. Let f ( x ) ∈ Z [ x ] be monic, with deg( f ) = n . We say f ( x ) is monogenic if f ( x ) is irreducible over Q and { 1 , α, α 2 , . . . , α n − 1 } is a basis for the ring of integers of K = Q ( α ) , where f ( α ) = 0 . In this article, we derive a new polynomial discriminant formula, and we use it to construct infinite families of monogenic quadrinomials, quintinomials and sextinomials for any degree n ≥ 3 , 4 , 5 , respectively. These results extend previous work of the author. We also give a brief discussion concerning the adaptation of our approach beyond sextinomials.
{"title":"Infinite families of monogenic quadrinomials, quintinomials and sextinomials","authors":"L. Jones","doi":"10.4064/cm8552-4-2021","DOIUrl":"https://doi.org/10.4064/cm8552-4-2021","url":null,"abstract":". Let f ( x ) ∈ Z [ x ] be monic, with deg( f ) = n . We say f ( x ) is monogenic if f ( x ) is irreducible over Q and { 1 , α, α 2 , . . . , α n − 1 } is a basis for the ring of integers of K = Q ( α ) , where f ( α ) = 0 . In this article, we derive a new polynomial discriminant formula, and we use it to construct infinite families of monogenic quadrinomials, quintinomials and sextinomials for any degree n ≥ 3 , 4 , 5 , respectively. These results extend previous work of the author. We also give a brief discussion concerning the adaptation of our approach beyond sextinomials.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70141127","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":"Scattering and blowup beyond the mass-energy threshold for the cubic NLS with a potential","authors":"Y. Wang","doi":"10.4064/cm8978-10-2022","DOIUrl":"https://doi.org/10.4064/cm8978-10-2022","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70144082","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 establish the test which allows to show that a mean does not admit a weak-Hardy property. As a result we prove that Hardy and weak-Hardy properties are equivalent in the class of homogeneous, symmetric, repetition invariant, and Jensen concave mean on R + . More precisely, for every mean M : S ∞ n =1 R n + → R as above, the inequality M ( a 1 ) + M a 1 , a 2 ) + · · · < ∞ holds for all a ∈ ℓ 1 ( R + ) if and only if there exists a positive, real constant C (depending only on M ) such that ) for every sequence a ∈ ℓ 1 ( R + ) .
{"title":"On negative results concerning weak-Hardy means","authors":"P. Pasteczka","doi":"10.4064/cm8749-4-2022","DOIUrl":"https://doi.org/10.4064/cm8749-4-2022","url":null,"abstract":". We establish the test which allows to show that a mean does not admit a weak-Hardy property. As a result we prove that Hardy and weak-Hardy properties are equivalent in the class of homogeneous, symmetric, repetition invariant, and Jensen concave mean on R + . More precisely, for every mean M : S ∞ n =1 R n + → R as above, the inequality M ( a 1 ) + M a 1 , a 2 ) + · · · < ∞ holds for all a ∈ ℓ 1 ( R + ) if and only if there exists a positive, real constant C (depending only on M ) such that ) for every sequence a ∈ ℓ 1 ( R + ) .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47041740","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}
Systems of dyadic cubes are the basic tools of harmonic analysis and geometry, and this notion had been extended to general metric spaces. In this paper, we construct systems of dyadic cubes of complete, doubling, uniformly perfect metric spaces, such that for any two points in the metric space, there exists a chain of three cubes whose diameters are comparable to the distance of the points. We also give an application of our construction to previous research of potential analysis and geometry of metric spaces.
{"title":"Systems of dyadic cubes of complete, doubling, uniformly perfect metric spaces without detours","authors":"Kohei Sasaya","doi":"10.4064/cm8702-7-2022","DOIUrl":"https://doi.org/10.4064/cm8702-7-2022","url":null,"abstract":"Systems of dyadic cubes are the basic tools of harmonic analysis and geometry, and this notion had been extended to general metric spaces. In this paper, we construct systems of dyadic cubes of complete, doubling, uniformly perfect metric spaces, such that for any two points in the metric space, there exists a chain of three cubes whose diameters are comparable to the distance of the points. We also give an application of our construction to previous research of potential analysis and geometry of metric spaces.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42899840","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 study left-invariant Laplacians on compact connected groups that are form-comparable perturbations of bi-invariant Laplacians. Our results show that Gaussian bounds for derivatives of heat kernels enjoyed by certain bi-invariant Laplacians hold for their form-comparable perturbations. We further show that the parabolic operators associated with such left-invariant Laplacians, in particular, with the bi-invariant Laplacians, are hypoelliptic in various senses.
{"title":"Perturbation results concerning Gaussian estimates and hypoellipticity for left-invariant Laplacians on compact groups","authors":"Qi Hou, L. Saloff‐Coste","doi":"10.4064/cm8696-8-2022","DOIUrl":"https://doi.org/10.4064/cm8696-8-2022","url":null,"abstract":"In this paper we study left-invariant Laplacians on compact connected groups that are form-comparable perturbations of bi-invariant Laplacians. Our results show that Gaussian bounds for derivatives of heat kernels enjoyed by certain bi-invariant Laplacians hold for their form-comparable perturbations. We further show that the parabolic operators associated with such left-invariant Laplacians, in particular, with the bi-invariant Laplacians, are hypoelliptic in various senses.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47224728","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 a construction of Gorenstein projective τ -tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective τ -tilting modules. Moreover, we show that a finite dimensional algebra Λ over an algebraically closed field is CM -τ -tilting finite if Tn(Λ) is CM -τ -tilting finite which gives a partial answer to a question on CM -τ -tilting finite algebras posed by Xie and Zhang.
{"title":"A construction of Gorenstein projective $tau $-tilting modules","authors":"Zhi-wei Li, Xiaojin Zhang","doi":"10.4064/cm8682-1-2022","DOIUrl":"https://doi.org/10.4064/cm8682-1-2022","url":null,"abstract":"We give a construction of Gorenstein projective τ -tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective τ -tilting modules. Moreover, we show that a finite dimensional algebra Λ over an algebraically closed field is CM -τ -tilting finite if Tn(Λ) is CM -τ -tilting finite which gives a partial answer to a question on CM -τ -tilting finite algebras posed by Xie and Zhang.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48344814","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}
Rodrigo Hern'andez-Guti'errez, Ver'onica Mart'inez-de-la-Vega, Jorge M. Mart'inez-Montejano, Jorge E. Vega
Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)={ A in C(X):Xsetminus Atext{ is connected}}$ when $X$ is a finite graph or a dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact, connected, locally connected or totally disconnected. Also, we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.
{"title":"The hyperspace of noncut subcontinua of graphs\u0000and dendrites","authors":"Rodrigo Hern'andez-Guti'errez, Ver'onica Mart'inez-de-la-Vega, Jorge M. Mart'inez-Montejano, Jorge E. Vega","doi":"10.4064/cm8947-9-2022","DOIUrl":"https://doi.org/10.4064/cm8947-9-2022","url":null,"abstract":"Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)={ A in C(X):Xsetminus Atext{ is connected}}$ when $X$ is a finite graph or a dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact, connected, locally connected or totally disconnected. Also, we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49002541","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 compute the spectral form of the Koopman representation induced by a natural boolean action of L(λ,T) identified earlier by the authors. Our computation establishes the sharpness of the constraints on spectral forms of Koopman representations of L(λ,T) previously found by the second author.
{"title":"The spectral form of Koopman representations of the group of measurable functions with values in the circle","authors":"J. Moore, Slawomir Solecki","doi":"10.4064/cm8784-6-2022","DOIUrl":"https://doi.org/10.4064/cm8784-6-2022","url":null,"abstract":"We compute the spectral form of the Koopman representation induced by a natural boolean action of L(λ,T) identified earlier by the authors. Our computation establishes the sharpness of the constraints on spectral forms of Koopman representations of L(λ,T) previously found by the second author.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49090543","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 classify maximal systems of arcs which intersect at most once on the 4-punctured sphere.
我们对在4点球面上最多相交一次的弧的极大系统进行了分类。
{"title":"Arcs intersecting at most once\u0000on the 4-punctured sphere","authors":"Pau‐Ling Tee","doi":"10.4064/cm8729-6-2023","DOIUrl":"https://doi.org/10.4064/cm8729-6-2023","url":null,"abstract":"We classify maximal systems of arcs which intersect at most once on the 4-punctured sphere.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42311147","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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