Pub Date : 2024-04-01DOI: 10.4208/ata.oa-2021-0014
Zhizhuo Zhang,Bo Wu, Zuguo Yu
The random walk is one of the most basic dynamic properties of complex networks, which has gradually become a research hotspot in recent years due to its many applications in actual networks. An important characteristic of the random walk is the mean time to absorption, which plays an extremely important role in the study of topology, dynamics and practical application of complex networks. Analyzing the mean time to absorption on the regular iterative self-similar network models is an important way to explore the influence of self-similarity on the properties of random walks on the network. The existing literatures have proved that even local self-similar structures can greatly affect the properties of random walks on the global network, but they have failed to prove whether these effects are related to the scale of these self-similar structures. In this article, we construct and study a class of Horizontal Partitioned Sierpinski Gasket network model based on the classic Sierpinski gasket network, which is composed of local self-similar structures, and the scale of these structures will be controlled by the partition coefficient $k.$ Then, the analytical expressions and approximate expressions of the mean time to absorption on the network model are obtained, which prove that the size of the self-similar structure in the network will directly restrict the influence of the self-similar structure on the properties of random walks on the network. Finally, we also analyzed the mean time to absorption of different absorption nodes on the network to find the location of the node with the highest absorption efficiency.
{"title":"The Mean Time to Absorption on Horizontal Partitioned Sierpinski Gasket Networks","authors":"Zhizhuo Zhang,Bo Wu, Zuguo Yu","doi":"10.4208/ata.oa-2021-0014","DOIUrl":"https://doi.org/10.4208/ata.oa-2021-0014","url":null,"abstract":"The random walk is one of the most basic dynamic properties of complex\u0000networks, which has gradually become a research hotspot in recent years due to its\u0000many applications in actual networks. An important characteristic of the random walk\u0000is the mean time to absorption, which plays an extremely important role in the study\u0000of topology, dynamics and practical application of complex networks. Analyzing the\u0000mean time to absorption on the regular iterative self-similar network models is an important way to explore the influence of self-similarity on the properties of random\u0000walks on the network. The existing literatures have proved that even local self-similar\u0000structures can greatly affect the properties of random walks on the global network,\u0000but they have failed to prove whether these effects are related to the scale of these\u0000self-similar structures. In this article, we construct and study a class of Horizontal Partitioned Sierpinski Gasket network model based on the classic Sierpinski gasket network, which is composed of local self-similar structures, and the scale of these structures will be controlled by the partition coefficient $k.$ Then, the analytical expressions\u0000and approximate expressions of the mean time to absorption on the network model\u0000are obtained, which prove that the size of the self-similar structure in the network will\u0000directly restrict the influence of the self-similar structure on the properties of random\u0000walks on the network. Finally, we also analyzed the mean time to absorption of different absorption nodes on the network to find the location of the node with the highest\u0000absorption efficiency.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140581170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.4208/ata.oa-2021-0004
Ziyu Zhao,Yicheng Liu, Xiao Wang
Noting that the particles communicate with each other within a finite distance, in the paper, we investigate the hierarchical model with free will and cut-off function (linear or nonlinear). As our observation, the cut-off size $r$ is a sensitive coefficient to achieve the flocking behavior. Besides we will use energy function to achieve some sufficient conditions to achieve a flock. The free-will represents the tendency of the particle itself to move. It will either facilitate or prevent cluster generation. Some numerical simulations will validate our conclusions.
{"title":"Dynamics Analysis for a Hierarchical System with Nonlinear Cut-Off Interaction and Free Will","authors":"Ziyu Zhao,Yicheng Liu, Xiao Wang","doi":"10.4208/ata.oa-2021-0004","DOIUrl":"https://doi.org/10.4208/ata.oa-2021-0004","url":null,"abstract":"Noting that the particles communicate with each other within a finite distance, in the paper, we investigate the hierarchical model with free will and cut-off\u0000function (linear or nonlinear). As our observation, the cut-off size $r$ is a sensitive coefficient to achieve the flocking behavior. Besides we will use energy function to achieve\u0000some sufficient conditions to achieve a flock. The free-will represents the tendency of\u0000the particle itself to move. It will either facilitate or prevent cluster generation. Some\u0000numerical simulations will validate our conclusions.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140581027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.4208/ata.oa-2022-0029
Chaojiang Xu, Yan Xu
In this note, we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an $L^2$ initial datum for positive time. So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.
{"title":"A Remark about Time-Analyticity of the Linear Landau Equation with Soft Potential","authors":"Chaojiang Xu, Yan Xu","doi":"10.4208/ata.oa-2022-0029","DOIUrl":"https://doi.org/10.4208/ata.oa-2022-0029","url":null,"abstract":"In this note, we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials. We prove that the solution to the Cauchy\u0000problem enjoys the analytic regularizing effect of the time variable with an $L^2$ initial\u0000datum for positive time. So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat\u0000equation.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140581160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/ata.oa-2017-0026
Demin Yao, Kai Zhao
In this paper, using the atomic decomposition of the Herz-Morrey-Hardy spaces with variable exponent, the wavelet characterization by means of a local version of the discrete tent spaces with variable exponent is established. As an application, the boundedness of the fractional integral operators from variable exponent Herz-Morrey-Hardy spaces into variable exponent Herz-Morrey spaces is obtained.
{"title":"Variable Exponent Herz-Morrey-Hardy Spaces Characterized by Wavelets and Its Application","authors":"Demin Yao, Kai Zhao","doi":"10.4208/ata.oa-2017-0026","DOIUrl":"https://doi.org/10.4208/ata.oa-2017-0026","url":null,"abstract":"In this paper, using the atomic decomposition of the Herz-Morrey-Hardy\u0000spaces with variable exponent, the wavelet characterization by means of a local version\u0000of the discrete tent spaces with variable exponent is established. As an application, the\u0000boundedness of the fractional integral operators from variable exponent Herz-Morrey-Hardy spaces into variable exponent Herz-Morrey spaces is obtained.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/ata.oa-2022-0012
Zaizheng Li,Qidi Zhang, Zhitao Zhang
We study the existence of standing waves of fractional Schrödinger equations with a potential term and a general nonlinear term: $$iu_t − (−∆) ^su − V(x)u + f(u) = 0, (t, x) ∈ mathbb{R}_+ × mathbb{R}^N,$$ where $s ∈ (0, 1),$ $N > 2s$ is an integer and $V(x) ≤ 0$ is radial. More precisely, we investigate the minimizing problem with $L^2$-constraint: $$E(alpha)={rm inf}left{frac{1}{2}int_{mathbb{R}^N}|(-Delta)^{frac{s}{2}}u|^2+V(x)|u|^2-2F(|u|)mid uin H^s(mathbb{R}^N),||u||^2_{L^2(mathbb{R}^N)}=alpharight}.$$ Under general assumptions on the nonlinearity term $f(u)$ and the potential term $V(x),$ we prove that there exists a constant $α_0 ≥ 0$ such that $E(α)$ can be achieved for all $α > α_0,$ and there is no global minimizer with respect to $E(α)$ for all $0 < α < α_0.$ Moreover, we propose some criteria determining $α_0 = 0$ or $α_0 > 0.$
{"title":"Standing Waves of Fractional Schrödinger Equations with Potentials and General Nonlinearities","authors":"Zaizheng Li,Qidi Zhang, Zhitao Zhang","doi":"10.4208/ata.oa-2022-0012","DOIUrl":"https://doi.org/10.4208/ata.oa-2022-0012","url":null,"abstract":"We study the existence of standing waves of fractional Schrödinger equations with a potential term and a general nonlinear term: $$iu_t − (−∆) ^su − V(x)u + f(u) = 0, (t, x) ∈ mathbb{R}_+ × mathbb{R}^N,$$ where $s ∈ (0, 1),$ $N > 2s$ is an integer and $V(x) ≤ 0$ is radial. More precisely, we\u0000investigate the minimizing problem with $L^2$-constraint: $$E(alpha)={rm inf}left{frac{1}{2}int_{mathbb{R}^N}|(-Delta)^{frac{s}{2}}u|^2+V(x)|u|^2-2F(|u|)mid uin H^s(mathbb{R}^N),||u||^2_{L^2(mathbb{R}^N)}=alpharight}.$$ Under general assumptions on the nonlinearity term $f(u)$ and the potential term $V(x),$ we prove that there exists a constant $α_0 ≥ 0$ such that $E(α)$ can be achieved for all $α > α_0,$ and there is no global minimizer with respect to $E(α)$ for all $0 < α < α_0.$ Moreover, we propose some criteria determining $α_0 = 0$ or $α_0 > 0.$","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139057550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/ata.oa-2020-0048
Laiyi Zhu, Xingjun Zhao
In the present note, we consider the problem: how many interpolation nodes can be deleted from the Newman-type rational function such that the convergence rate still achieve.
在本说明中，我们考虑的问题是：从纽曼型有理函数中删除多少个插值节点，仍能达到收敛率。
{"title":"Approximation Properties of Newman Type Interpolation Rational Functions with Fewer Nodes","authors":"Laiyi Zhu, Xingjun Zhao","doi":"10.4208/ata.oa-2020-0048","DOIUrl":"https://doi.org/10.4208/ata.oa-2020-0048","url":null,"abstract":"In the present note, we consider the problem: how many interpolation nodes\u0000can be deleted from the Newman-type rational function such that the convergence rate\u0000still achieve.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139066778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/ata.oa-2020-0039
Qianjun He, Mingquan Wei null, Dunyan Yan
In this paper, we calculate the sharp bound for the generalized m-linear ndimensional Hardy-Littlewood-Pólya operator on power weighted central and noncentral homogeneous Morrey spaces. As an application, the sharp bound for the Hardy-Littlewood-Pólya operator on power weighted central and noncentral homogeneous Morrey spaces is obtained. Finally, we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces, which generalizes the previous results.
{"title":"Sharp Bound for the Generalized $m$-Linear $n$-Dimensional Hardy-Littlewood-Pόlya Operator","authors":"Qianjun He, Mingquan Wei null, Dunyan Yan","doi":"10.4208/ata.oa-2020-0039","DOIUrl":"https://doi.org/10.4208/ata.oa-2020-0039","url":null,"abstract":"In this paper, we calculate the sharp bound for the generalized m-linear ndimensional Hardy-Littlewood-Pólya operator on power weighted central and noncentral homogeneous Morrey spaces. As an application, the sharp bound for the Hardy-Littlewood-Pólya operator on power weighted central and noncentral homogeneous Morrey spaces is obtained. Finally, we also find the sharp bound for the Hausdorff operator on power weighted central and noncentral homogeneous Morrey spaces, which generalizes the previous results.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80457796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/ata.oa-2018-0020
Xiangdong Yang
. The article considers the controllability of a diffusion equation with fractional integro-differential expressions. We prove that the resulting equation is null-controllable in arbitrary small time. Our method reduces essentially to the study of classical moment problems.
{"title":"Null-Controllability of a Diffusion Equation with Fractional Integro-Differential Expressions","authors":"Xiangdong Yang","doi":"10.4208/ata.oa-2018-0020","DOIUrl":"https://doi.org/10.4208/ata.oa-2018-0020","url":null,"abstract":". The article considers the controllability of a diffusion equation with fractional integro-differential expressions. We prove that the resulting equation is null-controllable in arbitrary small time. Our method reduces essentially to the study of classical moment problems.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135142855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/ata.oa-2019-0023
S. Dragomir
. In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions de(cid:133)ned on bodies B (cid:26) R n that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.
{"title":"Multiple Integral Inequalities for Schur Convex Functions on Symmetric and Convex Bodies","authors":"S. Dragomir","doi":"10.4208/ata.oa-2019-0023","DOIUrl":"https://doi.org/10.4208/ata.oa-2019-0023","url":null,"abstract":". In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions de(cid:133)ned on bodies B (cid:26) R n that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81504663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/ata.oa-2021-0030
Weidong Wang
{"title":"Busemann-Petty Type Problem for the General $L_p$-Centroid Bodies","authors":"Weidong Wang","doi":"10.4208/ata.oa-2021-0030","DOIUrl":"https://doi.org/10.4208/ata.oa-2021-0030","url":null,"abstract":"","PeriodicalId":29763,"journal":{"name":"Analysis in Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73062105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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