Pub Date : 2023-01-01DOI: 10.59277/mrar.2023.25.75.2.313
A. Dubickas
In this note, we give some lower and upper bounds for the gaps between consecutive elements of an integer sequence whose elements are expressible in the form P(m) + P(n), where P is a degree d polynomial with integer coefficients, and m, n are nonnegative
{"title":"GAPS IN A SUMSET OF A POLYNOMIAL WITH ITSELF","authors":"A. Dubickas","doi":"10.59277/mrar.2023.25.75.2.313","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.2.313","url":null,"abstract":"In this note, we give some lower and upper bounds for the gaps between consecutive elements of an integer sequence whose elements are expressible in the form P(m) + P(n), where P is a degree d polynomial with integer coefficients, and m, n are nonnegative","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"31 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75535585","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.3.441
ILEANA BUCUR, GAVRIIL PALTINEANU
In the first part of the paper, we present a theorem of type “partition of unity”, and three of its consequences, i.e., for algebras, linear subspaces and convex cones. In the second part, some theorems of localizability (or density) in a weighted space are presented. We mention that the weighted spaces are classes of continuous scalar functions on a locally compact space (for example, the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).
{"title":"A THEOREM OF TYPE “PARTITION OF UNITY” WITH","authors":"ILEANA BUCUR, GAVRIIL PALTINEANU","doi":"10.59277/mrar.2023.25.75.3.441","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.3.441","url":null,"abstract":"In the first part of the paper, we present a theorem of type “partition of unity”, and three of its consequences, i.e., for algebras, linear subspaces and convex cones. In the second part, some theorems of localizability (or density) in a weighted space are presented. We mention that the weighted spaces are classes of continuous scalar functions on a locally compact space (for example, the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135954193","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.3.455
M. M. AL-SHOMRANI, M. EZZAT MOHAMED
Let G be a finite group. A subgroup of G is called S-quasinormal in G if it permutes with each Sylow subgroup of G. In this paper, we investigate the structure of the finite group G when certain abelian subgroups of largest possible exponent of prime power orders are S-quasinormal in G.
{"title":"THE INFLUENCE OF SOME SUBGROUPS OF PRIME POWER","authors":"M. M. AL-SHOMRANI, M. EZZAT MOHAMED","doi":"10.59277/mrar.2023.25.75.3.455","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.3.455","url":null,"abstract":"Let G be a finite group. A subgroup of G is called S-quasinormal in G if it permutes with each Sylow subgroup of G. In this paper, we investigate the structure of the finite group G when certain abelian subgroups of largest possible exponent of prime power orders are S-quasinormal in G.","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135954542","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.3.495
ZORAN Z. PETROVIC, MAJA ROSLAVCEV
Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the term order is POT, or, “position over term”. In order to do this, we first prove that there is a Gr¨obner basis for finitely generated ideals in R[X], which is a special case of the main result. The proof is based on the results from [3]. In addition to this we show, in the case of valuation domains, that every Gr¨obner basis is actually a strong Gr¨obner basis
{"title":"GROBNER BASES FOR MODULES OVER PRUFER DOMAINS","authors":"ZORAN Z. PETROVIC, MAJA ROSLAVCEV","doi":"10.59277/mrar.2023.25.75.3.495","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.3.495","url":null,"abstract":"Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the term order is POT, or, “position over term”. In order to do this, we first prove that there is a Gr¨obner basis for finitely generated ideals in R[X], which is a special case of the main result. The proof is based on the results from [3]. In addition to this we show, in the case of valuation domains, that every Gr¨obner basis is actually a strong Gr¨obner basis","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135953725","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.3.381
SERMIN CAM CELIK, HAYDAR GORAL
In this note, we study the evaluations of Euler sums via trigonometric series. It is a commonly believed conjecture that for an even weight greater than seven, Euler sums cannot be evaluated in terms of the special values of the Riemann zeta function. For an even weight, we reduce the evaluations of Euler sums into the evaluations of double series and integrals of products of Clausen functions. We also re-evaluate Euler sums of odd weight using a new method based on trigonometric series.
{"title":"EULER–ZAGIER SUMS VIA TRIGONOMETRIC SERIES","authors":"SERMIN CAM CELIK, HAYDAR GORAL","doi":"10.59277/mrar.2023.25.75.3.381","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.3.381","url":null,"abstract":"In this note, we study the evaluations of Euler sums via trigonometric series. It is a commonly believed conjecture that for an even weight greater than seven, Euler sums cannot be evaluated in terms of the special values of the Riemann zeta function. For an even weight, we reduce the evaluations of Euler sums into the evaluations of double series and integrals of products of Clausen functions. We also re-evaluate Euler sums of odd weight using a new method based on trigonometric series.","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"295 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135953773","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.2.319
KH HERA SINGH, A. M. Buhphang, M. B. Rege
In this paper, we introduce the concepts niltorsionless modules and NSmodules (which are generalisations of NI-rings). We prove that the class of niltorsionless modules contains the classes of semiprime and in particular, the class of regular modules. We
{"title":"ARMENDARIZ MODULES AND NILTORSIONLESS MODULES","authors":"KH HERA SINGH, A. M. Buhphang, M. B. Rege","doi":"10.59277/mrar.2023.25.75.2.319","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.2.319","url":null,"abstract":"In this paper, we introduce the concepts niltorsionless modules and NSmodules (which are generalisations of NI-rings). We prove that the class of niltorsionless modules contains the classes of semiprime and in particular, the class of regular modules. We","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"75 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77241419","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.3.481
HUIHUI ZHENG, FANGSHU LI, TIANSHUI MA
In this paper, we mainly give some equivalent characterisations of Hopf cobraces, show that the full subcategory HCB(A) of Hopf co-braces is equivalent to the full subcategory C(A) of bijective 1-cocycles, and prove that the full subcategory HCB(A) is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many Hopf co-braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel’d doubles of finite dimensional Hopf algebras. And we also give a sufficient and necessary condition for a given bicrossed coproduct A ▷◁ H to be a Hopf co-brace if A or H is a Hopf co-brace
{"title":"HOPF CO-BRACE, BRAID EQUATION AND BICROSSED","authors":"HUIHUI ZHENG, FANGSHU LI, TIANSHUI MA","doi":"10.59277/mrar.2023.25.75.3.481","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.3.481","url":null,"abstract":"In this paper, we mainly give some equivalent characterisations of Hopf cobraces, show that the full subcategory HCB(A) of Hopf co-braces is equivalent to the full subcategory C(A) of bijective 1-cocycles, and prove that the full subcategory HCB(A) is also equivalent to the category M(A) of Hopf matched pairs. Moreover, we construct many Hopf co-braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel’d doubles of finite dimensional Hopf algebras. And we also give a sufficient and necessary condition for a given bicrossed coproduct A ▷◁ H to be a Hopf co-brace if A or H is a Hopf co-brace","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135953488","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.3.425
CHENGLONG FANG
In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey TriebelLizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator.
{"title":"CHARACTERIZATIONS OF COMMUTATORS OF SINGULAR INTEGRAL OPERATORS ON MORREY TRIEBEL-LIZORKIN","authors":"CHENGLONG FANG","doi":"10.59277/mrar.2023.25.75.3.425","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.3.425","url":null,"abstract":"In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey TriebelLizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator.","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135953720","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.2.279
N. Azimi Shahrabi, M. Akhavan-Malayeri
Let Γ(G) denote the set of commutators of a group G, G′ = ⟨Γ(G)⟩ and c(G) (or cw(G)) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. Recently we proved that if G is a finite p-group of maximal cl
{"title":"ON COMMUTATORS IN FINITE P-GROUPS OF ALMOST MAXIMAL CLASS","authors":"N. Azimi Shahrabi, M. Akhavan-Malayeri","doi":"10.59277/mrar.2023.25.75.2.279","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.2.279","url":null,"abstract":"Let Γ(G) denote the set of commutators of a group G, G′ = ⟨Γ(G)⟩ and c(G) (or cw(G)) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. Recently we proved that if G is a finite p-group of maximal cl","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88037644","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 : 2023-01-01DOI: 10.59277/mrar.2023.25.75.2.331
Maosen Xu, Zhixiang Wu
In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abe
{"title":"HOMOLOGY THEORY OF MULTIPLICATIVE HOM-LIE ALGEBRAS","authors":"Maosen Xu, Zhixiang Wu","doi":"10.59277/mrar.2023.25.75.2.331","DOIUrl":"https://doi.org/10.59277/mrar.2023.25.75.2.331","url":null,"abstract":"In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abe","PeriodicalId":49858,"journal":{"name":"Mathematical Reports","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84583444","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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