Assume that k ≤ d is a positive integer and 𝓒 is a finite collection of convex bodies in ℝd. We prove a Helly-type theorem: If for every subfamily 𝓒* ⊂ 𝓒 of size at most max{d + 1, 2(d – k + 1)} the set ⋂ 𝓒* contains a k-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.
假设 k ≤ d 是正整数，𝓒 是 ℝ d 中凸体的有限集合。我们将证明一个海尔利类型定理：如果对于每个大小至多为 max{d + 1, 2(d - k + 1)} 的子域 𝓒* ⊂ 𝓒 的集合 ⋂ 𝓒* 包含一个 k 维锥，那么 ⋂ 𝓒 也包含一个 k 维锥。证明的一个要素是另一个关于凸锥体线性空间维数的海利定理。
{"title":"Positive bases, cones, Helly-type theorems","authors":"Imre Bárány","doi":"10.1515/ms-2024-0054","DOIUrl":"https://doi.org/10.1515/ms-2024-0054","url":null,"abstract":"Assume that <jats:italic>k</jats:italic> ≤ <jats:italic>d</jats:italic> is a positive integer and 𝓒 is a finite collection of convex bodies in ℝ<jats:sup> <jats:italic>d</jats:italic> </jats:sup>. We prove a Helly-type theorem: If for every subfamily 𝓒<jats:sup>*</jats:sup> ⊂ 𝓒 of size at most max{<jats:italic>d</jats:italic> + 1, 2(<jats:italic>d</jats:italic> – <jats:italic>k</jats:italic> + 1)} the set ⋂ 𝓒<jats:sup>*</jats:sup> contains a <jats:italic>k</jats:italic>-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Filbert and Lilbert matrices are defined by terms of linear Fibonacci-like sequences. By considering terms of such linear recurrences and additional free parameters, we define a nonlinear variant of these matrices via ratios of q-forms of terms of Fibonacci-like sequences whose indices are in nonlinear forms. We derive explicit formulae for the matrices L and U come from LU-decomposition, their inverses, inverse of the main matrices and their determinants.
菲尔伯特矩阵和利尔伯特矩阵是由线性斐波那契类序列的项定义的。通过考虑这种线性递归的项和额外的自由参数，我们通过指数为非线性形式的斐波纳契类序列项的 q 形式比率定义了这些矩阵的非线性变体。我们从 LU 分解、它们的逆、主矩阵的逆和它们的行列式中推导出矩阵 L 和 U 的明确公式。
{"title":"A nonlinear Filbert-like matrix with three free parameters: From linearity to nonlinearity","authors":"Emrah Kiliç, Didem Ersanli","doi":"10.1515/ms-2024-0044","DOIUrl":"https://doi.org/10.1515/ms-2024-0044","url":null,"abstract":"Filbert and Lilbert matrices are defined by terms of <jats:italic>linear</jats:italic> Fibonacci-like sequences. By considering terms of such linear recurrences and additional free parameters, we define a nonlinear variant of these matrices via ratios of <jats:italic>q</jats:italic>-forms of terms of Fibonacci-like sequences whose indices are in nonlinear forms. We derive explicit formulae for the matrices <jats:italic>L</jats:italic> and <jats:italic>U</jats:italic> come from <jats:italic>LU</jats:italic>-decomposition, their inverses, inverse of the main matrices and their determinants.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One point compactification is studied in the light of ideal of subsets of ℕ. 𝓘-proper map is introduced and showed that a continuous map can be extended continuously to the one point 𝓘-compactification if and only if the map is 𝓘-proper. Nowhere tallness, introduced by P. Matet and J. Pawlikowski in [J. Symb. Log. 63(3) (1998), 1040–1054], plays an important role in this article to study various properties of 𝓘-proper maps. It is seen that one point 𝓘-compactification of a topological space may fail to be Hausdorff even if the underlying topological space is Hausdorff but a class {𝓘} of ideals has been identified for which one point 𝓘-compactification coincides with the one point compactification if the underlying topological space is metrizable. Let’s speak our minds that the results in this article will look elegant if one looks at it from a topological angle.
{"title":"Influence of ideals in compactifications","authors":"Manoranjan Singha, Sima Roy","doi":"10.1515/ms-2024-0056","DOIUrl":"https://doi.org/10.1515/ms-2024-0056","url":null,"abstract":"One point compactification is studied in the light of ideal of subsets of ℕ. 𝓘-proper map is introduced and showed that a continuous map can be extended continuously to the one point 𝓘-compactification if and only if the map is 𝓘-proper. Nowhere tallness, introduced by P. Matet and J. Pawlikowski in [J. Symb. Log. 63(3) (1998), 1040–1054], plays an important role in this article to study various properties of 𝓘-proper maps. It is seen that one point 𝓘-compactification of a topological space may fail to be Hausdorff even if the underlying topological space is Hausdorff but a class {𝓘} of ideals has been identified for which one point 𝓘-compactification coincides with the one point compactification if the underlying topological space is metrizable. Let’s speak our minds that the results in this article will look elegant if one looks at it from a topological angle.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions.
{"title":"On universality in short intervals for zeta-functions of certain cusp forms","authors":"Antanas Laurinčikas, Darius Šiaučiūnas","doi":"10.1515/ms-2024-0045","DOIUrl":"https://doi.org/10.1515/ms-2024-0045","url":null,"abstract":"In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Murad Khan Hassani, Nouressadat Touafek, Yasin Yazlik
In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.
在本文中，我们考虑了一类二阶非线性差分方程系统，这是对近期文献中一些结果的极大扩展。我们的主要结果表明，该类差分方程系统在理论上是可以闭式求解的。值得注意的是，上述系统的解与广义梅森数有关。此外，还给出了 a = b 和 p = 0 时上述差分方程组解的渐近行为。最后，还给出了数值示例来支持本文的理论结果。
{"title":"On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence","authors":"Murad Khan Hassani, Nouressadat Touafek, Yasin Yazlik","doi":"10.1515/ms-2024-0053","DOIUrl":"https://doi.org/10.1515/ms-2024-0053","url":null,"abstract":"In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when <jats:italic>a</jats:italic> = <jats:italic>b</jats:italic> and <jats:italic>p</jats:italic> = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In terms of a very-well-poised 6ϕ5 summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish four new q-supercongruences for truncated basic hypergeometric series. One of these results is a new q-analogue of the (F.2) supercongruence of Van Hamme.
{"title":"New q-analogues of Van Hamme’s (F.2) supercongruence and of some related supercongruences","authors":"Qiuxia Hu","doi":"10.1515/ms-2024-0048","DOIUrl":"https://doi.org/10.1515/ms-2024-0048","url":null,"abstract":"In terms of a very-well-poised <jats:sub>6</jats:sub> <jats:italic>ϕ</jats:italic> <jats:sub>5</jats:sub> summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish four new <jats:italic>q</jats:italic>-supercongruences for truncated basic hypergeometric series. One of these results is a new <jats:italic>q</jats:italic>-analogue of the (F.2) supercongruence of Van Hamme.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10dFm + Lk and Fn = 10dLm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
设 (Fn ) n≥0 和 (Ln ) n≥0 分别为斐波那契数列和卢卡斯数列。在本文中，我们确定所有斐波那契数都是一个斐波那契数和一个卢卡斯数的混合并集。我们所说的 a 和 b 的混合并集是指 ab 和 ba 的并集，其中 a 和 b 是任意两个非负整数。因此，这个问题的数学表达式引导我们在非负整数 (n, m, k) 中搜索两个二叉方程 Fn = 10 d Fm + Lk 和 Fn = 10 d Lm + Fk 的解，其中 d 分别表示 Lk 和 Fk 的位数。我们利用对数线性形式的下界和 Diophantine 近似中的还原法得出结果。
{"title":"Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers","authors":"Alaa Altassan, Murat Alan","doi":"10.1515/ms-2024-0042","DOIUrl":"https://doi.org/10.1515/ms-2024-0042","url":null,"abstract":"Let (<jats:italic>F<jats:sub>n</jats:sub> </jats:italic>)<jats:sub> <jats:italic>n</jats:italic>≥0</jats:sub> and (<jats:italic>L<jats:sub>n</jats:sub> </jats:italic>)<jats:sub> <jats:italic>n</jats:italic>≥0</jats:sub> be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic>, we mean the both concatenations <jats:overline> <jats:italic>ab</jats:italic> </jats:overline> and <jats:overline> <jats:italic>ba</jats:italic> </jats:overline> together, where <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations <jats:italic>F<jats:sub>n</jats:sub> </jats:italic> = 10<jats:sup> <jats:italic>d</jats:italic> </jats:sup> <jats:italic>F<jats:sub>m</jats:sub> </jats:italic> + <jats:italic>L<jats:sub>k</jats:sub> </jats:italic> and <jats:italic>F<jats:sub>n</jats:sub> </jats:italic> = 10<jats:sup> <jats:italic>d</jats:italic> </jats:sup> <jats:italic>L<jats:sub>m</jats:sub> </jats:italic> + <jats:italic>F<jats:sub>k</jats:sub> </jats:italic> in nonnegative integers (<jats:italic>n</jats:italic>, <jats:italic>m</jats:italic>, <jats:italic>k</jats:italic>), where <jats:italic>d</jats:italic> denotes the number of digits of <jats:italic>L<jats:sub>k</jats:sub> </jats:italic> and <jats:italic>F<jats:sub>k</jats:sub> </jats:italic>, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Irena Jadlovská, George E. Chatzarakis, Ercan Tunç
In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.
{"title":"Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients","authors":"Irena Jadlovská, George E. Chatzarakis, Ercan Tunç","doi":"10.1515/ms-2024-0049","DOIUrl":"https://doi.org/10.1515/ms-2024-0049","url":null,"abstract":"In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Božidar V. Popović, Ali İ. Genç, Miroslav M. Ristić
In this work, we construct a new bivariate statistical distribution by the conditionally specified model approach. The conditional distributions follow the well-known generalized exponential distribution which includes the ordinary exponential distribution and is more flexible than gamma and Weibull in some ways. The newly defined distribution has four parameters that increase the flexibility of the model in data fitting. By equating the dependence parameter to zero, the marginal distributions become independent generalized exponential distributions. The new bivariate distribution depends on the classical exponential integral function which is not difficult to evaluate numerically. The basic properties of the distribution such as distribution functions, moments and stress-strength reliability are derived. The parameters are estimated by the method of maximum likelihood. Two real data fitting applications prove its usefulness in case of negatively correlated bivariate data modelling.
{"title":"A bivariate distribution with generalized exponential conditionals","authors":"Božidar V. Popović, Ali İ. Genç, Miroslav M. Ristić","doi":"10.1515/ms-2024-0059","DOIUrl":"https://doi.org/10.1515/ms-2024-0059","url":null,"abstract":"In this work, we construct a new bivariate statistical distribution by the conditionally specified model approach. The conditional distributions follow the well-known generalized exponential distribution which includes the ordinary exponential distribution and is more flexible than gamma and Weibull in some ways. The newly defined distribution has four parameters that increase the flexibility of the model in data fitting. By equating the dependence parameter to zero, the marginal distributions become independent generalized exponential distributions. The new bivariate distribution depends on the classical exponential integral function which is not difficult to evaluate numerically. The basic properties of the distribution such as distribution functions, moments and stress-strength reliability are derived. The parameters are estimated by the method of maximum likelihood. Two real data fitting applications prove its usefulness in case of negatively correlated bivariate data modelling.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ján Mináč, Lyle Muller, Tung T. Nguyen, Nguyễn Duy Tân
Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293–306] about the size of this upper bound.
帕利图是二次残差分布与图论之间的一个很好的联系。这些图具有显著的性质，因此在多个数学分支中都很有用。因此，帕利图自然与 p 处的 Legendre 符号相关联，该符号是导体 p 的二次 Dirichlet 特性。这些图形与一般的二次狄利克特特征相关联。然后，我们将介绍它们的一些基本性质。特别是，我们将明确描述它们的频谱。然后，我们利用这些广义 Paley 图形来构造一些新的 Ramanujan 图形族。最后，利用 L 函数的特殊值，我们为它们的切格数提供了一个有效的上限。作为我们方法的副产品，我们解决了 [Cramer et al：The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293-306]中提出的关于这个上界大小的问题。
{"title":"On the Paley graph of a quadratic character","authors":"Ján Mináč, Lyle Muller, Tung T. Nguyen, Nguyễn Duy Tân","doi":"10.1515/ms-2024-0040","DOIUrl":"https://doi.org/10.1515/ms-2024-0040","url":null,"abstract":"Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number <jats:italic>p</jats:italic> we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo <jats:italic>p</jats:italic>. Therefore, Paley graphs are naturally associated with the Legendre symbol at <jats:italic>p</jats:italic> which is a quadratic Dirichlet character of conductor <jats:italic>p</jats:italic>. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of <jats:italic>L</jats:italic>-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: <jats:italic>The isoperimetric and Kazhdan constants associated to a Paley graph</jats:italic>, Involve 9 (2016), 293–306] about the size of this upper bound.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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