Pub Date : 2024-12-05DOI: 10.1016/j.amc.2024.129234
Xin Wang, Xiaoping Wang, Haitao Qi, Huanying Xu
In this paper, we introduce a high-precision Hermite neural network solver which employs Hermite interpolation technique to construct high-order explicit approximation schemes for fractional derivatives. By automatically satisfying the initial conditions, the construction process of the objective function is simplified, thereby reducing the complexity of the solution. Our neural networks are trained and fine-tuned to solve one-dimensional (1D) and two-dimensional (2D) time fractional Allen-Cahn equations with limited sampling points, yielding high-precision results. Additionally, we tackle the parameter inversion problem by accurately recovering model parameters from observed data, thereby validating the efficacy of the proposed algorithm. We compare the L2 relative error between the exact solution and the predicted solution to verify the robustness and accuracy of the algorithm. This analysis confirms the reliability of our method in capturing the fundamental dynamics of equations. Furthermore, we extend our analysis to three-dimensional (3D) cases, which is covered in the appendix, and provide a thorough evaluation of the performance of our method. This paper also conducts comprehensive analysis of the network structure. Numerical experiments indicate that the number of layers, the number of neurons in each layer, and the choice of learning rate play a crucial role in the performance of our solver.
{"title":"Numerical simulation of time fractional Allen-Cahn equation based on Hermite neural solver","authors":"Xin Wang, Xiaoping Wang, Haitao Qi, Huanying Xu","doi":"10.1016/j.amc.2024.129234","DOIUrl":"https://doi.org/10.1016/j.amc.2024.129234","url":null,"abstract":"In this paper, we introduce a high-precision Hermite neural network solver which employs Hermite interpolation technique to construct high-order explicit approximation schemes for fractional derivatives. By automatically satisfying the initial conditions, the construction process of the objective function is simplified, thereby reducing the complexity of the solution. Our neural networks are trained and fine-tuned to solve one-dimensional (1D) and two-dimensional (2D) time fractional Allen-Cahn equations with limited sampling points, yielding high-precision results. Additionally, we tackle the parameter inversion problem by accurately recovering model parameters from observed data, thereby validating the efficacy of the proposed algorithm. We compare the <mml:math altimg=\"si1.svg\"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> relative error between the exact solution and the predicted solution to verify the robustness and accuracy of the algorithm. This analysis confirms the reliability of our method in capturing the fundamental dynamics of equations. Furthermore, we extend our analysis to three-dimensional (3D) cases, which is covered in the appendix, and provide a thorough evaluation of the performance of our method. This paper also conducts comprehensive analysis of the network structure. Numerical experiments indicate that the number of layers, the number of neurons in each layer, and the choice of learning rate play a crucial role in the performance of our solver.","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"140 1","pages":""},"PeriodicalIF":4.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-03DOI: 10.1016/j.amc.2024.129216
Serafino Cicerone, Alessia Di Fonso, Gabriele Di Stefano, Alfredo Navarra, Francesco Piselli
Let G be a graph and X⊆V(G). Then, vertices x and y of G are X-visible if there exists a shortest x,y-path where no internal vertices belong to X. The set X is a mutual-visibility set of G if every two vertices of X are X-visible, while X is a total mutual-visibility set if any two vertices from V(G) are X-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) μ(G) (resp. μt(G)) of G. It is known that computing μ(G) is an NP-complete problem, as well as μt(G). In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, Fibonacci cubes, cube-connected cycles, and butterflies). Concerning computing μ(G), we provide approximation algorithms for hypercubes, Fibonacci cubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing μt(G) (in the literature, already studied in hypercubes), whereas we obtain exact formulae for both cube-connected cycles and butterflies.
{"title":"Mutual and total mutual visibility in hypercube-like graphs","authors":"Serafino Cicerone, Alessia Di Fonso, Gabriele Di Stefano, Alfredo Navarra, Francesco Piselli","doi":"10.1016/j.amc.2024.129216","DOIUrl":"https://doi.org/10.1016/j.amc.2024.129216","url":null,"abstract":"Let <ce:italic>G</ce:italic> be a graph and <mml:math altimg=\"si3.svg\"><mml:mi>X</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. Then, vertices <ce:italic>x</ce:italic> and <ce:italic>y</ce:italic> of <ce:italic>G</ce:italic> are <ce:italic>X</ce:italic>-visible if there exists a shortest <mml:math altimg=\"si4.svg\"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:math>-path where no internal vertices belong to <ce:italic>X</ce:italic>. The set <ce:italic>X</ce:italic> is a mutual-visibility set of <ce:italic>G</ce:italic> if every two vertices of <ce:italic>X</ce:italic> are <ce:italic>X</ce:italic>-visible, while <ce:italic>X</ce:italic> is a total mutual-visibility set if any two vertices from <mml:math altimg=\"si5.svg\"><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> are <ce:italic>X</ce:italic>-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) <mml:math altimg=\"si6.svg\"><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> (resp. <mml:math altimg=\"si7.svg\"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>) of <ce:italic>G</ce:italic>. It is known that computing <mml:math altimg=\"si6.svg\"><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> is an NP-complete problem, as well as <mml:math altimg=\"si7.svg\"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, Fibonacci cubes, cube-connected cycles, and butterflies). Concerning computing <mml:math altimg=\"si6.svg\"><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, we provide approximation algorithms for hypercubes, Fibonacci cubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing <mml:math altimg=\"si7.svg\"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> (in the literature, already studied in hypercubes), whereas we obtain exact formulae for both cube-connected cycles and butterflies.","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"42 1","pages":""},"PeriodicalIF":4.0,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-03DOI: 10.1016/j.amc.2024.129230
Juan Han, Liqiao Yang, Kit Ian Kou, Jifei Miao, Lizhi Liu
Matrix completion is a challenging problem in computer vision. Recently, quaternion representations of color images have achieved competitive performance in many fields. The information on the coupling between the three channels of the color image is better utilized since the color image is treated as a whole. Due to this, researcher interest in low-rank quaternion matrix completion (LRQMC) algorithms has grown significantly. In contrast to the traditional quaternion matrix completion algorithms that rely on quaternion singular value decomposition (QSVD), we propose a novel method based on quaternion Qatar Riyal decomposition (QQR). First, a novel approach (CQSVD-QQR) to computing an approximation of QSVD based on iterative QQR is put forward, which has lower computational complexity than QSVD. CQSVD-QQR can be employed to calculate the greatest r(r>0) singular values of a given quaternion matrix. Following that, we propose a novel quaternion matrix completion approach based on CQSVD-QQR which combines low-rank and sparse priors of color images. Furthermore, the convergence of the algorithm is analyzed. Our model outperforms those state-of-the-art approaches following experimental results on natural color images and color medical images.
{"title":"Low-rank quaternion matrix completion based on approximate quaternion SVD and sparse regularizer","authors":"Juan Han, Liqiao Yang, Kit Ian Kou, Jifei Miao, Lizhi Liu","doi":"10.1016/j.amc.2024.129230","DOIUrl":"https://doi.org/10.1016/j.amc.2024.129230","url":null,"abstract":"Matrix completion is a challenging problem in computer vision. Recently, quaternion representations of color images have achieved competitive performance in many fields. The information on the coupling between the three channels of the color image is better utilized since the color image is treated as a whole. Due to this, researcher interest in low-rank quaternion matrix completion (LRQMC) algorithms has grown significantly. In contrast to the traditional quaternion matrix completion algorithms that rely on quaternion singular value decomposition (QSVD), we propose a novel method based on quaternion Qatar Riyal decomposition (QQR). First, a novel approach (CQSVD-QQR) to computing an approximation of QSVD based on iterative QQR is put forward, which has lower computational complexity than QSVD. CQSVD-QQR can be employed to calculate the greatest <mml:math altimg=\"si1.svg\"><mml:mi>r</mml:mi><mml:mspace width=\"0.25em\"></mml:mspace><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"badbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> singular values of a given quaternion matrix. Following that, we propose a novel quaternion matrix completion approach based on CQSVD-QQR which combines low-rank and sparse priors of color images. Furthermore, the convergence of the algorithm is analyzed. Our model outperforms those state-of-the-art approaches following experimental results on natural color images and color medical images.","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"11 1","pages":""},"PeriodicalIF":4.0,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129212
Sichen Tang, Aili Fang
In the era of rapid development of the Internet, in order to reflect the evolution process of users' viewpoints on network relations, a Bayesian viewpoint evolution model based on Weibo data mining is proposed by studying the relationship between the viewpoints of the author and those of the forwarders on the Sina Weibo platform. Firstly, Python crawler technology was used to crawl the comments and forwarding data under the Weibo topic “#ChatGPT father says human-level AI is coming soon”. After data preprocessing and sentiment analysis, the user relationship network diagram was drawn with Gephi software. Secondly, the viewpoint evolution model is constructed and the viewpoint update formula based on Bayes rule is used to calculate the users' viewpoint evolution within the network relations of several kinds of different publication centers. The results show that: in the communication of public opinion, the evolution direction of the opinions of the media-centered network relations tends to be more consistent, which indicates the importance of the opinion guidance of the media in the communication of information. The analysis and technology provide a certain reference for the government and the media to control and guide the network public opinion.
{"title":"Analysis of viewpoint evolution based on WeiBo data mining","authors":"Sichen Tang, Aili Fang","doi":"10.1016/j.amc.2024.129212","DOIUrl":"10.1016/j.amc.2024.129212","url":null,"abstract":"<div><div>In the era of rapid development of the Internet, in order to reflect the evolution process of users' viewpoints on network relations, a Bayesian viewpoint evolution model based on Weibo data mining is proposed by studying the relationship between the viewpoints of the author and those of the forwarders on the Sina Weibo platform. Firstly, Python crawler technology was used to crawl the comments and forwarding data under the Weibo topic “#ChatGPT father says human-level AI is coming soon”. After data preprocessing and sentiment analysis, the user relationship network diagram was drawn with Gephi software. Secondly, the viewpoint evolution model is constructed and the viewpoint update formula based on Bayes rule is used to calculate the users' viewpoint evolution within the network relations of several kinds of different publication centers. The results show that: in the communication of public opinion, the evolution direction of the opinions of the media-centered network relations tends to be more consistent, which indicates the importance of the opinion guidance of the media in the communication of information. The analysis and technology provide a certain reference for the government and the media to control and guide the network public opinion.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129212"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142758853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129221
Zidie Zhang, Daiyong Wu, Nishan Li
Recognizing the threshold dynamics of highly developed animals with memory is significant for the governance of species within a specific domain. To investigate how the memory threshold affects population behavior, we formulate a spatially heterogeneous predator-prey system with memory-based diffusion and hunting cooperation on predators. In homogeneous environments, the occurrence conditions of Turing bifurcation and spatially inhomogeneous Hopf bifurcations respectively induced by the memory-based diffusion coefficient and the average memory period at coexistence constant steady states are investigated. Then, in heterogeneous environments, the stability of predator-free steady state is studied by the variational characterization of the principal eigenvalue, and the explicit expression of coexistence steady states is established by the implicit function theorem. In both homogeneous and heterogeneous environments, as the average memory period is beyond the thresholds, the spatially inhomogeneous periodic solutions occur by numerical simulations. Moreover, the increase of cooperative hunting can improve the predation rate of predators, thereby leading to the emergence of periodic solutions. It is worth noting that the introduction of heterogeneous environments results in a transition in spatial patterns from predator-free steady states to spatially inhomogeneous solutions, which biologically indicates that the heterogeneous environments are more conducive to predator invasion than homogeneous ones.
{"title":"Exploring threshold dynamics in a spatially heterogeneous ecosystem with memory-based diffusion and hunting cooperation on predators","authors":"Zidie Zhang, Daiyong Wu, Nishan Li","doi":"10.1016/j.amc.2024.129221","DOIUrl":"10.1016/j.amc.2024.129221","url":null,"abstract":"<div><div>Recognizing the threshold dynamics of highly developed animals with memory is significant for the governance of species within a specific domain. To investigate how the memory threshold affects population behavior, we formulate a spatially heterogeneous predator-prey system with memory-based diffusion and hunting cooperation on predators. In homogeneous environments, the occurrence conditions of Turing bifurcation and spatially inhomogeneous Hopf bifurcations respectively induced by the memory-based diffusion coefficient and the average memory period at coexistence constant steady states are investigated. Then, in heterogeneous environments, the stability of predator-free steady state is studied by the variational characterization of the principal eigenvalue, and the explicit expression of coexistence steady states is established by the implicit function theorem. In both homogeneous and heterogeneous environments, as the average memory period is beyond the thresholds, the spatially inhomogeneous periodic solutions occur by numerical simulations. Moreover, the increase of cooperative hunting can improve the predation rate of predators, thereby leading to the emergence of periodic solutions. It is worth noting that the introduction of heterogeneous environments results in a transition in spatial patterns from predator-free steady states to spatially inhomogeneous solutions, which biologically indicates that the heterogeneous environments are more conducive to predator invasion than homogeneous ones.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129221"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129217
Yu Mu , Wing-Cheong Lo , Yuanshun Tan , Zijian Liu
In the ecosystem, the chase of the predator with cooperation contributes to fear psychology in the prey, resulting in behavioral changes such as a decrease in the birth rate. We construct a spatially diffusive model with delay to investigate the combined perturbation of these factors. Initially, we establish the existence of positive solutions and examine the stability of steady-state solutions under varying conditions. The bifurcation dynamics of the positive solutions have been analyzed. Turing instability, arising from the random diffusion of the species, generates spatially irregular patterns characterized by patchy distribution of prey and predators in the spatial domain. Hopf bifurcation, resulting from the diffusive rate and delay, contributes to spatially periodic solutions where the number of species will spatially oscillate. The combined influence of diffusion and delay results in the emergence of Turing and Hopf bifurcation phenomena. In this case, the combined effect amplifies the spatially heterogeneous distribution of prey and predator. Our results reveal the heterogeneous behaviors of prey and predator under the coupled influence of cooperation hunting and fear effects. In this paper, we will also study a hybrid control scheme for controlling the generations of Turing patterns and Hopf bifurcation. Our theoretical results and numerical simulations demonstrate that the control scheme can mitigate the negative influence of combined factors and promote the species' stability.
{"title":"Hybrid control for the prey in a spatial prey-predator model with cooperative hunting and fear effect time lag","authors":"Yu Mu , Wing-Cheong Lo , Yuanshun Tan , Zijian Liu","doi":"10.1016/j.amc.2024.129217","DOIUrl":"10.1016/j.amc.2024.129217","url":null,"abstract":"<div><div>In the ecosystem, the chase of the predator with cooperation contributes to fear psychology in the prey, resulting in behavioral changes such as a decrease in the birth rate. We construct a spatially diffusive model with delay to investigate the combined perturbation of these factors. Initially, we establish the existence of positive solutions and examine the stability of steady-state solutions under varying conditions. The bifurcation dynamics of the positive solutions have been analyzed. Turing instability, arising from the random diffusion of the species, generates spatially irregular patterns characterized by patchy distribution of prey and predators in the spatial domain. Hopf bifurcation, resulting from the diffusive rate and delay, contributes to spatially periodic solutions where the number of species will spatially oscillate. The combined influence of diffusion and delay results in the emergence of Turing and Hopf bifurcation phenomena. In this case, the combined effect amplifies the spatially heterogeneous distribution of prey and predator. Our results reveal the heterogeneous behaviors of prey and predator under the coupled influence of cooperation hunting and fear effects. In this paper, we will also study a hybrid control scheme for controlling the generations of Turing patterns and Hopf bifurcation. Our theoretical results and numerical simulations demonstrate that the control scheme can mitigate the negative influence of combined factors and promote the species' stability.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129217"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142758850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129218
Danilo Korže , Aleksander Vesel
Let G be a graph and . Vertices are M-visible if there exists a shortest -path of G that does not pass through any vertex of . We say that M is a mutual-visibility set if each pair of vertices of M is M-visible, while the size of any largest mutual-visibility set of G is the mutual-visibility number of G. If some additional combinations for pairs of vertices are required to be M-visible, we obtain the total (every are M-visible), the outer (every and every are M-visible), and the dual (every are M-visible) mutual-visibility set of G. The cardinalities of the largest of the above defined sets are known as the total, the outer, and the dual mutual-visibility number of G, respectively.
We present results on the variety of mutual-visibility problems in hypercubes.
{"title":"Variety of mutual-visibility problems in hypercubes","authors":"Danilo Korže , Aleksander Vesel","doi":"10.1016/j.amc.2024.129218","DOIUrl":"10.1016/j.amc.2024.129218","url":null,"abstract":"<div><div>Let <em>G</em> be a graph and <span><math><mi>M</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Vertices <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>M</mi></math></span> are <em>M</em>-visible if there exists a shortest <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span>-path of <em>G</em> that does not pass through any vertex of <span><math><mi>M</mi><mo>∖</mo><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>}</mo></math></span>. We say that <em>M</em> is a mutual-visibility set if each pair of vertices of <em>M</em> is <em>M</em>-visible, while the size of any largest mutual-visibility set of <em>G</em> is the mutual-visibility number of <em>G</em>. If some additional combinations for pairs of vertices <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> are required to be <em>M</em>-visible, we obtain the total (every <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are <em>M</em>-visible), the outer (every <span><math><mi>x</mi><mo>∈</mo><mi>M</mi></math></span> and every <span><math><mi>y</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∖</mo><mi>M</mi></math></span> are <em>M</em>-visible), and the dual (every <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>∖</mo><mi>M</mi></math></span> are <em>M</em>-visible) mutual-visibility set of <em>G</em>. The cardinalities of the largest of the above defined sets are known as the total, the outer, and the dual mutual-visibility number of <em>G</em>, respectively.</div><div>We present results on the variety of mutual-visibility problems in hypercubes.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129218"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129087
Ming Huang , Si Qi Zhang , Yong Xiu Feng , Jin Long Yuan , Hong Han Bei
This paper introduces a proximal bundle scheme to solve generalized variational inequalities with inexact data. Under optimality conditions, the problem can be equivalently represented as seeking out the zero point of the sum of two multi-valued operators whose domains are the real Hilbert space. The two operators denoted by T and f, respectively, are the monotone operator and the subdifferential of a lower semi-continuous, non-differentiable, convex function. Our approach is based on the principles of the proximal point strategy, which involves incorporating inexact information into the subproblems and approximating them using a series of piecewise linear convex functions. Moreover, we put forward a novel stopping criterion to identify the adequacy of the current approximation. This approach serves to make the subproblems more manageable, and it has been proven that obtaining inexact information can ensure that the linearization error during the iteration process remains non-negative, thus avoiding triggering noise attenuation. Subsequently, we verify the convergence of the algorithm under relatively mild assumptions (the operator T is para-monotone and may be multi-valued). Ultimately, we present the findings of elementary numerical experiments to declare the method's efficacy.
{"title":"A proximal bundle approach for solving the generalized variational inequalities with inexact data","authors":"Ming Huang , Si Qi Zhang , Yong Xiu Feng , Jin Long Yuan , Hong Han Bei","doi":"10.1016/j.amc.2024.129087","DOIUrl":"10.1016/j.amc.2024.129087","url":null,"abstract":"<div><div>This paper introduces a proximal bundle scheme to solve generalized variational inequalities with inexact data. Under optimality conditions, the problem can be equivalently represented as seeking out the zero point of the sum of two multi-valued operators whose domains are the real Hilbert space. The two operators denoted by <em>T</em> and <em>f</em>, respectively, are the monotone operator and the subdifferential of a lower semi-continuous, non-differentiable, convex function. Our approach is based on the principles of the proximal point strategy, which involves incorporating inexact information into the subproblems and approximating them using a series of piecewise linear convex functions. Moreover, we put forward a novel stopping criterion to identify the adequacy of the current approximation. This approach serves to make the subproblems more manageable, and it has been proven that obtaining inexact information can ensure that the linearization error during the iteration process remains non-negative, thus avoiding triggering noise attenuation. Subsequently, we verify the convergence of the algorithm under relatively mild assumptions (the operator <em>T</em> is para-monotone and may be multi-valued). Ultimately, we present the findings of elementary numerical experiments to declare the method's efficacy.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129087"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142758851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129220
Shounan Lu , Yang Wang
Strategy update rules play an important role in repeated Prisoner's Dilemma games. This work proposes a modified strategy update rule based on the traditional Fermi function, in which individual past performance is taken into account in strategy update. Then, the consistency aspiration α serves as a benchmark to measure an individual's past performance, and the past performance score is dynamically adjusted during the evolution process according to the BM reinforcement learning rules. The computational results indicate that the proposed modified strategy update rules can significantly improve system cooperation t than the traditional version, and the network reciprocity effect is enhanced as the result of past performance is coupled into the strategy update rule. Moreover, different temptation to defection b exist a corresponding aspiration α result in maximizing system cooperation. Furthermore, an optimal sensitivity level β can also result in a maximizing system cooperation. As a whole, for α − β phase diagram, different a will correspond to an optimal value β that allows the system to achieve the maximum cooperation. Finally, the proposed mechanism is robust. Hopefully this can help to inspire further research on how to deal with social dilemmas.
{"title":"Past-performance-driven strategy updating promote cooperation in the spatial prisoner's dilemma game","authors":"Shounan Lu , Yang Wang","doi":"10.1016/j.amc.2024.129220","DOIUrl":"10.1016/j.amc.2024.129220","url":null,"abstract":"<div><div>Strategy update rules play an important role in repeated Prisoner's Dilemma games. This work proposes a modified strategy update rule based on the traditional Fermi function, in which individual past performance is taken into account in strategy update. Then, the consistency aspiration α serves as a benchmark to measure an individual's past performance, and the past performance score is dynamically adjusted during the evolution process according to the BM reinforcement learning rules. The computational results indicate that the proposed modified strategy update rules can significantly improve system cooperation t than the traditional version, and the network reciprocity effect is enhanced as the result of past performance is coupled into the strategy update rule. Moreover, different temptation to defection <em>b</em> exist a corresponding aspiration <em>α</em> result in maximizing system cooperation. Furthermore, an optimal sensitivity level β can also result in a maximizing system cooperation. As a whole, for α − β phase diagram, different <em>a</em> will correspond to an optimal value β that allows the system to achieve the maximum cooperation. Finally, the proposed mechanism is robust. Hopefully this can help to inspire further research on how to deal with social dilemmas.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129220"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142758967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-02DOI: 10.1016/j.amc.2024.129219
Yuya Yamakawa, Nobuo Yamashita
This paper presents a regularized Newton method (RNM) with generalized regularization terms for unconstrained convex optimization problems. The generalized regularization includes quadratic, cubic, and elastic net regularizations as special cases. Therefore, the proposed method serves as a general framework that includes not only the classical and cubic RNMs but also a novel RNM with elastic net regularization. We show that the proposed RNM has the global and local superlinear convergence, which are the same as those of the cubic RNM.
{"title":"Convergence analysis of a regularized Newton method with generalized regularization terms for unconstrained convex optimization problems","authors":"Yuya Yamakawa, Nobuo Yamashita","doi":"10.1016/j.amc.2024.129219","DOIUrl":"10.1016/j.amc.2024.129219","url":null,"abstract":"<div><div>This paper presents a regularized Newton method (RNM) with generalized regularization terms for unconstrained convex optimization problems. The generalized regularization includes quadratic, cubic, and elastic net regularizations as special cases. Therefore, the proposed method serves as a general framework that includes not only the classical and cubic RNMs but also a novel RNM with elastic net regularization. We show that the proposed RNM has the global <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span> and local superlinear convergence, which are the same as those of the cubic RNM.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"491 ","pages":"Article 129219"},"PeriodicalIF":3.5,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142758852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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