We prove that if {un}n≥ 0 is a nondegenerate Lucas sequence, then there are only finitely many effectively computable positive integers n such that |un|=f(m!), where f is either the sum-of-divisors function, or the sum-of-proper-divisors function, or the Euler phi function. We also give a theorem that holds for a more general class of integer sequences and illustrate our results through a few specific examples. This paper is motivated by a previous work of Iannucci and Luca who addressed the above problem with Catalan numbers and the sum-of-proper-divisors function.
{"title":"Some arithmetic functions of factorials in Lucas sequences","authors":"E. F. Bravo, Jhon J. Bravo","doi":"10.3336/gm.56.1.02","DOIUrl":"https://doi.org/10.3336/gm.56.1.02","url":null,"abstract":"We prove that if {un}n≥ 0 is a nondegenerate Lucas sequence, then there are only finitely many effectively computable positive integers n such that |un|=f(m!), where f is either the sum-of-divisors function, or the sum-of-proper-divisors function, or the Euler phi function. We also give a theorem that holds for a more general class of integer sequences and illustrate our results through a few specific examples. This paper is motivated by a previous work of Iannucci and Luca who addressed the above problem with Catalan numbers and the sum-of-proper-divisors function.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82535494","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 investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].
我们研究了有限域上有理函数场的经典Pólya和Turán猜想𝔽q。结合这两个猜想，我们研究了𝔽q[t]上二次字符对应的Dirichlet l -函数在s=1点处的截断符号，证明了𝔽q[t]上任意一元不可约多项式集的朗道定理的一个变体，并计算了𝔽q[t]上Liouville函数的某些变体的均值。
{"title":"Extremal behaviour of ± 1-valued completely multiplicative functions in function fields","authors":"Nikola Lelas","doi":"10.3336/gm.56.1.06","DOIUrl":"https://doi.org/10.3336/gm.56.1.06","url":null,"abstract":"We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85639479","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 introduce and characterize angular right symmetric and approximate angular right symmetric points of the algebra of all bounded linear operators defined on either real or complex Hilbert spaces.
引入并刻画了在实数或复希尔伯特空间上定义的所有有界线性算子的代数的角右对称点和近似角右对称点。
{"title":"Angular right symmetricity of bounded linear operators on Hilbert spaces","authors":"S. M. S. Nabavi Sales","doi":"10.3336/gm.56.1.09","DOIUrl":"https://doi.org/10.3336/gm.56.1.09","url":null,"abstract":"We introduce and characterize angular right symmetric and approximate angular right symmetric points of the algebra of all bounded linear operators defined on either real or complex Hilbert spaces.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84137215","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}
This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.
{"title":"Approximate inverse limits and (m,n)-dimensions","authors":"J. Peters","doi":"10.3336/gm.56.1.11","DOIUrl":"https://doi.org/10.3336/gm.56.1.11","url":null,"abstract":"This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86545938","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 propose a systematic and topological study of limits (lim_{nuto 0^+}G_mathbb{R}cdot(nu x)) of continuous families of adjoint orbits for a non-compact simple real Lie group (G_mathbb{R}). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of (mathrm{SL}_n(mathbb{R})) and (mathrm{SU}(p,q)) are computed in detail.
{"title":"Approximation of nilpotent orbits for simple Lie groups","authors":"Lucas Fresse, S. Mehdi","doi":"10.3336/gm.56.2.06","DOIUrl":"https://doi.org/10.3336/gm.56.2.06","url":null,"abstract":"We propose a systematic and topological study of limits (lim_{nuto 0^+}G_mathbb{R}cdot(nu x)) of continuous families of adjoint orbits for a non-compact simple real Lie group (G_mathbb{R}). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of (mathrm{SL}_n(mathbb{R})) and (mathrm{SU}(p,q)) are computed in detail.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72426004","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}
G. Dolinar, B. Kuzma, J. Marovt, B. Ungor, Jadranska ulica Si Ljubljana Slovenia Imfm, Razlagova Maribor Slovenia Business
Let 𝓡 be a ring with identity and let 𝓙𝓡 be a collection of subsets of 𝓡 such that their left and right annihilators are generated by the same idempotent. % from 𝓡. We extend the notion of the sharp, the left-sharp, and the right-sharp partial orders to 𝓙𝓡, present equivalent definitions of these orders, and study their properties. We also extend the concept of the core and the dual core orders to 𝓙𝓡, show that they are indeed partial orders when 𝓡 is a Baer *-ring, and connect them with one-sided sharp and star partial orders.
设一个具有恒等的环，设𝓙一个子集的集合，它们的左右湮灭子由相同的幂等子产生。% from 。我们将尖阶、左尖阶和右尖阶的概念推广到𝓙，给出了这些阶的等价定义，并研究了它们的性质。我们还将核阶和双核阶的概念推广到𝓙- edu，证明了当edu是Baer *环时它们确实是偏阶，并将它们与片面尖阶和星型偏阶联系起来。
{"title":"On some partial orders on a certain subset of the power set of rings","authors":"G. Dolinar, B. Kuzma, J. Marovt, B. Ungor, Jadranska ulica Si Ljubljana Slovenia Imfm, Razlagova Maribor Slovenia Business","doi":"10.3336/gm.55.2.01","DOIUrl":"https://doi.org/10.3336/gm.55.2.01","url":null,"abstract":"Let 𝓡 be a ring with identity and let 𝓙𝓡 be a collection of subsets of 𝓡 such that their left and right annihilators are generated by the same idempotent. % from 𝓡. We extend the notion of the sharp, the left-sharp, and the right-sharp partial orders to 𝓙𝓡, present equivalent definitions of these orders, and study their properties. We also extend the concept of the core and the dual core orders to 𝓙𝓡, show that they are indeed partial orders when 𝓡 is a Baer *-ring, and connect them with one-sided sharp and star partial orders.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77724324","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}
Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).
{"title":"A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z","authors":"M. Le, G. Soydan","doi":"10.3336/gm.55.2.03","DOIUrl":"https://doi.org/10.3336/gm.55.2.03","url":null,"abstract":"Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77546414","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 generalize a result of the first author who proved that the Čech system of open covers of a Hausdorff arc-like space cannot induce an approximate system of the nerves of these covers under any choices of the meshes and the projections.
{"title":"Čech systems and approximate inverse systems","authors":"Vlasta Matijević, L. Rubin","doi":"10.3336/gm.55.2.12","DOIUrl":"https://doi.org/10.3336/gm.55.2.12","url":null,"abstract":"We generalize a result of the first author who proved that the Čech system of open covers of a Hausdorff arc-like space cannot induce an approximate system of the nerves of these covers under any choices of the meshes and the projections.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77712637","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 significantly improve our previous results of reducing relative Thue inequalities to absolute ones.
在本文中，我们显著地改进了之前将相对Thue不等式简化为绝对Thue不等式的结果。
{"title":"otally real Thue inequalities over imaginary quadratic fields: an improvement","authors":"Istv'an Ga'al, Borka Jadrijevic, László Remete","doi":"10.3336/gm.55.2.02","DOIUrl":"https://doi.org/10.3336/gm.55.2.02","url":null,"abstract":"In this paper we significantly improve our previous results of reducing relative Thue inequalities to absolute ones.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74098370","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 define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.
{"title":"Dirichlet product and the multiple Dirichlet series over function fields","authors":"Y. Hamahata","doi":"10.3336/gm.55.2.06","DOIUrl":"https://doi.org/10.3336/gm.55.2.06","url":null,"abstract":"We define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73849772","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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