In this paper, we mainly consider operator equations $ AX = C $ and $ XB = D $ in the framework of Hilbert space. A new representation of the reduced solution of $ AX = C $ is given by a convergent operator sequence. The common solutions and common real positive solutions of the system of two operator equations $ AX = C $ and $ XB = D $ are studied. The detailed representations of these solutions are provided which extend the classical closed range case with a short proof.
本文主要考虑Hilbert空间框架下的算子方程$ AX = C $和$ XB = D $。用收敛算子序列给出了AX = C $的简化解的一种新表示。研究了两个算子方程组$ AX = C $和$ XB = D $的公共解和公共实正解。给出了这些解的详细表示，用简短的证明扩展了经典的闭范围情形。
{"title":"Real positive solutions of operator equations $ AX = C $ and $ XB = D $","authors":"Haiyan Zhang, Yanni Dou, Weiyan Yu","doi":"10.3934/math.2023777","DOIUrl":"https://doi.org/10.3934/math.2023777","url":null,"abstract":"In this paper, we mainly consider operator equations $ AX = C $ and $ XB = D $ in the framework of Hilbert space. A new representation of the reduced solution of $ AX = C $ is given by a convergent operator sequence. The common solutions and common real positive solutions of the system of two operator equations $ AX = C $ and $ XB = D $ are studied. The detailed representations of these solutions are provided which extend the classical closed range case with a short proof.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70202169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A denial-of-service (DoS) attack aims to exhaust the resources of the victim by sending attack packets and ultimately stop the legitimate packets by various techniques. The paper discusses the consequences of distributed denial-of-service (DDoS) attacks in various application areas of Internet of Things (IoT). In this paper, we have analyzed the performance of machine learning(ML)-based classifiers including bagging and boosting techniques for the binary classification of attack traffic. For the analysis, we have used the benchmark CICDDoS2019 dataset which deals with DDoS attacks based on User Datagram Protocol (UDP) and Transmission Control Protocol (TCP) in order to study new kinds of attacks. Since these protocols are widely used for communication in IoT networks, this data has been used for studying DDoS attacks in the IoT domain. Since the data is highly unbalanced, data balancing is done using an ensemble sampling approach comprising random under-sampler and ADAptive SYNthetic (ADASYN) oversampling technique. Feature selection is achieved using two methods, i.e., (a) Pearson correlation coefficient and (b) Extra Tree classifier. Further, performance is evaluated for ML classifiers viz. Random Forest (RF), Naïve Bayes (NB), support vector machine (SVM), AdaBoost, eXtreme Gradient Boosting (XGBoost) and Gradient Boosting (GB) algorithms. It is found that RF has given the best performance with the least training and prediction time. Further, it is found that feature selection using extra trees classifier is more efficient as compared to the Pearson correlation coefficient method in terms of total time required in training and prediction for most classifiers. It is found that RF has given best performance with least time along with feature selection using Pearson correlation coefficient in attack detection.
DoS (denial-of-service)攻击的目的是通过发送攻击报文，耗尽被攻击对象的资源，并通过各种技术手段最终阻断合法的报文。本文讨论了分布式拒绝服务(DDoS)攻击在物联网(IoT)各个应用领域的后果。在本文中，我们分析了基于机器学习(ML)的分类器的性能，包括用于攻击流量二进制分类的bagging和boosting技术。在分析中，我们使用了CICDDoS2019基准数据集，该数据集处理基于用户数据报协议(UDP)和传输控制协议(TCP)的DDoS攻击，以研究新的攻击类型。由于这些协议广泛用于物联网网络中的通信，因此这些数据已用于研究物联网领域的DDoS攻击。由于数据高度不平衡，数据平衡使用由随机欠采样器和自适应合成(ADASYN)过采样技术组成的集成采样方法来完成。特征选择使用两种方法实现，即(a) Pearson相关系数和(b) Extra Tree分类器。此外，还评估了ML分类器的性能，即随机森林(RF)、Naïve贝叶斯(NB)、支持向量机(SVM)、AdaBoost、极端梯度增强(XGBoost)和梯度增强(GB)算法。结果表明，在训练和预测时间最少的情况下，射频算法的性能最好。此外，就大多数分类器的训练和预测所需的总时间而言，与Pearson相关系数方法相比，使用额外树分类器进行特征选择的效率更高。研究发现，结合使用Pearson相关系数的特征选择，射频在攻击检测中能够以最少的时间获得最佳性能。
{"title":"Detection of DDoS attack in IoT traffic using ensemble machine learning techniques","authors":"N. Pandey, P. K. Mishra","doi":"10.3934/nhm.2023061","DOIUrl":"https://doi.org/10.3934/nhm.2023061","url":null,"abstract":"A denial-of-service (DoS) attack aims to exhaust the resources of the victim by sending attack packets and ultimately stop the legitimate packets by various techniques. The paper discusses the consequences of distributed denial-of-service (DDoS) attacks in various application areas of Internet of Things (IoT). In this paper, we have analyzed the performance of machine learning(ML)-based classifiers including bagging and boosting techniques for the binary classification of attack traffic. For the analysis, we have used the benchmark CICDDoS2019 dataset which deals with DDoS attacks based on User Datagram Protocol (UDP) and Transmission Control Protocol (TCP) in order to study new kinds of attacks. Since these protocols are widely used for communication in IoT networks, this data has been used for studying DDoS attacks in the IoT domain. Since the data is highly unbalanced, data balancing is done using an ensemble sampling approach comprising random under-sampler and ADAptive SYNthetic (ADASYN) oversampling technique. Feature selection is achieved using two methods, i.e., (a) Pearson correlation coefficient and (b) Extra Tree classifier. Further, performance is evaluated for ML classifiers viz. Random Forest (RF), Naïve Bayes (NB), support vector machine (SVM), AdaBoost, eXtreme Gradient Boosting (XGBoost) and Gradient Boosting (GB) algorithms. It is found that RF has given the best performance with the least training and prediction time. Further, it is found that feature selection using extra trees classifier is more efficient as compared to the Pearson correlation coefficient method in terms of total time required in training and prediction for most classifiers. It is found that RF has given best performance with least time along with feature selection using Pearson correlation coefficient in attack detection.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228479","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 is concerned with the existence, uniqueness and exponential stability of mild solutions for a class of impulsive stochastic differential equations driven by Poisson jumps and time-varying delays. Utilizing the successive approximation method, we obtain the criteria of existence and uniqueness of mild solutions for the considered impulsive stochastic differential equations. Then, the exponential stability in the $ p $th moment of the mild solution is also devised for considered equations by establishing an improved impulsive-integral inequality, which improves some known existing ones. Finally, an example and numerical simulations are given to illustrate the efficiency of the obtained theoretical results.
{"title":"Some results on the existence and stability of impulsive delayed stochastic differential equations with Poisson jumps","authors":"Dongdong Gao, Da Kuang, Jianli Li","doi":"10.3934/math.2023780","DOIUrl":"https://doi.org/10.3934/math.2023780","url":null,"abstract":"This paper is concerned with the existence, uniqueness and exponential stability of mild solutions for a class of impulsive stochastic differential equations driven by Poisson jumps and time-varying delays. Utilizing the successive approximation method, we obtain the criteria of existence and uniqueness of mild solutions for the considered impulsive stochastic differential equations. Then, the exponential stability in the $ p $th moment of the mild solution is also devised for considered equations by establishing an improved impulsive-integral inequality, which improves some known existing ones. Finally, an example and numerical simulations are given to illustrate the efficiency of the obtained theoretical results.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70202460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A two-dimensional multi-term time fractional diffusion equation $ {D}_{t}^{mathit{boldsymbol{alpha}}} u(x, y, t)- Delta u(x, y, t) = f(x, y, t) $ is considered in this paper, where $ {D}_{t}^{mathit{boldsymbol{alpha}}} $ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.
本文考虑二维多项时间分数扩散方程$ {D}_{t}^{mathit{boldsymbol{alpha}}} u(x, y, t)- Delta u(x, y, t) = f(x, y, t) $，其中$ {D}_{t}^{mathit{boldsymbol{alpha}}} $为多项时间Caputo分数导数。为了在数值上求解方程，在渐变时间网格上对每个分数阶导数使用L1离散化，并在均匀空间网格上对空间导数使用标准有限差分法。我们给出了求解多项时间分数扩散问题的完全离散L1-ADI格式的严格的稳定性和收敛性分析。数值结果表明，该方法的估计误差很小。
{"title":"Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation","authors":"Kexin Li, Hu Chen, Shusen Xie","doi":"10.3934/nhm.2023064","DOIUrl":"https://doi.org/10.3934/nhm.2023064","url":null,"abstract":"A two-dimensional multi-term time fractional diffusion equation $ {D}_{t}^{mathit{boldsymbol{alpha}}} u(x, y, t)- Delta u(x, y, t) = f(x, y, t) $ is considered in this paper, where $ {D}_{t}^{mathit{boldsymbol{alpha}}} $ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228188","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 dynamical systems on networks, Kirchhoff's first law describes the local conservation of a quantity across edges. Predominantly, Kirchhoff's first law has been conceived as a phenomenological law of continuum physics. We establish its algebraic form as a property that is inherited from fundamental axioms of a network's geometry, instead of a law observed in physical nature. To this end, we extend calculus to networks, modeled as abstract metric spaces, and derive Kirchhoff's first law for hyperbolic conservation laws. In particular, our results show that hyperbolic conservation laws on networks can be stated without explicit Kirchhoff-type boundary conditions.
{"title":"A proof of Kirchhoff's first law for hyperbolic conservation laws on networks","authors":"Alexandre M. Bayen, Alexander Keimer, Nils Müller","doi":"10.3934/nhm.2023078","DOIUrl":"https://doi.org/10.3934/nhm.2023078","url":null,"abstract":"<abstract><p>In dynamical systems on networks, Kirchhoff's first law describes the local conservation of a quantity across edges. Predominantly, Kirchhoff's first law has been conceived as a phenomenological law of continuum physics. We establish its algebraic form as a property that is inherited from fundamental axioms of a network's geometry, instead of a law observed in physical nature. To this end, we extend calculus to networks, modeled as abstract metric spaces, and derive Kirchhoff's first law for hyperbolic conservation laws. In particular, our results show that hyperbolic conservation laws on networks can be stated without explicit Kirchhoff-type boundary conditions.</p></abstract>","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666801","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}
Eman S. Attia, Ashraf A. M. Khalaf, Fathi E. Abd El-Samie, Saied M. Abd El-atty, Konstantinos A. Lizos, Osama Alfarraj
Recent nanomedical applications, particularly targeted drug delivery system (TDDS) scenarios, have made use of molecular communications via diffusion (MCvD) based on nanosystems. In order to improve the performance of such nanosystems, nanonetworks-based molecular communication is investigated. Employing a nanorelay approach and cooperative molecular communications, we provide a method for optimizing the performance of the nanosystem while taking blood flow effects into consideration in terms of drift velocity. Unlike the earlier studies, the position of the nanorelay and the allocated amount of molecular drugs will be optimized. We provide closed-form expressions for molecular channel capacity and the error probability of a molecular frame. According to the simulation results, it is possible to significantly reduce error probability of a molecular frame and thus increase channel capacity by optimizing the drift velocity, detection threshold and the location of the nanorelay within the proposed nanosystem.
{"title":"Performance analysis of nanosystem based on cooperative relay for nanonetworks","authors":"Eman S. Attia, Ashraf A. M. Khalaf, Fathi E. Abd El-Samie, Saied M. Abd El-atty, Konstantinos A. Lizos, Osama Alfarraj","doi":"10.3934/nhm.2023072","DOIUrl":"https://doi.org/10.3934/nhm.2023072","url":null,"abstract":"<abstract> <p>Recent nanomedical applications, particularly targeted drug delivery system (TDDS) scenarios, have made use of molecular communications via diffusion (MCvD) based on nanosystems. In order to improve the performance of such nanosystems, nanonetworks-based molecular communication is investigated. Employing a nanorelay approach and cooperative molecular communications, we provide a method for optimizing the performance of the nanosystem while taking blood flow effects into consideration in terms of drift velocity. Unlike the earlier studies, the position of the nanorelay and the allocated amount of molecular drugs will be optimized. We provide closed-form expressions for molecular channel capacity and the error probability of a molecular frame. According to the simulation results, it is possible to significantly reduce error probability of a molecular frame and thus increase channel capacity by optimizing the drift velocity, detection threshold and the location of the nanorelay within the proposed nanosystem.</p> </abstract>","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135496619","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}
Recently, researchers have proposed a lot of methods to boost the performance of convolutional neural networks (CNNs) for classifying remote sensing images (RSI). However, the methods' performance improvements were insignificant, while time and hardware costs increased dramatically due to re-modeling. To tackle this problem, this study sought a simple, lightweight, yet more accurate solution for RSI semantic classification (RSI-SC). At first, we proposed a set of mathematical derivations to analyze and identify the best way among different technical roadmaps. Afterward, we selected a simple route that can significantly boost a single CNN's performance while maintaining simplicity and reducing costs in time and hardware. The proposed method, called RE-EfficientNet, only consists of a lightweight EfficientNet-B3 and a concise training algorithm named RE-CNN. The novelty of RE-EfficientNet and RE-CNN includes the following: First, EfficientNet-B3 employs transfer learning from ImageNet-1K and excludes any complicated re-modeling. It can adequately utilize the easily accessible pre-trained weights for time savings and avoid the pre-training effect being weakened due to re-modeling. Second, RE-CNN includes an effective combination of data augmentation (DA) transformations and two modified training tricks (TTs). It can alleviate the data distribution shift from DA-processed training sets and make the TTs more effective through modification according to the inherent nature of RSI. Extensive experimental results on two RSI sets prove that RE-EfficientNet can surpass all 30 cutting-edge methods published before 2023. It gives a remarkable improvement of 0.50% to 0.75% in overall accuracy (OA) and a 75% or more reduction in parameters. The ablation experiment also reveals that RE-CNN can improve CNN OA by 0.55% to 1.10%. All the results indicate that RE-EfficientNet is a simple, lightweight and more accurate solution for RSI-SC. In addition, we argue that the ideas proposed in this work about how to choose an appropriate model and training algorithm can help us find more efficient approaches in the future.
{"title":"Simple is best: A single-CNN method for classifying remote sensing images","authors":"Huaxiang Song, Yong Zhou","doi":"10.3934/nhm.2023070","DOIUrl":"https://doi.org/10.3934/nhm.2023070","url":null,"abstract":"Recently, researchers have proposed a lot of methods to boost the performance of convolutional neural networks (CNNs) for classifying remote sensing images (RSI). However, the methods' performance improvements were insignificant, while time and hardware costs increased dramatically due to re-modeling. To tackle this problem, this study sought a simple, lightweight, yet more accurate solution for RSI semantic classification (RSI-SC). At first, we proposed a set of mathematical derivations to analyze and identify the best way among different technical roadmaps. Afterward, we selected a simple route that can significantly boost a single CNN's performance while maintaining simplicity and reducing costs in time and hardware. The proposed method, called RE-EfficientNet, only consists of a lightweight EfficientNet-B3 and a concise training algorithm named RE-CNN. The novelty of RE-EfficientNet and RE-CNN includes the following: First, EfficientNet-B3 employs transfer learning from ImageNet-1K and excludes any complicated re-modeling. It can adequately utilize the easily accessible pre-trained weights for time savings and avoid the pre-training effect being weakened due to re-modeling. Second, RE-CNN includes an effective combination of data augmentation (DA) transformations and two modified training tricks (TTs). It can alleviate the data distribution shift from DA-processed training sets and make the TTs more effective through modification according to the inherent nature of RSI. Extensive experimental results on two RSI sets prove that RE-EfficientNet can surpass all 30 cutting-edge methods published before 2023. It gives a remarkable improvement of 0.50% to 0.75% in overall accuracy (OA) and a 75% or more reduction in parameters. The ablation experiment also reveals that RE-CNN can improve CNN OA by 0.55% to 1.10%. All the results indicate that RE-EfficientNet is a simple, lightweight and more accurate solution for RSI-SC. In addition, we argue that the ideas proposed in this work about how to choose an appropriate model and training algorithm can help us find more efficient approaches in the future.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228453","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}
Sharayu Moharir, Ananya S. Omanwar, Neeraja Sahasrabudhe
We consider a growing population of individuals with binary opinions, namely, 0 or 1, that evolve in discrete time. The underlying interaction network is complete. At every time step, a fixed number of individuals are added to the population. The opinion of the new individuals may or may not depend on the current configuration of opinions in the population. Further, in each time step, a fixed number of individuals are chosen and they update their opinion in three possible ways: they organically switch their opinion with some probability and with some probability they adopt the majority or the minority opinion. We study the asymptotic behaviour of the fraction of individuals with either opinion and characterize conditions under which it converges to a deterministic limit. We analyze the behaviour of the limiting fraction as a function of the probability of new individuals having opinion 1 as well as with respect to the ratio of the number of people being added to the population and the number of people being chosen to update opinions. We also discuss the nature of fluctuations around the limiting fraction and study the transitions in scaling depending on the system parameters. Further, for this opinion dynamics model on a finite time horizon, we obtain optimal external influencing strategies in terms of when to influence to get the maximum expected fraction of individuals with opinion 1 at the end of the finite time horizon.
{"title":"Diffusion of binary opinions in a growing population with heterogeneous behaviour and external influence","authors":"Sharayu Moharir, Ananya S. Omanwar, Neeraja Sahasrabudhe","doi":"10.3934/nhm.2023056","DOIUrl":"https://doi.org/10.3934/nhm.2023056","url":null,"abstract":"We consider a growing population of individuals with binary opinions, namely, 0 or 1, that evolve in discrete time. The underlying interaction network is complete. At every time step, a fixed number of individuals are added to the population. The opinion of the new individuals may or may not depend on the current configuration of opinions in the population. Further, in each time step, a fixed number of individuals are chosen and they update their opinion in three possible ways: they organically switch their opinion with some probability and with some probability they adopt the majority or the minority opinion. We study the asymptotic behaviour of the fraction of individuals with either opinion and characterize conditions under which it converges to a deterministic limit. We analyze the behaviour of the limiting fraction as a function of the probability of new individuals having opinion 1 as well as with respect to the ratio of the number of people being added to the population and the number of people being chosen to update opinions. We also discuss the nature of fluctuations around the limiting fraction and study the transitions in scaling depending on the system parameters. Further, for this opinion dynamics model on a finite time horizon, we obtain optimal external influencing strategies in terms of when to influence to get the maximum expected fraction of individuals with opinion 1 at the end of the finite time horizon.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228015","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 demonstrates several sufficient frameworks for the mono-cluster flocking, the non-emergence of mono-cluster flocking and the multi-cluster flocking of the thermodynamic Cucker–Smale model with a unit-speed constraint (say TCSUS). First, in a different way than [2], we present the admissible data for the mono-cluster flocking of TCSUS to occur. Second, we prove that when the coupling strength is less than some positive value, mono-cluster flocking does not occur in the TCSUS system with an integrable communication weight. Third, motivated from the study on coupling strengths where the mono-cluster flocking does not occur, we investigate appropriate sufficient frameworks to derive the multi-cluster flocking of the TCSUS system.
{"title":"Non-emergence of mono-cluster flocking and multi-cluster flocking of the thermodynamic Cucker–Smale model with a unit-speed constraint","authors":"Hyunjin Ahn","doi":"10.3934/nhm.2023066","DOIUrl":"https://doi.org/10.3934/nhm.2023066","url":null,"abstract":"This paper demonstrates several sufficient frameworks for the mono-cluster flocking, the non-emergence of mono-cluster flocking and the multi-cluster flocking of the thermodynamic Cucker–Smale model with a unit-speed constraint (say TCSUS). First, in a different way than [2], we present the admissible data for the mono-cluster flocking of TCSUS to occur. Second, we prove that when the coupling strength is less than some positive value, mono-cluster flocking does not occur in the TCSUS system with an integrable communication weight. Third, motivated from the study on coupling strengths where the mono-cluster flocking does not occur, we investigate appropriate sufficient frameworks to derive the multi-cluster flocking of the TCSUS system.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228358","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, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.
{"title":"An adaptive grid method for a singularly perturbed convection-diffusion equation with a discontinuous convection coefficient","authors":"Xiong-fa Mai, Ciwen Zhu, Libin Liu","doi":"10.3934/nhm.2023067","DOIUrl":"https://doi.org/10.3934/nhm.2023067","url":null,"abstract":"In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.","PeriodicalId":54732,"journal":{"name":"Networks and Heterogeneous Media","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70228362","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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