K. N. Adédji, Marija Bliznac Trebješanin, A. Filipin, A. Togbé
Let (a) and (b=ka) be positive integers with (kin {2, 3, 6},) such that (ab+4) is a perfect square. In this paper, we study the extensibility of the (D(4))-pairs ({a, ka}.) More precisely, we prove that by considering families of positive integers (c) depending on (a,) if ({a, b, c, d}) is a set of positive integers which has the property that the product of any two of its elements increased by (4) is a perfect square, then (d) is given by d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))). As a corollary, we prove that any (D(4))-quadruple tht contains the pair ({a, ka}) is regular.
{"title":"On the (D(4))-pairs ({a, ka}) with (kin {2,3,6})","authors":"K. N. Adédji, Marija Bliznac Trebješanin, A. Filipin, A. Togbé","doi":"10.3336/gm.58.1.03","DOIUrl":"https://doi.org/10.3336/gm.58.1.03","url":null,"abstract":"Let (a) and (b=ka) be positive integers with (kin {2, 3, 6},) such that (ab+4) is a perfect square. In this paper, we study the extensibility of the (D(4))-pairs ({a, ka}.) More precisely, we prove that by considering families of positive integers (c) depending on (a,) if ({a, b, c, d}) is a set of positive integers which has the property that the product of any two of its elements increased by (4) is a perfect square, then (d) is given by\u0000\u0000 d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))).\u0000\u0000As a corollary, we prove that any (D(4))-quadruple tht contains the pair ({a, ka}) is regular.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74547754","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}
A cyclotomic polynomial (Phi_n(x)) is said to be flat if its nonzero coefficients involve only (pm1). In this paper, for odd primes (p lt q lt r) with (qequiv 1pmod p) and (9requiv pm1pmod {pq}), we prove that (Phi_{pqr}(x)) is flat if and only if (p=5), (qgeq 41), and (qequiv 1pmod 5).
{"title":"On the coefficients of a class of ternary cyclotomic polynomials","authors":"Bin Zhang","doi":"10.3336/gm.58.1.02","DOIUrl":"https://doi.org/10.3336/gm.58.1.02","url":null,"abstract":"A cyclotomic polynomial (Phi_n(x)) is said to be flat if its nonzero coefficients involve only (pm1).\u0000In this paper, for odd primes (p lt q lt r)\u0000with (qequiv 1pmod p) and (9requiv pm1pmod {pq}), we\u0000prove that (Phi_{pqr}(x)) is flat if and only if (p=5), (qgeq\u000041), and (qequiv 1pmod 5).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85713189","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}
Using properties of binary quadratic Diophantine equations, we prove that if (r=p^{m} q^{n}), where (p, q) are distinct odd primes and (m, n) are positive integers, then the equation (x^{2}-left(r^{2}+1right) y^{2}=r^{2}) has at most one positive integer solution ((x, y)) with (y lt r-1).
{"title":"A note on Dujella's unicity conjecture","authors":"M. Le, A. Srinivasan","doi":"10.3336/gm.58.1.04","DOIUrl":"https://doi.org/10.3336/gm.58.1.04","url":null,"abstract":"Using properties of binary quadratic Diophantine equations, we prove that if (r=p^{m} q^{n}), where (p, q) are distinct odd primes and (m, n) are positive integers, then the equation (x^{2}-left(r^{2}+1right) y^{2}=r^{2}) has at most one positive integer solution ((x, y)) with (y lt r-1).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74472345","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}
Vesna Gotovac DJogaš, K. Helisova, L. Klebanov, J. Stanek, I. Volchenkova
A new definition of random sets is proposed in the presented paper. It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets. Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases. We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations. A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements. The distance is then used to define a class of Borel subsets of B.A. Consequently, random sets are defined as measurable mappings taking values in the B.A. This approach enables us to use more general family of distances between realisations of random sets which allows us to make new statistical tests concerning equality of some characteristics of random set distributions. As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.
{"title":"A new definition of random set","authors":"Vesna Gotovac DJogaš, K. Helisova, L. Klebanov, J. Stanek, I. Volchenkova","doi":"10.3336/gm.58.1.10","DOIUrl":"https://doi.org/10.3336/gm.58.1.10","url":null,"abstract":"A new definition of random sets is proposed in the presented paper.\u0000It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets.\u0000Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases.\u0000We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations.\u0000A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements.\u0000The distance is then used to define a class of Borel subsets of B.A.\u0000Consequently, random sets are defined as measurable mappings taking values in the B.A.\u0000This approach enables us to use more general family of distances between realisations of random sets\u0000which allows us to make new statistical tests concerning equality of some characteristics of random set distributions.\u0000As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91342691","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}
The Baxter numbers (B_n) enumerate a lot of algebraic and combinatorial objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra and the pairs of twin binary trees on (n) nodes. The Turán inequalities and higher order Turán inequalities are related to the Laguerre-Pólya ((mathcal{L})-(mathcal{P})) class of real entire functions, and the (mathcal{L})-(mathcal{P}) class has a close relation with the Riemann hypothesis. The Turán type inequalities have received much attention. In this paper, we are mainly concerned with Turán type inequalities, or more precisely, the log-behavior, and the higher order Turán inequalities associated with the Baxter numbers. We prove the Turán inequalities (or equivalently, the log-concavity) of the sequences ({B_{n+1}/B_n}_{ngeqslant 0}) and ({hspace{-2.5pt}sqrt[n]{B_n}}_{ngeqslant 1}). Monotonicity of the sequence ({hspace{-2.5pt}sqrt[n]{B_n}}_{ngeqslant 1}) is also obtained. Finally, we prove that the sequences ({B_n/n!}_{ngeqslant 2}) and ({B_{n+1}B_n^{-1}/n!}_{ngeqslant 2}) satisfy the higher order Turán inequalities.
{"title":"Inequalities associated with the Baxter numbers","authors":"J. Zhao","doi":"10.3336/gm.58.1.01","DOIUrl":"https://doi.org/10.3336/gm.58.1.01","url":null,"abstract":"The Baxter numbers (B_n) enumerate a lot of algebraic and combinatorial objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra and the pairs of twin binary trees on (n) nodes.\u0000The Turán inequalities and higher order Turán inequalities are related to the Laguerre-Pólya ((mathcal{L})-(mathcal{P})) class of real entire functions, and the (mathcal{L})-(mathcal{P}) class has a close relation with the Riemann hypothesis. The Turán type inequalities have received much attention.\u0000In this paper, we are mainly concerned with Turán type inequalities, or more precisely, the log-behavior, and the higher order Turán inequalities associated with the Baxter numbers. We prove the Turán inequalities (or equivalently, the log-concavity) of the sequences ({B_{n+1}/B_n}_{ngeqslant 0}) and ({hspace{-2.5pt}sqrt[n]{B_n}}_{ngeqslant 1}).\u0000Monotonicity of the sequence ({hspace{-2.5pt}sqrt[n]{B_n}}_{ngeqslant 1}) is also obtained. Finally, we prove that the sequences ({B_n/n!}_{ngeqslant 2}) and ({B_{n+1}B_n^{-1}/n!}_{ngeqslant 2}) satisfy the higher order Turán inequalities.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81558591","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}
In this paper, we define the directional edge escaping points set of function iteration under a given plane partition and then prove that the upper bound of Hausdorff dimension of the directional edge escaping points set of (S(z)=a e^{z}+b e^{-z}), where (a, bin mathbb{C}) and (|a|^{2}+|b|^{2}neq 0), is no more than 1.
本文定义了给定平面划分下函数迭代的方向边转义点集，并证明了(S(z)=a e^{z}+b e^{-z})的方向边转义点集的Hausdorff维数上界不大于1，其中(a, bin mathbb{C})和(|a|^{2}+|b|^{2}neq 0)。
{"title":"The Hausdorff dimension of directional edge escaping points set","authors":"Xiaojie Huang, Zhixiu Liu, Yuntong Li","doi":"10.3336/gm.57.2.05","DOIUrl":"https://doi.org/10.3336/gm.57.2.05","url":null,"abstract":"In this paper, we define the directional edge escaping points set of function iteration under a given plane partition and then prove that the upper bound of Hausdorff dimension of the directional edge escaping points set of (S(z)=a e^{z}+b e^{-z}), where (a, bin mathbb{C}) and (|a|^{2}+|b|^{2}neq 0), is no more than 1.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75533129","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}
The aim of this paper is to give some Datko and Barbashin type characterizations for the uniform (h)-instability of evolution families in Banach spaces, by using some important sets of growth rates. We prove four characterization theorems of Datko type and two characterization theorems of Barbashin type for uniform (h)-instability. Variants for uniform (h)-instability of some well-known results in stability theory (Barbashin (1967), Datko (1972)) are obtained.
{"title":"Some Datko and Barbashin type characterizations for the uniform (h)-instability of evolution families","authors":"Tian Yue","doi":"10.3336/gm.57.2.07","DOIUrl":"https://doi.org/10.3336/gm.57.2.07","url":null,"abstract":"The aim of this paper is to give some Datko and Barbashin type characterizations for the uniform (h)-instability of evolution families in Banach spaces, by using some important sets of growth rates. We prove four characterization theorems of Datko type and two characterization theorems of Barbashin type for uniform (h)-instability. Variants for uniform (h)-instability of some well-known results in stability theory (Barbashin (1967), Datko (1972)) are obtained.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87437340","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}
We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points (F) of the sum of divisors function (sigma : {mathbb{F}}_2[x] mapsto {mathbb{F}}_2[x]) (defined mutatis mutandi like the usual sum of divisors over the integers) of the form (F := A^2 cdot S), (S) square-free, with (omega(S) leq 3), coprime with (A), for (A) even, of whatever degree, under some conditions. This gives a characterization of (5) of the (11) known fixed points of (sigma) in ({mathbb{F}}_2[x]).
{"title":"Fixed points of the sum of divisors function on ({{mathbb{F}}}_2[x])","authors":"L. Gallardo","doi":"10.3336/gm.57.2.04","DOIUrl":"https://doi.org/10.3336/gm.57.2.04","url":null,"abstract":"We work on an analogue of a classical arithmetic problem over polynomials. More precisely,\u0000we study the fixed points (F) of the sum of divisors function (sigma : {mathbb{F}}_2[x] mapsto {mathbb{F}}_2[x])\u0000(defined mutatis mutandi like the usual sum of divisors over the integers)\u0000 of the form (F := A^2 cdot S), (S) square-free, with (omega(S) leq 3), coprime with (A), for (A) even, of whatever degree, under some conditions. This gives a characterization of (5) of the (11) known fixed points of (sigma) in ({mathbb{F}}_2[x]).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89118514","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}
In this paper we prove that there does not exist a set of four non-zero polynomials from (mathbb{Z}[X]), not all constant, such that the product of any two of its distinct elements decreased by (3) is a square of a polynomial from (mathbb{Z}[X]).
{"title":"On the existence of (D(-3))-quadruples over (mathbb{Z})","authors":"A. Filipin, Ana Jurasic","doi":"10.3336/gm.57.2.03","DOIUrl":"https://doi.org/10.3336/gm.57.2.03","url":null,"abstract":"In this paper we prove that there does not exist a set of four non-zero polynomials from (mathbb{Z}[X]), not all constant, such that the product of any two of its distinct elements decreased by (3) is a square of a polynomial from (mathbb{Z}[X]).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76809989","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}
In this paper we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type (C^{(1)}_ell), (ellgeq2), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.
{"title":"New partition identities from (C^{(1)}_ell)-modules","authors":"S. Capparelli, A. Meurman, Andrej Primc, M. Primc","doi":"10.3336/gm.57.2.01","DOIUrl":"https://doi.org/10.3336/gm.57.2.01","url":null,"abstract":"In this paper we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type (C^{(1)}_ell), (ellgeq2), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80912433","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：