The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.
本文的主要目的是证明关于Walsh - Kaczmarz系统的Riesz对数核的非负性不成立。
{"title":"On the Negativity of the Walsh–Kaczmarz–Riesz Logarithmic Kernels","authors":"G. Gát, Gábor Lucskai","doi":"10.1556/314.2021.00018","DOIUrl":"https://doi.org/10.1556/314.2021.00018","url":null,"abstract":"The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"211 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116154047","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 work out a Riemann–von Mangoldt type formula for the summatory function := , where is an arithmetical semigroup (a Beurling generalized system of integers) and is the corresponding von Mangoldt function attaining with a prime element and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function , belonging to , to the number of zeroes of in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.
{"title":"A Riemann–von Mangoldt-Type Formula for the Distribution of Beurling Primes","authors":"S. R'ev'esz","doi":"10.1556/314.2021.00019","DOIUrl":"https://doi.org/10.1556/314.2021.00019","url":null,"abstract":"In this paper we work out a Riemann–von Mangoldt type formula for the summatory function := , where is an arithmetical semigroup (a Beurling generalized system of integers) and is the corresponding von Mangoldt function attaining with a prime element and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function , belonging to , to the number of zeroes of in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116703276","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}
Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.Finally, for the rings of integers modulo n, we determine the 2x2 matrices which are (or not) full and their numbers.
{"title":"On Zero Determinant Matrices that are Full","authors":"G. Călugăreanu, Horia F. Pop","doi":"10.1556/314.2021.00008","DOIUrl":"https://doi.org/10.1556/314.2021.00008","url":null,"abstract":"Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.Finally, for the rings of integers modulo n, we determine the 2x2 matrices which are (or not) full and their numbers.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114873809","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}
Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).
{"title":"From Binary Groups to Terminal Rings","authors":"S. D. Scott","doi":"10.1556/314.2021.00013","DOIUrl":"https://doi.org/10.1556/314.2021.00013","url":null,"abstract":"Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131269489","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 continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transformwhere the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, thenandwhere is the second derivative of as a real function.Applications for power function and logarithm are also provided.
{"title":"Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces","authors":"S. Dragomir","doi":"10.1556/314.2021.00011","DOIUrl":"https://doi.org/10.1556/314.2021.00011","url":null,"abstract":"For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transformwhere the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, thenandwhere is the second derivative of as a real function.Applications for power function and logarithm are also provided.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122350663","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 authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such circles of curvature at the points of a parabola in the isotropic plane.
{"title":"Circles of Curvature at Points of Parabola in Isotropic Plane","authors":"V. Volenec, Marija Šimić Horvath, E. Jurkin","doi":"10.1556/314.2021.00012","DOIUrl":"https://doi.org/10.1556/314.2021.00012","url":null,"abstract":"The authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such circles of curvature at the points of a parabola in the isotropic plane.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132177801","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 k ≥ 1. A Sperner k-family is a maximum-sized subset of a finite poset that contains no chain with k + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set Sk(P) of Sperner k-families of a fifinite poset P and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of Sk(P),” adding, “This has apparently not been done even for the case k = 1.”In this article, the case k = 1 is done.
{"title":"The “k = 1” Case of a Problem of Greene and Kleitman from 1976: Join-Irreducible Elements in the Lattice of Sperner 1-Families","authors":"J. Farley","doi":"10.1556/314.2021.00010","DOIUrl":"https://doi.org/10.1556/314.2021.00010","url":null,"abstract":"Let k ≥ 1. A Sperner k-family is a maximum-sized subset of a finite poset that contains no chain with k + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set Sk(P) of Sperner k-families of a fifinite poset P and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of Sk(P),” adding, “This has apparently not been done even for the case k = 1.”In this article, the case k = 1 is done.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"126 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121488460","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 prove certain Menon-type identities associated with the subsets of the set {1, 2,..., n} and related to the functions f, fk , Ф and Ф k , defined and investigated by Nathanson.
{"title":"Menon-Type Identities Concerning Subsets of the Set {1, 2,..., n}","authors":"L. Tóth","doi":"10.1556/314.2022.00008","DOIUrl":"https://doi.org/10.1556/314.2022.00008","url":null,"abstract":"We prove certain Menon-type identities associated with the subsets of the set {1, 2,..., n} and related to the functions f, fk\u0000 , Ф and Ф\u0000 k\u0000 , defined and investigated by Nathanson.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"140 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123321646","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 ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.
{"title":"Interval Chains and Completeness in Ultrapowers of Ordered Sets","authors":"Z. Boros, P'eter V. T'oth","doi":"10.1556/314.2022.00004","DOIUrl":"https://doi.org/10.1556/314.2022.00004","url":null,"abstract":"The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129081648","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 many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.
{"title":"Lower Estimate of Clique Size via Edge Coloring","authors":"Balázs Király, S. Szabó","doi":"10.1556/314.2020.00002","DOIUrl":"https://doi.org/10.1556/314.2020.00002","url":null,"abstract":"In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.","PeriodicalId":383314,"journal":{"name":"Mathematica Pannonica","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129515359","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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