We present a self-contained proof of the Khinchin inequality for multiple sums, which avoids advanced results from Probability theory. Not only our new proof is more accessible, but also it sheds lights on some properties of the inequality which may yield further generalizations.
{"title":"The Khinchin inequality for multiple sums revisited","authors":"Anselmo Raposo, Katiuscia B. Teixeira","doi":"10.36045/j.bbms.220831","DOIUrl":"https://doi.org/10.36045/j.bbms.220831","url":null,"abstract":"We present a self-contained proof of the Khinchin inequality for multiple sums, which avoids advanced results from Probability theory. Not only our new proof is more accessible, but also it sheds lights on some properties of the inequality which may yield further generalizations.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277282","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 paper contains the complete classification of nil rings with decomposable torsion-free additive group of rank two and description of nil rings with indecomposable torsion-free additive group of rank two. Moreover, it is shown that an indecomposable non-nil torsion-free abelian group $A$ of rank two supports only nilpotent rings exactly if $A$ is the additive group of a nil ring. Some decomposition criteria for torsion-free abelian groups of rank two are also included.
{"title":"On the Classification of Torsion-Free Nil Rings of Rank Two","authors":"Ryszard R. Andruszkiewicz, Mateusz Woronowicz","doi":"10.36045/j.bbms.221123","DOIUrl":"https://doi.org/10.36045/j.bbms.221123","url":null,"abstract":"The paper contains the complete classification of nil rings with decomposable torsion-free additive group of rank two and description of nil rings with indecomposable torsion-free additive group of rank two. Moreover, it is shown that an indecomposable non-nil torsion-free abelian group $A$ of rank two supports only nilpotent rings exactly if $A$ is the additive group of a nil ring. Some decomposition criteria for torsion-free abelian groups of rank two are also included.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277084","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 want to bound the symbol length of classes in ${_{2^{m-1}}Br}(F)$ which are represented by tensor products of 5 or 6 cyclic algebras of degree $2^m$. The main ingredients are the chain lemma for quadratic forms, a form of a generalized Clifford invariant and Pfister's and Rost's descriptions of 12- and 14-dimensional forms in $I^3 F$.
{"title":"Chain Lemma, Quadratic Forms and Symbol Length","authors":"Adam Chapman, Ilan Levin","doi":"10.36045/j.bbms.230123","DOIUrl":"https://doi.org/10.36045/j.bbms.230123","url":null,"abstract":"We want to bound the symbol length of classes in ${_{2^{m-1}}Br}(F)$ which are represented by tensor products of 5 or 6 cyclic algebras of degree $2^m$. The main ingredients are the chain lemma for quadratic forms, a form of a generalized Clifford invariant and Pfister's and Rost's descriptions of 12- and 14-dimensional forms in $I^3 F$.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277086","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 establish a sufficient and necessary condition for a function involving the inverse hyperbolic tangent function as a best possible upper bound for the complete elliptic integral of the first kind. Equivalently, we obtain a lower bound involving the arithmetic mean and logarithmic mean for the Gauss arithmetic-geometric mean. This provides a positive answer to a conjecture proposed by Yang, Song and Chu in 2014.
{"title":"A best possible upper bound for the complete elliptic integral of the first kind","authors":"Zhong-Xuan Mao, Lan-Xiang Yu, Jun-Yi Li, Jing-Feng Tian","doi":"10.36045/j.bbms.230228","DOIUrl":"https://doi.org/10.36045/j.bbms.230228","url":null,"abstract":"In this paper, we establish a sufficient and necessary condition for a function involving the inverse hyperbolic tangent function as a best possible upper bound for the complete elliptic integral of the first kind. Equivalently, we obtain a lower bound involving the arithmetic mean and logarithmic mean for the Gauss arithmetic-geometric mean. This provides a positive answer to a conjecture proposed by Yang, Song and Chu in 2014.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277085","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}
Jan Okniński raised the question whether the primes dividing the size $n$ of a finite indecomposable set-theoretic solution to the Yang-Baxter equation are related to the primes dividing the order of the associated permutation group. With Cedó he proved that both prime sets are equal if $n$ is square-free. We characterize equality and prove that surjective morphisms of solutions admit a canonical factorization into a covering and a morphism given by a brace ideal. The existence of solutions with non-equality of the prime sets is reduced to irretractable solutions. It is proved that non-equality is possible, and a minimal example is constructed..
Jan Okniński提出了这样一个问题:划分Yang-Baxter方程有限不可分解集合论解的大小$n$的素数是否与划分相关置换群的顺序有关。他用Cedó证明了如果$n$是无平方的，两个素数集相等。我们刻画了等式，并证明了解的满射态射允许正则分解为一个覆盖和一个由双理想给出的态射。将素数集不相等解的存在性简化为不可伸缩解。证明了不相等是可能的，并构造了一个极小的例子。
{"title":"Primes in coverings of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation","authors":"Wolfgang Rump","doi":"10.36045/j.bbms.230429","DOIUrl":"https://doi.org/10.36045/j.bbms.230429","url":null,"abstract":"Jan Okniński raised the question whether the primes dividing the size $n$ of a finite indecomposable set-theoretic solution to the Yang-Baxter equation are related to the primes dividing the order of the associated permutation group. With Cedó he proved that both prime sets are equal if $n$ is square-free. We characterize equality and prove that surjective morphisms of solutions admit a canonical factorization into a covering and a morphism given by a brace ideal. The existence of solutions with non-equality of the prime sets is reduced to irretractable solutions. It is proved that non-equality is possible, and a minimal example is constructed..","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277080","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 Lagrangian submanifolds in the nearly Kähler manifold $G times G$. First we review the construction of a nearly Kähler structure on the Lie group product $G times G$, where $G$ is a Lie group with a bi-invariant metric. This construction was proposed by K. Sekigawa to E. Abbena and S. Garbiero. An example of this construction is the homogeneous nearly Kähler manifold $mathbb{S}^{3}times mathbb{S}^{3}$, where $G=mathbb{S}^{3}$ with its standard metric. It is known that this construction on $G times G$ gives a nearly Kähler structure. To get our main result, we extend the notion of angle functions of a Lagrangian submanifold proposed by B. Dioos, L. Vrancken and X. Wang in the case of $mathbb{S}^{3}times mathbb{S}^{3}$. These angle functions are useful to characterize minimal Lagrangian submanifolds in the NK manifold $G times G$. We prove our main result: a Lagrangian submanifold is minimal if and only if the sum of its angle functions is constant. We give five examples of Lagrangian submanifolds: three canonical examples and other two examples using an element in the center of $G$.
{"title":"A characterization of minimal Lagrangian submanifolds of the nearly Kähler $G times G$","authors":"Rodrigo Aguilar-Suárez, Gabriel Ruiz-Hernández","doi":"10.36045/j.bbms.220331","DOIUrl":"https://doi.org/10.36045/j.bbms.220331","url":null,"abstract":"We investigate Lagrangian submanifolds in the nearly Kähler manifold $G times G$. First we review the construction of a nearly Kähler structure on the Lie group product $G times G$, where $G$ is a Lie group with a bi-invariant metric. This construction was proposed by K. Sekigawa to E. Abbena and S. Garbiero. An example of this construction is the homogeneous nearly Kähler manifold $mathbb{S}^{3}times mathbb{S}^{3}$, where $G=mathbb{S}^{3}$ with its standard metric. It is known that this construction on $G times G$ gives a nearly Kähler structure. To get our main result, we extend the notion of angle functions of a Lagrangian submanifold proposed by B. Dioos, L. Vrancken and X. Wang in the case of $mathbb{S}^{3}times mathbb{S}^{3}$. These angle functions are useful to characterize minimal Lagrangian submanifolds in the NK manifold $G times G$. We prove our main result: a Lagrangian submanifold is minimal if and only if the sum of its angle functions is constant. We give five examples of Lagrangian submanifolds: three canonical examples and other two examples using an element in the center of $G$.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277083","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 $P$-space is a topological space whose every $G_{delta}$-set is open. In this article, basic properties of $P$-spaces are investigated in the absence of the Axiom of Choice. New weaker forms of the Axiom of Choice, all relevant to $P$-spaces or to countable intersections of $G_{delta}$-sets, are introduced for applications. Special subrings of rings of continuous real functions are applied. New notions of a quasi Baire space and a strongly (quasi) Baire space are introduced. Several independence results are obtained. For instance, it is shown in $mathbf{ZF}$ that if $G_{delta}$-modifications of Tychonoff spaces are $P$-spaces, then every denumerable family of denumerable sets has a multiple choice function. In $mathbf{ZF}$, a zero-dimensional subspace of $mathbb{R}$ may fail to be strongly zero-dimensional, and countable intersections of $G_{delta}$-sets of $mathbb{R}$ may fail to be $G_{delta}$-sets. New open problems are posed. Partial answers to some of them are given.
{"title":"On $P$-spaces and $G_{delta}$-sets in the absence of the Axiom of Choice","authors":"Kyriakos Keremedis, AliReza Olfati, Eliza Wajch","doi":"10.36045/j.bbms.230117","DOIUrl":"https://doi.org/10.36045/j.bbms.230117","url":null,"abstract":"A $P$-space is a topological space whose every $G_{delta}$-set is open. In this article, basic properties of $P$-spaces are investigated in the absence of the Axiom of Choice. New weaker forms of the Axiom of Choice, all relevant to $P$-spaces or to countable intersections of $G_{delta}$-sets, are introduced for applications. Special subrings of rings of continuous real functions are applied. New notions of a quasi Baire space and a strongly (quasi) Baire space are introduced. Several independence results are obtained. For instance, it is shown in $mathbf{ZF}$ that if $G_{delta}$-modifications of Tychonoff spaces are $P$-spaces, then every denumerable family of denumerable sets has a multiple choice function. In $mathbf{ZF}$, a zero-dimensional subspace of $mathbb{R}$ may fail to be strongly zero-dimensional, and countable intersections of $G_{delta}$-sets of $mathbb{R}$ may fail to be $G_{delta}$-sets. New open problems are posed. Partial answers to some of them are given.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136277078","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}
{"title":"Asymptotics of the eigenvalues of a fourth order differential operator with eigenvalue dependent and periodic boundary conditions","authors":"B. Moletsane, Bertin Zinsou","doi":"10.36045/j.bbms.210416a","DOIUrl":"https://doi.org/10.36045/j.bbms.210416a","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72426173","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}
{"title":"Asymptotic Behavior of a Periodic Sequence of Nonexpansive Mappings with Applications","authors":"M. Hashemi, H. Khatibzadeh","doi":"10.36045/j.bbms.220607","DOIUrl":"https://doi.org/10.36045/j.bbms.220607","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82430803","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}
{"title":"Surgery on ${rm Aut}(F_2)$","authors":"Sylvain Barré, Mikael Pichot","doi":"10.36045/j.bbms.210809","DOIUrl":"https://doi.org/10.36045/j.bbms.210809","url":null,"abstract":"","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85593883","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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