{"title":"Inequalities involving $pi(x)$","authors":"H. Alzer, M. Kwong, J. Sándor","doi":"10.4171/rsmup/98","DOIUrl":"https://doi.org/10.4171/rsmup/98","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84108603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $A$ be a commutative noetherian ring, $frak a$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that $Ext^i_A(A/frak a,M)$ is finitely generated for all $ileq m+n$. We define a class $cS_n(frak a)$ of modules and we assume that $H_{frak a}^s(M)incS_{n}(frak a)$ for all $sleq m$. We show that $H_{frak a}^s(M)$ is $frak a$-cofinite for all $sleq m$ if either $n=1$ or $ngeq 2$ and $Ext_A^{i}(A/frak a,H_{frak a}^{t+s-i}(M))$ is finitely generated for all $1leq tleq n-1$, $ileq t-1$ and $sleq m$. If $A$ is a ring of dimension $d$ and $MincS_n(frak a)$ for any ideal $frak a$ of dimension $leq d-1$, then we prove that $MincS_n(frak a)$ for any ideal $frak a$ of $A$.
{"title":"A criterion for cofiniteness of modules","authors":"M. Khazaei, R. Sazeedeh","doi":"10.4171/rsmup/128","DOIUrl":"https://doi.org/10.4171/rsmup/128","url":null,"abstract":"Let $A$ be a commutative noetherian ring, $frak a$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that $Ext^i_A(A/frak a,M)$ is finitely generated for all $ileq m+n$. We define a class $cS_n(frak a)$ of modules and we assume that $H_{frak a}^s(M)incS_{n}(frak a)$ for all $sleq m$. We show that $H_{frak a}^s(M)$ is $frak a$-cofinite for all $sleq m$ if either $n=1$ or $ngeq 2$ and $Ext_A^{i}(A/frak a,H_{frak a}^{t+s-i}(M))$ is finitely generated for all $1leq tleq n-1$, $ileq t-1$ and $sleq m$. If $A$ is a ring of dimension $d$ and $MincS_n(frak a)$ for any ideal $frak a$ of dimension $leq d-1$, then we prove that $MincS_n(frak a)$ for any ideal $frak a$ of $A$.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87895529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A new approach to de Rham–Witt complexes, after Bhatt, Lurie, and Mathew","authors":"L. Illusie","doi":"10.4171/rsmup/86","DOIUrl":"https://doi.org/10.4171/rsmup/86","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82039959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the following nonlinear fractional critical equation with zero Dirichlet boundary condition Asu = Ku n+2s n−2s , u > 0 in Ω and u = 0 on ∂Ω, whereK is a positive function, Ω is a regular bounded domain of R, n ≥ 2 and As, s ∈ (0, 1) represents the spectral fractional Laplacian operator (−∆) in Ω with zero Dirichlet boundary condition. We prove a version of Morse lemmas at infinity for this problem. We also exhibit a relevant application of our novel result. More precisely, we characterize the critical points at infinity of the associated variational problem and we prove an existence result for s = 1 2 and n = 3. Mathematics Subject Classification (2010). Primary: 35J65; Secondary: 35R11, 58J20, 58C30.
本文考虑以下零Dirichlet边界条件的非线性分数阶临界方程Asu = Ku n+2s n−2s, u > 0在Ω上，u = 0在∂Ω上，其中ek是一个正函数，Ω是R, n≥2的正则有界域，As, s∈(0,1)表示零Dirichlet边界条件下Ω上的谱分数阶拉普拉斯算子(-∆)。对于这个问题，我们证明了无穷远处摩尔斯引理的一个版本。我们还展示了我们的新结果的相关应用。更准确地说，我们刻画了相关变分问题在无穷远处的临界点，并证明了s = 1 2和n = 3的存在性结果。数学学科分类(2010)。主:35 j65;次级:35R11、58J20、58C30。
{"title":"A Morse lemma at infinity for nonlinear elliptic fractional equations","authors":"W. Abdelhedi, H. Hajaiej, Zeinab Mhamdi","doi":"10.4171/RSMUP/82","DOIUrl":"https://doi.org/10.4171/RSMUP/82","url":null,"abstract":"In this paper, we consider the following nonlinear fractional critical equation with zero Dirichlet boundary condition Asu = Ku n+2s n−2s , u > 0 in Ω and u = 0 on ∂Ω, whereK is a positive function, Ω is a regular bounded domain of R, n ≥ 2 and As, s ∈ (0, 1) represents the spectral fractional Laplacian operator (−∆) in Ω with zero Dirichlet boundary condition. We prove a version of Morse lemmas at infinity for this problem. We also exhibit a relevant application of our novel result. More precisely, we characterize the critical points at infinity of the associated variational problem and we prove an existence result for s = 1 2 and n = 3. Mathematics Subject Classification (2010). Primary: 35J65; Secondary: 35R11, 58J20, 58C30.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78671783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
– In this paper the sufficient conditions of the modularity of the lattice of all partially composition Fitting classes were found.
本文给出了所有部分复合拟合类的格的模性的充分条件。
{"title":"On the modularity property of the lattice of partially composition Fitting classes","authors":"N. Yang, N. N. Vorob’ev, A. R. Filimonova","doi":"10.4171/RSMUP/83","DOIUrl":"https://doi.org/10.4171/RSMUP/83","url":null,"abstract":"– In this paper the sufficient conditions of the modularity of the lattice of all partially composition Fitting classes were found.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73027280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Normalizers of classical groups arising under extension of the base ring","authors":"N. H. Nhat, T. N. Hoi","doi":"10.4171/RSMUP/75","DOIUrl":"https://doi.org/10.4171/RSMUP/75","url":null,"abstract":"","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"84 1","pages":"153-165"},"PeriodicalIF":0.0,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76843016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $infty$-category $mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a homotopy commutative algebra in $mathscr{M}$ we show that $E$-nilpotent completion, $E$-localization, and a suitable formal completion agree on bounded below objects when $E$ satisfies some reasonable conditions.
{"title":"Localizations and completions of stable $infty$-categories","authors":"L. Mantovani","doi":"10.4171/rsmup/122","DOIUrl":"https://doi.org/10.4171/rsmup/122","url":null,"abstract":"We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $infty$-category $mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a homotopy commutative algebra in $mathscr{M}$ we show that $E$-nilpotent completion, $E$-localization, and a suitable formal completion agree on bounded below objects when $E$ satisfies some reasonable conditions.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75686155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The compact connected abelian groups of dimension 1 are represented and classified in an efficient and explicit way. Main tools are Pontryagin Duality and the Resolution Theorem for compact abelian groups. Mathematics Subject Classification (2010). Primary: 22C05; Secondary: 20K15, 22B05
{"title":"Compact connected abelian groups of dimension 1","authors":"Wayne Lewis, A. Mader","doi":"10.4171/RSMUP/85","DOIUrl":"https://doi.org/10.4171/RSMUP/85","url":null,"abstract":"The compact connected abelian groups of dimension 1 are represented and classified in an efficient and explicit way. Main tools are Pontryagin Duality and the Resolution Theorem for compact abelian groups. Mathematics Subject Classification (2010). Primary: 22C05; Secondary: 20K15, 22B05","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81098918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss some elementary questions in the calculus of variations related to the Lavrentiev phenomenon and the uniqueness of the minimizers. Mathematics Subject Classification (2010). Primary: 49J05; Secondary: 49K05.
{"title":"Some elementary questions in the calculus of variations","authors":"C. Mantegazza","doi":"10.4171/RSMUP/73","DOIUrl":"https://doi.org/10.4171/RSMUP/73","url":null,"abstract":"We discuss some elementary questions in the calculus of variations related to the Lavrentiev phenomenon and the uniqueness of the minimizers. Mathematics Subject Classification (2010). Primary: 49J05; Secondary: 49K05.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"8 1","pages":"107-115"},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87685197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A ringed partially ordered set with zero is a pair (L,F ), where L is a partially ordered set with a least element 0L and F : L → Ring is a covariant functor. Here the partially ordered set L is given a category structure in the usual way and Ring denotes the category of associative rings with identity. Let RingedParOrd0 be the category of ringed partially ordered sets with zero. There is a functor H : Ring → RingedParOrd0 that associates to any ring R a ringed partially ordered set with zero (Hom(R), FR). The functor H has a left inverse Z : RingedParOrd0 → Ring. The category RingedParOrd0 is a fibred category. Mathematics Subject Classification (2010). Primary 18D30.
{"title":"A natural fibration for rings","authors":"A. Bosi, A. Facchini","doi":"10.4171/RSMUP/76","DOIUrl":"https://doi.org/10.4171/RSMUP/76","url":null,"abstract":"A ringed partially ordered set with zero is a pair (L,F ), where L is a partially ordered set with a least element 0L and F : L → Ring is a covariant functor. Here the partially ordered set L is given a category structure in the usual way and Ring denotes the category of associative rings with identity. Let RingedParOrd0 be the category of ringed partially ordered sets with zero. There is a functor H : Ring → RingedParOrd0 that associates to any ring R a ringed partially ordered set with zero (Hom(R), FR). The functor H has a left inverse Z : RingedParOrd0 → Ring. The category RingedParOrd0 is a fibred category. Mathematics Subject Classification (2010). Primary 18D30.","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"154 1","pages":"167-180"},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85402891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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