{"title":"On direct and inverse problems related to some dilated sumsets","authors":"Ramandeep Kaur, Sandeep Singh","doi":"10.5802/crmath.537","DOIUrl":"https://doi.org/10.5802/crmath.537","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"88 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139869302","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 $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $hin H$ where, the sumset $h^{(r)}A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)}A$, i.e., for $|H^{(r)}A|$ and the structure of the underlying sets $A$ and $H$ when $|H^{(r)}A|$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$, since $H^{(r)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.
{"title":"Generalized H-fold sumset and Subsequence sum","authors":"Mohan, Ram Krishna Pandey","doi":"10.5802/crmath.483","DOIUrl":"https://doi.org/10.5802/crmath.483","url":null,"abstract":"Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $hin H$ where, the sumset $h^{(r)}A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)}A$, i.e., for $|H^{(r)}A|$ and the structure of the underlying sets $A$ and $H$ when $|H^{(r)}A|$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$, since $H^{(r)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"6 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140509254","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":"Salem numbers of automorphisms of K3 surfaces with Picard number 4","authors":"Kwangwoo Lee","doi":"10.5802/crmath.533","DOIUrl":"https://doi.org/10.5802/crmath.533","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"44 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138950883","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":"On Bousfield’s conjectures for the unstable Adams spectral sequence for SO and U","authors":"Thế Cường Nguyễn","doi":"10.5802/crmath.531","DOIUrl":"https://doi.org/10.5802/crmath.531","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"16 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138948503","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 framework for shallow approximations of incompressible flows","authors":"Nathan Shourick, I. Cheddadi, Pierre Saramito","doi":"10.5802/crmath.526","DOIUrl":"https://doi.org/10.5802/crmath.526","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"59 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138951696","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}
Michel Duprez, V. Lleras, A. Lozinski, Killian Vuillemot
{"title":"ϕ-FEM for the heat equation: optimal convergence on unfitted meshes in space","authors":"Michel Duprez, V. Lleras, A. Lozinski, Killian Vuillemot","doi":"10.5802/crmath.497","DOIUrl":"https://doi.org/10.5802/crmath.497","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"25 17","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138950175","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":"Equivalence of K-functionals and modulus of smoothness generated by a Dunkl type operator on the interval (-1,1)","authors":"Faouaz Saasi, R. Daher","doi":"10.5802/crmath.517","DOIUrl":"https://doi.org/10.5802/crmath.517","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"66 1-2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139262612","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":"On the boundedness of a family of oscillatory singular integrals","authors":"H. Al-Qassem, Leslie C. Cheng, Yibiao Pan","doi":"10.5802/crmath.523","DOIUrl":"https://doi.org/10.5802/crmath.523","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"29 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139264298","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":"Remarks on complexities and entropies for singularity categories","authors":"Ryo Takahashi","doi":"10.5802/crmath.482","DOIUrl":"https://doi.org/10.5802/crmath.482","url":null,"abstract":"","PeriodicalId":395483,"journal":{"name":"Comptes Rendus. Mathématique","volume":"8 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139263710","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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