In this paper we study the sum , where denotes the number of divisors of n, and {np } is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is . As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form.
{"title":"The Average Number of Divisors in Certain Arithmetic Sequences","authors":"Liubomir Chiriac","doi":"10.1556/314.2022.00019","DOIUrl":"https://doi.org/10.1556/314.2022.00019","url":null,"abstract":"In this paper we study the sum \u0000 \u0000 , where \u0000 \u0000 denotes the number of divisors of n, and {np\u0000 } is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is \u0000 \u0000 . As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"36 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123530161","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 study the path behavior of the symmetric walk on some special comb-type subsets of ℤ2 which are obtained from ℤ2 by generalizing the comb having finitely many horizontal lines instead of one.
{"title":"Random Walks on the Two-Dimensional K-Comb Lattice","authors":"Endre Cs'aki, A. Foldes","doi":"10.1556/314.2023.00001","DOIUrl":"https://doi.org/10.1556/314.2023.00001","url":null,"abstract":"We study the path behavior of the symmetric walk on some special comb-type subsets of ℤ2 which are obtained from ℤ2 by generalizing the comb having finitely many horizontal lines instead of one.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127137529","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 derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.
{"title":"Some New Inequalities Involving the Generalized Hardy Operator","authors":"Kristina Krulić Himmelreich","doi":"10.1556/314.2022.00016","DOIUrl":"https://doi.org/10.1556/314.2022.00016","url":null,"abstract":"In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129679939","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 bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.
{"title":"Bipartite Domination in Graphs","authors":"Anna Bachstein, W. Goddard, Michael A. Henning","doi":"10.1556/314.2022.00015","DOIUrl":"https://doi.org/10.1556/314.2022.00015","url":null,"abstract":"The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114112887","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}
This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.
{"title":"Polynomial Interpolation on Sequences","authors":"F. Tugores, L. Tugores","doi":"10.1556/314.2022.00013","DOIUrl":"https://doi.org/10.1556/314.2022.00013","url":null,"abstract":"This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133083250","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}
For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, y ∈ L, x ∧ y = a, and x ∨ y = b, then y ∈ S. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.
对于有限长度n的格L，设RCSub(L)为其空集和L上取相对补而封闭的子格的集合。即X L属于一个子集RCSub (L)当且仅当X是join-closed, meet-closed,每当{a, X, b}⊆年代,y∈L,∧y = X, X∨y = b,然后y∈S证明(1)偏序集RCSub (L)对集包含的晶格长度n + 1,(2)如果RCSub (L)是排名晶格和L是模块化的,然后在Andras p L 2-distributive Huhn的意义,和(3)如果L是分配,然后RCSub排名(L)是一个晶格。
{"title":"A Property of Lattices of Sublattices Closed Under Taking Relative Complements and Its Connection to 2-Distributivity","authors":"G. Czédli","doi":"10.1556/314.2022.00014","DOIUrl":"https://doi.org/10.1556/314.2022.00014","url":null,"abstract":"For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, y ∈ L, x ∧ y = a, and x ∨ y = b, then y ∈ S. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131209596","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, centralizing (semi-centralizing) and commuting (semi-commuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semi-centralizing) derivations of prime semirings. Further, we observe that if there exists a skew-commuting (skew-centralizing) derivation D of S, then D = 0. It is also proved that for any two derivations d 1 and d 2 of a prime semiring S with char S ≠ 2 and x d 1 x d 2 = 0, for all x ∈ S implies either d 1 = 0 or d 2 = 0.
{"title":"On Derivations and Lie Ideals of Semirings","authors":"M. Dadhwal, Neelam","doi":"10.1556/314.2022.00011","DOIUrl":"https://doi.org/10.1556/314.2022.00011","url":null,"abstract":"In this paper, centralizing (semi-centralizing) and commuting (semi-commuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semi-centralizing) derivations of prime semirings. Further, we observe that if there exists a skew-commuting (skew-centralizing) derivation D of S, then D = 0. It is also proved that for any two derivations d\u0000 1 and d\u0000 2 of a prime semiring S with char S ≠ 2 and x\u0000 \u0000 d\u0000 1\u0000 \u0000 x\u0000 \u0000 d\u0000 2\u0000 = 0, for all x ∈ S implies either d\u0000 1 = 0 or d\u0000 2 = 0.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123767305","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":"In Memoriam István Győri","authors":"F. Hartung, M. Pituk","doi":"10.1556/314.2022.00012","DOIUrl":"https://doi.org/10.1556/314.2022.00012","url":null,"abstract":"","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124048118","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 offer new properties of the special Gini mean S(a, b) = aa/(a+b) ⋅ bb/(a+b), in connections with other special means of two arguments.
我们给出了特殊基尼均值S(a, b) = aa /(a + b)·bb /(a + b)的新性质，并与其他特殊的两个参数相联系。
{"title":"On a Special Gini Mean","authors":"J. Sándor","doi":"10.1556/314.2022.00010","DOIUrl":"https://doi.org/10.1556/314.2022.00010","url":null,"abstract":"<jats:p>We offer new properties of the special Gini mean <jats:italic>S</jats:italic>(<jats:italic>a, b</jats:italic>) = <jats:italic>a<jats:sup>a</jats:sup>\u0000 </jats:italic>\u0000 <jats:sup>/(</jats:sup>\u0000 <jats:italic>\u0000 <jats:sup>a</jats:sup>\u0000 </jats:italic>\u0000 <jats:sup>+</jats:sup>\u0000 <jats:italic>\u0000 <jats:sup>b</jats:sup>\u0000 </jats:italic>\u0000 <jats:sup>)</jats:sup> ⋅ <jats:italic>b<jats:sup>b</jats:sup>\u0000 </jats:italic>\u0000 <jats:sup>/(</jats:sup>\u0000 <jats:italic>\u0000 <jats:sup>a</jats:sup>\u0000 </jats:italic>\u0000 <jats:sup>+</jats:sup>\u0000 <jats:italic>\u0000 <jats:sup>b</jats:sup>\u0000 </jats:italic>\u0000 <jats:sup>)</jats:sup>, in connections with other special means of two arguments.</jats:p>","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131486014","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}
Proctor and Scoppetta conjectured that (1) there exists an infinite locally finite poset that satisfies their conditions VT and NTC but not SIS; (2) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MF but not both VT and FT; and (3) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MD but not NCC. In this note, the conjecture of Proctor and Scoppetta, which is related to d-complete posets, is proven.
{"title":"Proof of a Conjecture of Proctor and Scoppetta Related to d-Complete Posets","authors":"J. Farley","doi":"10.1556/314.2022.00009","DOIUrl":"https://doi.org/10.1556/314.2022.00009","url":null,"abstract":"Proctor and Scoppetta conjectured that\u0000 \u0000 \u0000 \u0000 (1) there exists an infinite locally finite poset that satisfies their conditions VT and NTC but not SIS;\u0000 \u0000 \u0000 (2) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MF but not both VT and FT; and\u0000 \u0000 \u0000 (3) there exists an infinite locally finite poset satisfying their conditions D3-C and D3MD but not NCC.\u0000 \u0000 \u0000 \u0000 In this note, the conjecture of Proctor and Scoppetta, which is related to d-complete posets, is proven.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"143 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133259367","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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