{"title":"Hecke-type double sums and mock theta functions","authors":"Su-Ping Cui, Haixia Du, Nancy S. S. Gu","doi":"10.4064/cm8886-10-2022","DOIUrl":"https://doi.org/10.4064/cm8886-10-2022","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70144207","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 standard graded artinian monomial complete intersection algebra A = k [ x 1 , x 2 , . . . , x n ] / ( x a 1 1 , x a 2 2 , . . . , x a n n ), with k a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of k is greater than the socle degree of A , namely a 1 + a 2 + · · · + a n − n .
{"title":"A new proof of Stanley’s theorem on the strong Lefschetz property","authors":"Hong Phuong, Quang Hoa Tran","doi":"10.4064/cm8987-11-2022","DOIUrl":"https://doi.org/10.4064/cm8987-11-2022","url":null,"abstract":". A standard graded artinian monomial complete intersection algebra A = k [ x 1 , x 2 , . . . , x n ] / ( x a 1 1 , x a 2 2 , . . . , x a n n ), with k a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of k is greater than the socle degree of A , namely a 1 + a 2 + · · · + a n − n .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49656153","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":"Asymptotic smoothness and universality\u0000in Banach spaces","authors":"R. Causey, G. Lancien","doi":"10.4064/cm8923-12-2022","DOIUrl":"https://doi.org/10.4064/cm8923-12-2022","url":null,"abstract":"For $1<pleqslant infty$, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $textsf{A}_p$ and $textsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of these classes is Borel in the class of separable Banach spaces. Then we build small families of Banach spaces that are both injectively and surjectively universal for these classes. Finally, we prove the optimality of this universality result, by proving in particular that none of these classes admits a universal space.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43811983","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 $mathcal{E}$ be the $sigma$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $mathcal{E}$, their associated cardinal characteristics (i.e. additivity, covering, uniformity and cofinality) are pairwise different.
{"title":"A friendly iteration forcing that the four cardinal characteristics of $mathcal E$ can be pairwise different","authors":"Miguel Alvarado Cardona","doi":"10.4064/cm8917-2-2023","DOIUrl":"https://doi.org/10.4064/cm8917-2-2023","url":null,"abstract":"Let $mathcal{E}$ be the $sigma$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $mathcal{E}$, their associated cardinal characteristics (i.e. additivity, covering, uniformity and cofinality) are pairwise different.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49258068","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 : 2022-06-17DOI: 10.4064/CM-60-61-1-77-92
M. V. Siadat
In this dissertation we explore the $[L^{mathrm{p}}, L^{q}]$-boundedness of certain integral operators on weighted spaces on cones in ${mathbb R}^{n}.$ These integral operators are of the type $displaystyle int_{V}k(x, y)f(y)dy$ defined on a homogeneous cone $V$. The results of this dissertation are then applied to an important class of operators such as Riemann-Liouville's fractional integral operators, Weyl's fractional integral operators and Laplace's operators. As special cases of the above, we obtain an ${mathbb R}^{n}$ -generalization of the celebrated Hardy's inequality on domains of positivity. We also prove dual results.
{"title":"Norm inequalities for integral operators on cones","authors":"M. V. Siadat","doi":"10.4064/CM-60-61-1-77-92","DOIUrl":"https://doi.org/10.4064/CM-60-61-1-77-92","url":null,"abstract":"In this dissertation we explore the $[L^{mathrm{p}}, L^{q}]$-boundedness of certain integral operators on weighted spaces on cones in ${mathbb R}^{n}.$ These integral operators are of the type $displaystyle int_{V}k(x, y)f(y)dy$ defined on a homogeneous cone $V$. The results of this dissertation are then applied to an important class of operators such as Riemann-Liouville's fractional integral operators, Weyl's fractional integral operators and Laplace's operators. As special cases of the above, we obtain an ${mathbb R}^{n}$ -generalization of the celebrated Hardy's inequality on domains of positivity. We also prove dual results.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"60 1","pages":"77-92"},"PeriodicalIF":0.4,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42346715","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 article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;delta]$, under the hypothesis that $R$ is $s$-unital and $ker(delta)$ contains a nonzero idempotent. This result generalizes a result by "Oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.
{"title":"Non-unital Ore extensions","authors":"Patrik Lundstrom, Johan Oinert, J. Richter","doi":"10.4064/cm8941-11-2022","DOIUrl":"https://doi.org/10.4064/cm8941-11-2022","url":null,"abstract":"In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;delta]$, under the hypothesis that $R$ is $s$-unital and $ker(delta)$ contains a nonzero idempotent. This result generalizes a result by \"Oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48205391","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 characterize the L p ( R 2 ) boundeness of the geometric maximal operator M a,b associated to the basis B a,b ( a, b > 0) which is composed of rectangles R whose eccentricity and orientation is of the form ( e R , ω R ) = (cid:18) 1 n a , π 4 n b (cid:19) for some n ∈ N ∗ . The proof involves generalized Perron trees , as constructed in [12].
我们刻画了几何极大算子M a,b与基ba,b (a,b > 0)相关联的L p (r2)有界性，该算子由矩形R组成，其偏心率和方向为(e R， ω R) = (cid:18) 1 n a， π 4 n b (cid:19)，对于某些n∈n∗。证明涉及到在[12]中构造的广义Perron树。
{"title":"Application of Perron trees to\u0000geometric maximal operators","authors":"A. Gauvan","doi":"10.4064/cm8693-8-2022","DOIUrl":"https://doi.org/10.4064/cm8693-8-2022","url":null,"abstract":"We characterize the L p ( R 2 ) boundeness of the geometric maximal operator M a,b associated to the basis B a,b ( a, b > 0) which is composed of rectangles R whose eccentricity and orientation is of the form ( e R , ω R ) = (cid:18) 1 n a , π 4 n b (cid:19) for some n ∈ N ∗ . The proof involves generalized Perron trees , as constructed in [12].","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44128492","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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