The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this article, we study a discrete predator-prey system with Holling type II functional response and Allee effect. First, the number of fixed points of the system, local stability, and global stability is discussed. The population changes of predator and prey under strong or weak Allee effects are proved using the nullclines and direction field, respectively. Second, using the bifurcation theory, the bifurcation conditions for the system to undergo transcritical bifurcation and Neimark-Sacker bifurcation at the equilibrium point are obtained. Finally, the dynamic behavior of the system is analyzed by numerical simulation of bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram. The results show that the system will produce complex dynamic phenomena such as periodic state, quasi-periodic state, and chaos.
捕食者的强阿利效应和弱阿利效应之间的过渡提供了生态学中一个简单的机制转换。本文研究了一个具有霍林 II 型功能响应和阿利效应的离散捕食者-猎物系统。首先，讨论了系统的固定点数、局部稳定性和全局稳定性。分别利用空线和方向场证明了强或弱阿利效应下捕食者和猎物的种群变化。其次，利用分岔理论，得到了系统在平衡点发生跨临界分岔和 Neimark-Sacker 分岔的分岔条件。最后，通过分岔图、相图和最大李雅普诺夫指数图的数值模拟分析了系统的动态行为。结果表明，系统会产生周期态、准周期态和混沌等复杂的动态现象。
{"title":"Complex dynamics of a nonlinear discrete predator-prey system with Allee effect","authors":"Jing Wang, Ceyu Lei","doi":"10.1515/math-2024-0013","DOIUrl":"https://doi.org/10.1515/math-2024-0013","url":null,"abstract":"The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this article, we study a discrete predator-prey system with Holling type II functional response and Allee effect. First, the number of fixed points of the system, local stability, and global stability is discussed. The population changes of predator and prey under strong or weak Allee effects are proved using the nullclines and direction field, respectively. Second, using the bifurcation theory, the bifurcation conditions for the system to undergo transcritical bifurcation and Neimark-Sacker bifurcation at the equilibrium point are obtained. Finally, the dynamic behavior of the system is analyzed by numerical simulation of bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram. The results show that the system will produce complex dynamic phenomena such as periodic state, quasi-periodic state, and chaos.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529762","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 Δ=ANBAMABBDelta =left(begin{array}{cc}A& {}_{A}N_{B} {}_{B}M_{A}& Bend{array}right) be a Morita ring, where M⊗AN=0=N⊗BMM{otimes }_{A}N=0=N{otimes }_{B}M. Let XX be left AA-module and YY be left BB-module. We prove that
让 Δ = A N B A M A B B Delta =left(begin{array}{cc}A& {}_{A}N_{B} {}_{B}M_{A}& Bend{array}right) 是一个莫里塔环，其中 M ⊗ A N = 0 = N ⊗ B M M{otimes }_{A}N=0=N{otimes }_{B}M 。设 X X 是左 A A 模块，Y Y 是左 B B 模块。我们证明 ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) left(X,M{otimes }_{A}X,1,0)oplus left(N{otimes }_{B}Y,Y,0,1) 是一个淤积模块，当且仅当 X X 是一个淤积 A A - 模块、 Y Y 是淤积的 B B -模块，M ⊗ A X M{otimes }_{A}X 由 Y Y 生成，N ⊗ B Y N{otimes }_{B}Y 由 X X 生成。因此，我们得到，如果 M A {M}_{A} 和 N B {N}_{B} 是平的，那么 ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) left(X,M{otimes }_{A}X,1,0)oplus left(N{otimes }_{B}Y,Y,0、当且仅当 X X 是倾斜 A A - 模块，Y Y 是倾斜 B B - 模块，M ⊗ A X M{otimes }_{A}X 由 Y Y 生成，N ⊗ B Y N{otimes }_{B}Y 由 X X 生成时，X X 是倾斜 Δ Δ Delta - 模块。
A new distributionally robust ratio optimization model is proposed under the known first and second moments of the uncertain distributions. In this article, both standard deviation (SD) and conditional value-at-risk (CVaR) are used to measure the risk, avoiding both fat-tail and volatility. The new model can be reduced to a simple distributionally robust model under assumptions on the measurements of reward, CVaR and SD. Furthermore, it can be rewritten as a tractable semi-definite programming problem by the duality theorem under partially known information of the uncertain parameters. Finally, the model is tested on portfolio problems and verified from numerical results that it can give a reasonable decision under only the first and second moments.
{"title":"A new distributionally robust reward-risk model for portfolio optimization","authors":"Yijia Zhou, Lijun Xu","doi":"10.1515/math-2024-0010","DOIUrl":"https://doi.org/10.1515/math-2024-0010","url":null,"abstract":"A new distributionally robust ratio optimization model is proposed under the known first and second moments of the uncertain distributions. In this article, both standard deviation (SD) and conditional value-at-risk (CVaR) are used to measure the risk, avoiding both fat-tail and volatility. The new model can be reduced to a simple distributionally robust model under assumptions on the measurements of reward, CVaR and SD. Furthermore, it can be rewritten as a tractable semi-definite programming problem by the duality theorem under partially known information of the uncertain parameters. Finally, the model is tested on portfolio problems and verified from numerical results that it can give a reasonable decision under only the first and second moments.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148598","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 study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ1φtt−κ(φx+ψ)x+(g∗φxx)(t)=0,−bψxx+κ(φx+ψ)=0,begin{array}{rcl}{rho }_{1}{varphi }_{tt}-kappa {left({varphi }_{x}+psi )}_{x}+left(gast {varphi }_{xx})left(t)& =& 0, -b{psi }_{xx}+kappa left({varphi }_{x}+psi )& =& 0,end{array} where the convolution memory function gg belongs to a class of L1(0,∞){L}^{1}left(0,infty )
在本研究中，我们考虑的是无转动惯量的粘弹性剪切梁模型。具体来说，我们研究 ρ 1 φ t t - κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 、 - b ψ x x + κ ( φ x + ψ ) = 0 , begin{array}{rcl}{rho }_{1}{varphi }_{tt}-kappa {left({varphi }_{x}+psi )}_{x}+left(gast {varphi }_{xx})left(t)&；=& 0,（ -b{psi }_{xx}+kappa left({varphi }_{x}+psi )& =&；0,end{array} 其中卷積記憶函數 g g 屬於 L 1 ( 0 , ∞ ) {L}^{1}left(0,infty)函數的一類，它滿足 g ′ ( t ) ≤ - ξ ( t ) ϒ ( g ( t ) , ∀ t ≥ 0 , g^{prime} le -xi left(t)Upsilon left(gleft(t)),hspace{1.0em}forall tge 0, 其中ξ xi是一個正的非遞增的可微分函數，而ϒ Upsilon是一個靠近原點的遞增的凸函數。僅僅利用這個關於g g在無限處行為的一般假設，我們提供了最佳的和明確的一般能量衰減率，當 ÕLu_3D2↩ ( s ) = s p Upsilon left(s)={s}^{p} 和 p p 覆蓋了全部可允許的範圍 [ 1 , 2 ]時，我們可以從中恢復指數率和多项式率。 left[1,2) 。鉴于这种普遍性，我们的结果改进了文献中早期的一些相关结果。
In this study, we are concerned with the existence and exponential stability issue of a delayed differential neoclassical growth model with discontinuous control strategy. By employing the Filippov’s theory and dichotomy theory, together with the Lyapunov functional method, novel criteria on existence and exponential stability are established for the addressed model. The established theoretical results extend and supplement the related results in the existing literature. Moreover, a simulation example is presented to verify the practicability of the proposed results.
{"title":"Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy","authors":"Qian Wang, Wei Wang, Qian Zhan","doi":"10.1515/math-2024-0006","DOIUrl":"https://doi.org/10.1515/math-2024-0006","url":null,"abstract":"In this study, we are concerned with the existence and exponential stability issue of a delayed differential neoclassical growth model with discontinuous control strategy. By employing the Filippov’s theory and dichotomy theory, together with the Lyapunov functional method, novel criteria on existence and exponential stability are established for the addressed model. The established theoretical results extend and supplement the related results in the existing literature. Moreover, a simulation example is presented to verify the practicability of the proposed results.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148553","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 article, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of (α,β)left(alpha ,beta )-mixing random variables are presented. An application to simple linear errors-in-variables model is provided. Simulation studies are also carried out to support the theoretical results.
{"title":"Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model","authors":"Wenjing Hu, Wei Wang, Yi Wu","doi":"10.1515/math-2024-0003","DOIUrl":"https://doi.org/10.1515/math-2024-0003","url":null,"abstract":"In this article, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0003_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(alpha ,beta )</jats:tex-math> </jats:alternatives> </jats:inline-formula>-mixing random variables are presented. An application to simple linear errors-in-variables model is provided. Simulation studies are also carried out to support the theoretical results.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148554","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}
Asymptotic formulae are established for the number of natural numbers mm with largest square-free divisor not exceeding mϑ{m}^{{vartheta }}, for any fixed positive parameter ϑ{vartheta }. Related counting functions are also considered.
建立了最大无平方除数不超过 m ϑ {m}^{vartheta }} 的自然数 m m 个数的渐近公式。 对于任意固定正参数 ϑ {vartheta }. .还考虑了相关的计数函数。
{"title":"On the distribution of powered numbers","authors":"Jörg Brüdern, Olivier Robert","doi":"10.1515/math-2024-0007","DOIUrl":"https://doi.org/10.1515/math-2024-0007","url":null,"abstract":"Asymptotic formulae are established for the number of natural numbers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0007_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>m</m:mi> </m:math> <jats:tex-math>m</jats:tex-math> </jats:alternatives> </jats:inline-formula> with largest square-free divisor not exceeding <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0007_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"italic\">ϑ</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{m}^{{vartheta }}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, for any fixed positive parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0007_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"italic\">ϑ</m:mi> </m:math> <jats:tex-math>{vartheta }</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Related counting functions are also considered.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148604","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}
Based on the concept of (p,q)left(p,q)-compact operator for p∈[1,∞]pin left[1,infty ] and q∈[1,p*]qin left[1,{p}^{* }], we introduce and study the notion of (p,q)left(p,q)-compact holomorphic mapping between Banach spaces. We prove that the space formed by such mappings is a surjective pq∕(p+q)pq/left(p+q)-Banach bounded-holomorphic ideal that can be generated by composition with the ideal of (p,q)left(p,q)
基于对于 p∈ [ 1 , ∞ ] pin left[1,infty ] 和 q∈ [ 1 , p * ] qin left[1,{p}^{* }] 的 ( p , q ) left(p,q) -compact 算子的概念，我们引入并研究了巴拿赫空间之间的 ( p , q ) left(p,q) -compact 全态映射的概念。我们证明，由这种映射形成的空间是一个投射 p q ∕ ( p + q ) pq/left(p+q) -Banach 有界全形理想，它可以通过与 ( p , q ) left(p,q) -compact 算子的理想组成而生成。此外，我们还研究了穆希卡对此类映射的线性化、它与 ( u * v * + t v * + t u * ) ∕ t u * v * left({u}^{* }{v}^{* }+t{v}^{* }+t{u}^{* })/t{u}^{* }{v}^{* } 的关系。 -巴拿赫有界全形构成理想的 ( t , u , v ) left(t,u,v)-核全形映射为 t , u , v∈ [ 1 , ∞ ] t,u,vin left[1,infty ] 、通过( p , q * , 1 ) left(p,{q}^{* },1)-核算子的理想的注入全域、( p , q ) left(p,q)-D{mathbb{D}}上紧凑全态映射的莫比乌斯不变性、以及通过(p, q ) left(p,q)-D{mathbb{D}}的全紧凑因子化来实现其全态转置 及其通过紧凑全态映射、( p , q ) left(p,q)-紧凑算子和紧凑算子的全紧凑因式分解。
{"title":"(p, q)-Compactness in spaces of holomorphic mappings","authors":"Antonio Jiménez-Vargas, David Ruiz-Casternado","doi":"10.1515/math-2023-0183","DOIUrl":"https://doi.org/10.1515/math-2023-0183","url":null,"abstract":"Based on the concept of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0183_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(p,q)</jats:tex-math> </jats:alternatives> </jats:inline-formula>-compact operator for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0183_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>pin left[1,infty ]</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0183_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>q</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> <jats:tex-math>qin left[1,{p}^{* }]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we introduce and study the notion of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0183_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(p,q)</jats:tex-math> </jats:alternatives> </jats:inline-formula>-compact holomorphic mapping between Banach spaces. We prove that the space formed by such mappings is a surjective <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0183_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>p</m:mi> <m:mi>q</m:mi> <m:mo>∕</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>pq/left(p+q)</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Banach bounded-holomorphic ideal that can be generated by composition with the ideal of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0183_eq_006.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(p,q)</jats:tex-math> </jats:alternatives> </jats:inline-formu","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141058719","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}
Khaled Alhazmy, Fuad Ali Ahmed Almahdi, Najib Mahdou, El Houssaine Oubouhou
Let RR be a commutative ring with identity and j{mathscr{j}} an ideal of RR. An ideal II of RR is said to be a j{mathscr{j}}-ideal if I⊈jIhspace{0.33em} nsubseteq hspace{0.33em}{mathscr{j}}. We define RR to be a
设 R R 是一个具有同一性的交换环，而 j {mathscr{j}} 是 R R 的一个理想。如果 R R 的理想 I I ⊈ j Ihspace{0}} 是一个 j {mathscr{j}} 的理想，那么这个理想就是 R R 的理想 I I 。 -理想，如果 I ⊈ j Ihspace{0.33em}nsubseteq hspace{0.33em}{mathscr{j}} .我们定义 R R 是一个 j {mathscr{j}} 。 -如果每个 j {mathscr{j} 都是 R R 的ideal，那么 R R 就是一个 j {mathscr{j}} 的诺特环。 -的ideal 都是有限生成的。在这项工作中，我们将研究 j {mathscr{j}} -诺特环的一些性质。 -诺特环的一些性质。更准确地说，我们通过共振来研究 j {mathscr{j}} -诺特环。 -Noetherian 环的科恩型定理、平延伸、可分解环、三维延伸环、合并重复、多项式环延伸和幂级数环延伸。
{"title":"About j{mathscr{j}}-Noetherian rings","authors":"Khaled Alhazmy, Fuad Ali Ahmed Almahdi, Najib Mahdou, El Houssaine Oubouhou","doi":"10.1515/math-2024-0014","DOIUrl":"https://doi.org/10.1515/math-2024-0014","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>R</m:mi> </m:math> <jats:tex-math>R</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a commutative ring with identity and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">j</m:mi> </m:math> <jats:tex-math>{mathscr{j}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> an ideal of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>R</m:mi> </m:math> <jats:tex-math>R</jats:tex-math> </jats:alternatives> </jats:inline-formula>. An ideal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_006.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>I</m:mi> </m:math> <jats:tex-math>I</jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_007.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>R</m:mi> </m:math> <jats:tex-math>R</jats:tex-math> </jats:alternatives> </jats:inline-formula> is said to be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_008.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">j</m:mi> </m:math> <jats:tex-math>{mathscr{j}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-ideal if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_009.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>I</m:mi> <m:mspace width=\"0.33em\"/> <m:mo>⊈</m:mo> <m:mspace width=\"0.33em\"/> <m:mi mathvariant=\"script\">j</m:mi> </m:math> <jats:tex-math>Ihspace{0.33em} nsubseteq hspace{0.33em}{mathscr{j}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We define <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_010.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>R</m:mi> </m:math> <jats:tex-math>R</jats:tex-math> </jats:alternatives> </jats:inline-formula> to be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0014_eq_011.png\"/> <m:mat","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141058614","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 article, we first put forward the concept of deferred f-double natural density for double sequences, where f is an unbounded modulus. Then, we combine f-density with deferred statistical convergence for double sequences and investigate deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f. Moreover, we extend these concepts to deferred f-statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred f-statistical convergence of order αalpha and strongly deferred f-summability of order αalpha for double sequences and obtain some conclusions.
在本文中，我们首先提出了双序列的延迟 f-双自然密度概念，其中 f 是无界模数。然后，我们将 f-density 与双序列的延迟统计收敛结合起来，研究了关于模 f 的延迟 f 统计收敛和强延迟 Cesàro 可求和性。此外，我们将这些概念扩展到维杰曼意义上的随机变量双序列的延迟 f 统计收敛，并证明了一些结论。最后，我们考虑了双序列的阶α alpha 的延迟 f 统计收敛性和阶α alpha 的强延迟 f 可求和性的概念，并得出了一些结论。
{"title":"On deferred f-statistical convergence for double sequences","authors":"Yahui Zhu, Ang Shen, Zhongzhi Wang, Weicai Peng","doi":"10.1515/math-2023-0174","DOIUrl":"https://doi.org/10.1515/math-2023-0174","url":null,"abstract":"In this article, we first put forward the concept of deferred <jats:italic>f</jats:italic>-double natural density for double sequences, where <jats:italic>f</jats:italic> is an unbounded modulus. Then, we combine <jats:italic>f</jats:italic>-density with deferred statistical convergence for double sequences and investigate deferred <jats:italic>f</jats:italic>-statistical convergence and strongly deferred <jats:italic>Cesàro</jats:italic> summability with respect to modulus <jats:italic>f</jats:italic>. Moreover, we extend these concepts to deferred <jats:italic>f</jats:italic>-statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred <jats:italic>f</jats:italic>-statistical convergence of order <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0174_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>α</m:mi> </m:math> <jats:tex-math>alpha </jats:tex-math> </jats:alternatives> </jats:inline-formula> and strongly deferred <jats:italic>f</jats:italic>-summability of order <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0174_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>α</m:mi> </m:math> <jats:tex-math>alpha </jats:tex-math> </jats:alternatives> </jats:inline-formula> for double sequences and obtain some conclusions.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801293","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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