","doi":"10.5802/crmath.479","DOIUrl":"https://doi.org/10.5802/crmath.479","url":null,"abstract":"where γ s =2 -s π -n 2 Γ(n-s 2) Γ(s 2). We also consider the behavior of the best constant 𝒞 n,s of weak type estimate for Riesz potentials, and we prove 𝒞 n,s =O(γ s s) as s→0.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135268166","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 a non-archimedean locally compact field and let G n be the group PGL n (F). In this paper we construct a tower (X ˜ k ) k⩾0 of graphs fibred over the one-skeleton of the Bruhat–Tits building of G n . We prove that a non-spherical and irreducible generic complex representation of G n can be realized as a quotient of the compactly supported cohomology of the graph X ˜ k for k large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.
设F是一个非阿基米德局部紧化场，并设gn是群PGL n (F)。在本文中，我们在gn的Bruhat-Tits建筑的一个骨架上构建了一个塔(X ~ k) k小于0的图形。证明了G n的非球面不可约一般复表示可以作为图X ~ k的紧支持上同调的商来实现，且k足够大。此外，当表示是倒立的，那么它在这样的模型中有一个独特的实现。
{"title":"Compactly supported cohomology of a tower of graphs and generic representations of PGL n over a local field","authors":"Anis Rajhi","doi":"10.5802/crmath.485","DOIUrl":"https://doi.org/10.5802/crmath.485","url":null,"abstract":"Let F be a non-archimedean locally compact field and let G n be the group PGL n (F). In this paper we construct a tower (X ˜ k ) k⩾0 of graphs fibred over the one-skeleton of the Bruhat–Tits building of G n . We prove that a non-spherical and irreducible generic complex representation of G n can be realized as a quotient of the compactly supported cohomology of the graph X ˜ k for k large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"3 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267542","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 give necessary conditions for the positivity of Littlewood–Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if S λ (V) appears as a summand in the decomposition into irreducibles of S μ (S ν (V)), then ν’s diagram is contained in λ’s diagram.
给出了Littlewood-Richardson系数和SXP系数正的必要条件。我们推导出多倍体系数为正的必要条件。明确地，我们的主要结果表明，如果S λ (V)在S μ (S ν (V))的不可约分解中以求和形式出现，则ν的图包含在λ的图中。
{"title":"Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients","authors":"Álvaro Gutiérrez, Mercedes H. Rosas","doi":"10.5802/crmath.468","DOIUrl":"https://doi.org/10.5802/crmath.468","url":null,"abstract":"We give necessary conditions for the positivity of Littlewood–Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if S λ (V) appears as a summand in the decomposition into irreducibles of S μ (S ν (V)), then ν’s diagram is contained in λ’s diagram.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"12 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220359","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 $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including identification of parameter regimes under which the density is bounded, asymptotic approximations of tail probabilities, and fractional moments; in particular, we see that the mean is undefined. In the case that $X$ and $Y$ are independent symmetric variance-gamma random variables, an exact formula is also given for the cumulative distribution function of the ratio $X/Y$.
{"title":"The variance-gamma ratio distribution","authors":"Robert E. Gaunt, Siqi Li","doi":"10.5802/crmath.495","DOIUrl":"https://doi.org/10.5802/crmath.495","url":null,"abstract":"Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including identification of parameter regimes under which the density is bounded, asymptotic approximations of tail probabilities, and fractional moments; in particular, we see that the mean is undefined. In the case that $X$ and $Y$ are independent symmetric variance-gamma random variables, an exact formula is also given for the cumulative distribution function of the ratio $X/Y$.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267545","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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