Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.
{"title":"Degeneration phenomenon in linear ordinary differential equations","authors":"Vakhtang Lomadze","doi":"10.1515/gmj-2024-2007","DOIUrl":"https://doi.org/10.1515/gmj-2024-2007","url":null,"abstract":"Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"19 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918169","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 the present paper, we examine the perturbation of continuous frames and Riesz-type frames in Hilbert C*{C^{*}}-modules. We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to continuous frames in Hilbert C*{C^{*}}-modules. We obtain a necessary condition under which the perturbation of a Riesz-type frame of Hilbert C*{C^{*}}-modules remains to be a Riesz-type frame. Also, we examine the effect of duality on the perturbation of continuous frames in Hilbert C*{C^{*}}-modules, and we prove that if the operator frame of a continuous frame F is near to the combination of the synthesis operator of a continuous Bessel mapping G and the analysis operator of F, then G is a continuous frame.
在本文中，我们研究了希尔伯特 C * {C^{*}} 模块中连续框架和里兹型框架的扰动。 -模块中的连续帧和里兹型帧的扰动。我们将希尔伯特空间帧的卡萨扎-克里斯滕森一般扰动定理推广到希尔伯特 C * {C^{*}} 模块中的连续帧。 -模块中的连续帧。我们得到了一个必要条件，在这个条件下，希尔伯特 C * {C^{*}} 模块的李斯型帧的扰动仍然是一个李斯型帧。 -模块的里兹型框架的扰动仍然是里兹型框架的必要条件。此外，我们还考察了对偶性对希尔伯特 C * {C^{*}} 模块中连续帧的扰动的影响。 -模块的扰动的影响，并证明如果连续帧 F 的算子帧接近于连续贝塞尔映射 G 的合成算子与 F 的分析算子的组合，那么 G 就是一个连续帧。
{"title":"On perturbation of continuous frames in Hilbert C *-modules","authors":"Hadi Ghasemi, Tayebe Lal Shateri","doi":"10.1515/gmj-2023-2111","DOIUrl":"https://doi.org/10.1515/gmj-2023-2111","url":null,"abstract":"In the present paper, we examine the perturbation of continuous frames and Riesz-type frames in Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules. We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to continuous frames in Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules. We obtain a necessary condition under which the perturbation of a Riesz-type frame of Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules remains to be a Riesz-type frame. Also, we examine the effect of duality on the perturbation of continuous frames in Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules, and we prove that if the operator frame of a continuous frame <jats:italic>F</jats:italic> is near to the combination of the synthesis operator of a continuous Bessel mapping <jats:italic>G</jats:italic> and the analysis operator of <jats:italic>F</jats:italic>, then <jats:italic>G</jats:italic> is a continuous frame.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"27 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918315","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 extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℭ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2024-2006_eq_0150.png" /> <jats:tex-math>{mathfrak{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo></m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>x</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> <m:mi>k</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2024-2006_eq_0106.png" /> <jats:tex-math>|mathfrak{C}x^{k}|_{t,1}leq U|x|_{l_{p}}^{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2024-2006_eq_0107.png" /> <jats:tex-math>k=1,2</jats:tex-math> </jats:alternatives> </jats:inline-formula>), where <jats:italic>x</jats:italic> is a sequence, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2024-2006_eq_0149.png" /> <jats:tex-math>{mathfrak{C}x^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a tensor, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2024-2006_eq_0155.png" /> <jats:tex-math>{|cdot|_{t,1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3
在本文中，我们将哈代不等式扩展到无限张量。为此，我们引入 Cesàro 张量 ℭ {mathfrak{C}} ，并将其视为从序列空间到张量空间的张量映射。 ，并将它们视为从序列空间到张量空间的张量映射。事实上，我们证明了形式为 ∥ ℭ x k ∥ t , 1 ≤ U ∥ x ∥ l p k|mathfrak{C}x^{k}|_{t,1}leq U|x|_{l_{p}}^{k} 的不等式。 ( k = 1 , 2 k=1,2 ), 其中 x 是一个序列，ℭ x k {mathfrak{C}x^{k}} 是一个张量，并且 ∥ ⋅ ∥ t , 1 {|cdot|_{t,1}} , ∥ ⋅ ∥ l p {|cdot|_{l_{p}}} 分别是张量规范和序列规范。常数 U 与 x 无关，我们寻求 U 的最小值。
{"title":"A generalization of Hardy’s inequality to infinite tensors","authors":"Morteza Saheli, Davoud Foroutannia, Sara Yusefian","doi":"10.1515/gmj-2024-2006","DOIUrl":"https://doi.org/10.1515/gmj-2024-2006","url":null,"abstract":"In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℭ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0150.png\" /> <jats:tex-math>{mathfrak{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo></m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>x</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> <m:mi>k</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0106.png\" /> <jats:tex-math>|mathfrak{C}x^{k}|_{t,1}leq U|x|_{l_{p}}^{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0107.png\" /> <jats:tex-math>k=1,2</jats:tex-math> </jats:alternatives> </jats:inline-formula>), where <jats:italic>x</jats:italic> is a sequence, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0149.png\" /> <jats:tex-math>{mathfrak{C}x^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a tensor, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0155.png\" /> <jats:tex-math>{|cdot|_{t,1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"81 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918167","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 Busemann–Petty problem of arbitrary measure for symmetric star bodies is proposed and studied by Zvavitch, which is a generalization of the classical Busemann–Petty problem. In this paper, we study the Busemann–Petty-type problem for homogeneous measure for general star bodies.
{"title":"Busemann--Petty-type problem for μ-intersection bodies","authors":"Chao Li, Gangyi Chen","doi":"10.1515/gmj-2024-2009","DOIUrl":"https://doi.org/10.1515/gmj-2024-2009","url":null,"abstract":"The Busemann–Petty problem of arbitrary measure for symmetric star bodies is proposed and studied by Zvavitch, which is a generalization of the classical Busemann–Petty problem. In this paper, we study the Busemann–Petty-type problem for homogeneous measure for general star bodies.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"54 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918317","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}
David Barilla, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani
In this paper, we consider a fractional p-Laplacian elliptic Dirichlet problem that possesses one control parameter and has a Lipschitz nonlinearity order of p-1{p-1}. The multiplicity of the weak solutions is proved by means of the variational method and critical point theory. We investigate the existence of at least three solutions to the problem.
{"title":"Fractional p-Laplacian elliptic Dirichlet problems","authors":"David Barilla, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani","doi":"10.1515/gmj-2024-2008","DOIUrl":"https://doi.org/10.1515/gmj-2024-2008","url":null,"abstract":"In this paper, we consider a fractional <jats:italic>p</jats:italic>-Laplacian elliptic Dirichlet problem that possesses one control parameter and has a Lipschitz nonlinearity order of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2008_eq_0274.png\" /> <jats:tex-math>{p-1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The multiplicity of the weak solutions is proved by means of the variational method and critical point theory. We investigate the existence of at least three solutions to the problem.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"40 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918316","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 studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.
{"title":"On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales","authors":"Viktoriia Tsan, Oleksandr Stanzhytskyi, Olha Martynyuk","doi":"10.1515/gmj-2024-2003","DOIUrl":"https://doi.org/10.1515/gmj-2024-2003","url":null,"abstract":"This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"52 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771538","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 deal with the notion of Va{V_{a}}-deformed free convolution, introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of Va{V_{a}}-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by Va{V_{a}}-deformed free Poisson distribution.
本文将讨论 V a {V_{a}} 变形自由卷积的概念。 -变形自由卷积的概念。D. Krystek 和 L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin.Dimens.Anal.Quantum Probab.Relat.Top.8 2005, 3, 515-545], 从与 Cauchy-Stieltjes 核（CSK）族及其相应方差函数理论相关的角度出发。我们确定了 V a {V_{a}} 的幂下的方差函数公式。 -变形自由卷积下的方差函数公式。然后，我们提供了由 V a {V_{a}} 变形自由泊松分布生成的 CSK 族元素的近似值。 -变形自由泊松分布产生的 CSK 族元素的近似值。
{"title":"V_a -deformed free convolution and variance function","authors":"Raouf Fakhfakh","doi":"10.1515/gmj-2024-2004","DOIUrl":"https://doi.org/10.1515/gmj-2024-2004","url":null,"abstract":"In this paper, we deal with the notion of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution, introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free Poisson distribution.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"181 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658777","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 and G be finite groups such that A acts coprimely on G by automorphisms. We prove that if every self-centralizing non-nilpotent A-invariant subgroup of G is a TI-subgroup or a subnormal subgroup, then every non-nilpotent A-invariant subgroup of G is subnormal and G is p-nilpotent or p-closed for any prime divisor p of |G|{|G|}. If every self-centralizing non-metacyclic A-invariant subgroup of G is a TI-subgroup or a subnormal subgroup, then every non-metacyclic A-invariant subgroup of G is subnormal and G is solvable.
设 A 和 G 都是有限群，且 A 通过自动形共同作用于 G。我们证明，如果 G 的每个自中心化非零能 A 不变子群都是 TI 子群或子正常子群，那么 G 的每个非零能 A 不变子群都是子正常的，并且对于 | G | {|G|} 的任何素除数 p，G 都是 p 零能或 p 封闭的。如果 G 的每个自中心化非元胞 A 不变子群都是 TI 子群或子正常子群，那么 G 的每个非元胞 A 不变子群都是子正常的，并且 G 是可解的。
{"title":"Finite groups in which some particular invariant subgroups are TI-subgroups or subnormal subgroups","authors":"Yifan Liu, Jiangtao Shi","doi":"10.1515/gmj-2024-2001","DOIUrl":"https://doi.org/10.1515/gmj-2024-2001","url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>G</jats:italic> be finite groups such that <jats:italic>A</jats:italic> acts coprimely on <jats:italic>G</jats:italic> by automorphisms. We prove that if every self-centralizing non-nilpotent <jats:italic>A</jats:italic>-invariant subgroup of <jats:italic>G</jats:italic> is a TI-subgroup or a subnormal subgroup, then every non-nilpotent <jats:italic>A</jats:italic>-invariant subgroup of <jats:italic>G</jats:italic> is subnormal and <jats:italic>G</jats:italic> is <jats:italic>p</jats:italic>-nilpotent or <jats:italic>p</jats:italic>-closed for any prime divisor <jats:italic>p</jats:italic> of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2001_eq_0039.png\" /> <jats:tex-math>{|G|}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. If every self-centralizing non-metacyclic <jats:italic>A</jats:italic>-invariant subgroup of <jats:italic>G</jats:italic> is a TI-subgroup or a subnormal subgroup, then every non-metacyclic <jats:italic>A</jats:italic>-invariant subgroup of <jats:italic>G</jats:italic> is subnormal and <jats:italic>G</jats:italic> is solvable.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139590014","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 explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X. The results are formulated in terms of some topological conditions made on X or on its dual X*{X^{*}}. In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided.
在本文中，我们探讨了作用于非反射巴拿赫空间 X 的无界广义弗雷德霍姆算子的谱性质。结果是根据对 X 或其对偶 X * {X^{*}} 的一些拓扑条件得出的。 .此外，我们还引入了所谓 g-g-Riesz 线性算子的概念，作为 Riesz 算子的扩展。所获得的结果被用来讨论广义本质谱行为的发生。此外，我们还通过 g-Riesz 扰动提供了广义本质谱与左（或右）本质谱之间的关系。
{"title":"Generalized essential spectra involving the class of g-g-Riesz operators","authors":"Imen Ferjani, Omaima Kchaou, Bilel Krichen","doi":"10.1515/gmj-2024-2002","DOIUrl":"https://doi.org/10.1515/gmj-2024-2002","url":null,"abstract":"In this paper, we explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space <jats:italic>X</jats:italic>. The results are formulated in terms of some topological conditions made on <jats:italic>X</jats:italic> or on its dual <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>X</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2002_eq_0247.png\" /> <jats:tex-math>{X^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"37 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647865","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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