In this paper, we introduced novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We used our new characterizations of majorization to derive an improved entropy inequality.
{"title":"A note on equivalent conditions for majorization","authors":"Roberto Bruno, Ugo Vaccaro","doi":"10.3934/math.2024419","DOIUrl":"https://doi.org/10.3934/math.2024419","url":null,"abstract":"In this paper, we introduced novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We used our new characterizations of majorization to derive an improved entropy inequality.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140983701","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 $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if $ G $ is the normal closure of $ S $ in $ G. $ In this case, any element of $ G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ gin G $ and $ S $ is a normally generating subset of $ G, $ then we write $ | g|_{S} $ for the length of a shortest word in $ mbox{Conj}_{G}(S^{pm 1}): = {h^{-1}sh | hin G, sin S , mbox{or} , s{^{-1}}in S } $ needed to express $ g. $ For any normally generating subset $ S $ of $ G, $ we write $ |G|_{S} = mbox{sup}{|g|_{S} , |, , gin G}. $ Moreover, we write $ Delta(G) $ for the supremum of all $ |G|_{S}, $ where $ S $ is a finite normally generating subset of $ G, $ and we call $ Delta(G) $ the conjugacy diameter of $ G. $ In this paper, we derive the conjugacy diameters of the semidihedral $ 2 $-groups, the generalized quaternion groups and the modular $ p $-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.
让 $ G $ 是一个群。如果 $ G $ 是 $ S $ 在 $ G $ 中的正常闭包, 那么 $ G $ 的一个子集 $ S $ 就被称为正常生成 $ G $.如果 $ gin G $ 和 $ S $ 是 $ G 的正常生成子集，那么我们可以写 $ | g|_{S} $ 表示 $ mbox{Conj}_{G}(S^{pm 1}) 中最短单词的长度： = {h^{-1}sh | hin G, sin S , mbox{or}对于 $ G 的任何正常生成子集 $ S $, $ 我们写 $|G|_{S} = mbox{sup}{|g|_{S} , |, , gin G}.$ 此外，我们把所有 $|G|_{S} 的上集写成 $Delta(G)$，其中 $ S $ 是 $ G 的有限常生成子集，$ 我们称 $ Delta(G) $ 为 $ G 的共轭直径。 $ 在本文中，我们推导了半二面体 $ 2 $ 群、广义四元数群和模数 $ p $ 群的共轭直径。这是在确定了二面群的共轭直径之后的一个自然步骤。
{"title":"The conjugacy diameters of non-abelian finite $ p $-groups with cyclic maximal subgroups","authors":"Fawaz Aseeri, J. Kaspczyk","doi":"10.3934/math.2024524","DOIUrl":"https://doi.org/10.3934/math.2024524","url":null,"abstract":"Let $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if $ G $ is the normal closure of $ S $ in $ G. $ In this case, any element of $ G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ gin G $ and $ S $ is a normally generating subset of $ G, $ then we write $ | g|_{S} $ for the length of a shortest word in $ mbox{Conj}_{G}(S^{pm 1}): = {h^{-1}sh | hin G, sin S , mbox{or} , s{^{-1}}in S } $ needed to express $ g. $ For any normally generating subset $ S $ of $ G, $ we write $ |G|_{S} = mbox{sup}{|g|_{S} , |, , gin G}. $ Moreover, we write $ Delta(G) $ for the supremum of all $ |G|_{S}, $ where $ S $ is a finite normally generating subset of $ G, $ and we call $ Delta(G) $ the conjugacy diameter of $ G. $ In this paper, we derive the conjugacy diameters of the semidihedral $ 2 $-groups, the generalized quaternion groups and the modular $ p $-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140505170","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}
We presented a novel $ mathcal{N} = 2 $ $ mathbb{Z}_2^2 $-graded supersymmetric quantum mechanics ($ {mathbb{Z}_2^2} $-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and was the first example of the quantum mechanical realization of an eight-dimensional irreducible representation (irrep) of the $ mathcal{N} = 2 $ $ mathbb{Z}_2^2 $-supersymmetry algebra. The $ {mathbb{Z}_2^2} $-SQM was obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional $ {mathbb{Z}_2^2} $-supersymmetric Lagrangian of $ mathcal{N} = 1 $, which we constructed in our previous work. The ground states of the $ {mathbb{Z}_2^2} $-SQM were also investigated.
{"title":"$ mathcal{N} = 2 $ double graded supersymmetric quantum mechanics via dimensional reduction","authors":"N. Aizawa, Ren Ito, Toshiya Tanaka","doi":"10.3934/math.2024513","DOIUrl":"https://doi.org/10.3934/math.2024513","url":null,"abstract":"We presented a novel $ mathcal{N} = 2 $ $ mathbb{Z}_2^2 $-graded supersymmetric quantum mechanics ($ {mathbb{Z}_2^2} $-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and was the first example of the quantum mechanical realization of an eight-dimensional irreducible representation (irrep) of the $ mathcal{N} = 2 $ $ mathbb{Z}_2^2 $-supersymmetry algebra. The $ {mathbb{Z}_2^2} $-SQM was obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional $ {mathbb{Z}_2^2} $-supersymmetric Lagrangian of $ mathcal{N} = 1 $, which we constructed in our previous work. The ground states of the $ {mathbb{Z}_2^2} $-SQM were also investigated.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140513415","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 study, some mappings related to the Fejér-type inequalities for harmonically convex functions are defined over $ left[ 0, 1right] $. Some Fejér-type inequalities for harmonically convex functions are proved using these mappings. Properties of these mappings are considered and consequently, refinements are obtained of some known results.
{"title":"Fejér type inequalities for harmonically convex functions","authors":"Muhammad Amer Latif","doi":"10.3934/math.2022835","DOIUrl":"https://doi.org/10.3934/math.2022835","url":null,"abstract":"In this study, some mappings related to the Fejér-type inequalities for harmonically convex functions are defined over $ left[ 0, 1right] $. Some Fejér-type inequalities for harmonically convex functions are proved using these mappings. Properties of these mappings are considered and consequently, refinements are obtained of some known results.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46984565","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}
M. Nur, M. Bahri, A. Islamiyati, Harmanus Batkunde
The aim of this paper is to investigate completness of $ A $ that equipped with usual norm on $ p $-summable sequences space where $ A $ is subspace in $ p $-summable sequences space and $ 1le p < infty $. We also introduce a new inner product on $ A $ and prove completness of $ A $ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $ A $. In particular, we discuss the angle between $ 1 $-dimensional subspace and $ (s-1) $-dimensional subspace where $ sge 2 $ of $ A $.
本文的目的是研究具有通常范数的$ A $在$ p $ -可和序列空间上的完备性，其中$ A $是$ p $ -可和序列空间和$ 1le p < infty $中的子空间。在$ A $上引入了一个新的内积，并用一个新的范数证明了$ A $的完备性。此外，我们还讨论了$ A $中两个向量与两个子空间之间的夹角。特别地，我们讨论了$ 1 $维子空间与$ (s-1) $维子空间之间的夹角，其中$ A $的$ sge 2 $。
{"title":"Angle in the space of $ p $-summable sequences","authors":"M. Nur, M. Bahri, A. Islamiyati, Harmanus Batkunde","doi":"10.3934/math.2022155","DOIUrl":"https://doi.org/10.3934/math.2022155","url":null,"abstract":"The aim of this paper is to investigate completness of $ A $ that equipped with usual norm on $ p $-summable sequences space where $ A $ is subspace in $ p $-summable sequences space and $ 1le p < infty $. We also introduce a new inner product on $ A $ and prove completness of $ A $ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $ A $. In particular, we discuss the angle between $ 1 $-dimensional subspace and $ (s-1) $-dimensional subspace where $ sge 2 $ of $ A $.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48080725","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 introduce the concept and representation of modified $ lambda $-differential Lie triple systems. Next, we define the cohomology of modified $ lambda $-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $ lambda $-differential Lie triple systems.
{"title":"Cohomologies of modified $ lambda $-differential Lie triple systems and applications","authors":"Wen Teng, Fengshan Long, Yu Zhang","doi":"10.3934/math.20231280","DOIUrl":"https://doi.org/10.3934/math.20231280","url":null,"abstract":"In this paper, we introduce the concept and representation of modified $ lambda $-differential Lie triple systems. Next, we define the cohomology of modified $ lambda $-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $ lambda $-differential Lie triple systems.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42483532","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}
This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.
{"title":"The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative","authors":"Takiko Sasaki, Shuhei Takamatsu, H. Takamura","doi":"10.3934/math.20231300","DOIUrl":"https://doi.org/10.3934/math.20231300","url":null,"abstract":"This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47127415","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}
T. Lukashiv, I. Malyk, Maryna K. Chepeleva, P. Nazarov
This article aims to investigate sufficient conditions for the stability of the trivial solution of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system. The novelty of this work is in the consideration of jump concentration points, which are not considered in classical works. The assumption of the existence of concentration points leads to additional constraints on jumps, jump times and relations between them.
{"title":"Stability of stochastic dynamic systems of a random structure with Markov switching in the presence of concentration points","authors":"T. Lukashiv, I. Malyk, Maryna K. Chepeleva, P. Nazarov","doi":"10.3934/math.20231245","DOIUrl":"https://doi.org/10.3934/math.20231245","url":null,"abstract":"This article aims to investigate sufficient conditions for the stability of the trivial solution of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system. The novelty of this work is in the consideration of jump concentration points, which are not considered in classical works. The assumption of the existence of concentration points leads to additional constraints on jumps, jump times and relations between them.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45589800","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 study $ n $-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg group and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $ n $-dimensional weighted Hardy operator and weighted Cesàro operator are also obtained.
{"title":"Mixed radial-angular bounds for Hardy-type operators on Heisenberg group","authors":"Zhongci Hang, Xiang Li, D. Yan","doi":"10.3934/math.20231070","DOIUrl":"https://doi.org/10.3934/math.20231070","url":null,"abstract":"In this paper, we study $ n $-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg group and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $ n $-dimensional weighted Hardy operator and weighted Cesàro operator are also obtained.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41319140","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}
R. Campoamor-Stursberg, Eduardo Fernández-Saiz, F. J. Herranz
Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ mathfrak{b}_2 $ and $ mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ mathfrak{h}_6 $, according to the embedding chain $ mathfrak{b}_2subset mathfrak{h}_4subset mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.
{"title":"Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations","authors":"R. Campoamor-Stursberg, Eduardo Fernández-Saiz, F. J. Herranz","doi":"10.3934/math.20231225","DOIUrl":"https://doi.org/10.3934/math.20231225","url":null,"abstract":"Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ mathfrak{b}_2 $ and $ mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ mathfrak{h}_6 $, according to the embedding chain $ mathfrak{b}_2subset mathfrak{h}_4subset mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43783486","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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