Pub Date : 2025-04-04DOI: 10.1016/j.amc.2025.129447
Sreepriya P., Denny K.D., G.D. Reddy
Klann and Ramlau [16] hypothesized fractional Tikhonov regularization as an interpolation between generalized inverse and Tikhonov regularization. In fact, fractional schemes can be viewed as a generalization of the Tikhonov scheme. One of the motives of this work is the major pitfall of the a priori parameter choice rule, which primarily relies on source conditions that are often unknown. It necessitates the need for advocating a data-driven approach (a posteriori choice strategy). We briefly overview fractional scheme in learning theory and propose a modified Engl type [9] discrepancy principle, thus integrating supervised learning into the field of inverse problems. In due course of the investigation, we effectively explored the relation between learning from examples and the inverse problems. We demonstrate the regularization properties and establish the convergence rate of this scheme. Finally, the theoretical results are corroborated using two well known examples in learning theory.
{"title":"A class of parameter choice rules for fractional Tikhonov regularization scheme in learning theory","authors":"Sreepriya P., Denny K.D., G.D. Reddy","doi":"10.1016/j.amc.2025.129447","DOIUrl":"10.1016/j.amc.2025.129447","url":null,"abstract":"<div><div>Klann and Ramlau <span><span>[16]</span></span> hypothesized fractional Tikhonov regularization as an interpolation between generalized inverse and Tikhonov regularization. In fact, fractional schemes can be viewed as a generalization of the Tikhonov scheme. One of the motives of this work is the major pitfall of the a priori parameter choice rule, which primarily relies on source conditions that are often unknown. It necessitates the need for advocating a data-driven approach (a posteriori choice strategy). We briefly overview fractional scheme in learning theory and propose a modified Engl type <span><span>[9]</span></span> discrepancy principle, thus integrating supervised learning into the field of inverse problems. In due course of the investigation, we effectively explored the relation between learning from examples and the inverse problems. We demonstrate the regularization properties and establish the convergence rate of this scheme. Finally, the theoretical results are corroborated using two well known examples in learning theory.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129447"},"PeriodicalIF":3.5,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767455","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 : 2025-04-04DOI: 10.1016/j.amc.2025.129454
Ce Liang , Weiyuan Ma , Chenjun Ma , Ling Guo
This paper investigates data-driven learning techniques for fractional chaotic systems (FCS), specifically those utilizing the Caputo derivative. Three machine learning (ML) methods are employed for parameter estimation: feedforward neural networks (FNN), long short-term memory (LSTM), and gated recurrent units (GRU). Optimization problems are formulated, and the well-known algorithms, Backpropagation Through Time (BPTT) and Adam, are employed to train the weights and parameters of the ML models. Systematic numerical testing reveals that LSTM demonstrates superior recognition performance for undisturbed data, while GRU achieves higher accuracy in the presence of disturbances. This study presents a highly accurate approach for solving parameter inverse problems, with the potential for extending these methods to other fractional systems.
{"title":"Harnessing machine learning for identifying parameters in fractional chaotic systems","authors":"Ce Liang , Weiyuan Ma , Chenjun Ma , Ling Guo","doi":"10.1016/j.amc.2025.129454","DOIUrl":"10.1016/j.amc.2025.129454","url":null,"abstract":"<div><div>This paper investigates data-driven learning techniques for fractional chaotic systems (FCS), specifically those utilizing the Caputo derivative. Three machine learning (ML) methods are employed for parameter estimation: feedforward neural networks (FNN), long short-term memory (LSTM), and gated recurrent units (GRU). Optimization problems are formulated, and the well-known algorithms, Backpropagation Through Time (BPTT) and Adam, are employed to train the weights and parameters of the ML models. Systematic numerical testing reveals that LSTM demonstrates superior recognition performance for undisturbed data, while GRU achieves higher accuracy in the presence of disturbances. This study presents a highly accurate approach for solving parameter inverse problems, with the potential for extending these methods to other fractional systems.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129454"},"PeriodicalIF":3.5,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767562","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 : 2025-04-04DOI: 10.1016/j.amc.2025.129442
Zong-Yao Sun , Xing-Xing Wang , Jiao-Jiao Li , Chih-Chiang Chen
The event-triggered adaptive control issue for a class of nonlinearly parameterized higher-order systems with sensor fault is investigated in this article. The difficulty lies in how to replace the actual signal with the sensor fault output signal for control design and how to use adaptive technology to deal with the resulting uncertainties and the inherent unstructured uncertainties of the system at the same time. By introducing subtle state transformations, revamping integral Lyapunov-functions and combining parameter separation technology, we successfully deal with the uncertainties and sensor fault existing in the system by constructing a continuous adaptive event-triggered controller. Different from the existing results, the controller designed in this paper by means of state feedback and event-triggered mechanism ensures all signals are semiglobally uniformly ultimately bounded and the theoretical analysis also shows that Zeno phenomenon does not exist. Finally, the control effect is tested by selecting two simulation examples reasonably.
{"title":"Event-triggered adaptive control for nonlinearly parameterized higher-order systems with sensor fault","authors":"Zong-Yao Sun , Xing-Xing Wang , Jiao-Jiao Li , Chih-Chiang Chen","doi":"10.1016/j.amc.2025.129442","DOIUrl":"10.1016/j.amc.2025.129442","url":null,"abstract":"<div><div>The event-triggered adaptive control issue for a class of nonlinearly parameterized higher-order systems with sensor fault is investigated in this article. The difficulty lies in how to replace the actual signal with the sensor fault output signal for control design and how to use adaptive technology to deal with the resulting uncertainties and the inherent unstructured uncertainties of the system at the same time. By introducing subtle state transformations, revamping integral Lyapunov-functions and combining parameter separation technology, we successfully deal with the uncertainties and sensor fault existing in the system by constructing a continuous adaptive event-triggered controller. Different from the existing results, the controller designed in this paper by means of state feedback and event-triggered mechanism ensures all signals are semiglobally uniformly ultimately bounded and the theoretical analysis also shows that Zeno phenomenon does not exist. Finally, the control effect is tested by selecting two simulation examples reasonably.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129442"},"PeriodicalIF":3.5,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767454","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 : 2025-04-03DOI: 10.1016/j.amc.2025.129452
Jiao Yu
In this paper, the classic Favard inequality in classical calculus is extended to quantum calculus, resulting in the quantum integral form of the Favard-type inequality. Furthermore, the quantum integral form under weighted conditions is considered. As , it degenerates into the classical Favard inequality.
{"title":"Quantum integral Favard-type inequality","authors":"Jiao Yu","doi":"10.1016/j.amc.2025.129452","DOIUrl":"10.1016/j.amc.2025.129452","url":null,"abstract":"<div><div>In this paper, the classic Favard inequality in classical calculus is extended to quantum calculus, resulting in the quantum integral form of the Favard-type inequality. Furthermore, the quantum integral form under weighted conditions is considered. As <span><math><mi>q</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span>, it degenerates into the classical Favard inequality.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129452"},"PeriodicalIF":3.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760212","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 : 2025-04-02DOI: 10.1016/j.amc.2025.129449
Guangbin Ren, Xin Zhao
This article centers around the octonion wavelet transform, exploring its transformation function derived from the admissible octonionic mother wavelet ψ, incorporating translation, rotation, and dilation components. We establish the inverse transform and the Plancherel formula, unveiling the inner product relationship of transformed functions. The Uncertainty Principle for the octonion wavelet transform reveals inherent bounds in wavelet analysis within the octonionic framework. However, it is essential to note that these discoveries are specific to the alternative properties of octonions and cannot be extended to general Cayley-Dickson algebras, where the sedenion wavelet transform lacks the isometry property observed in the octonionic setting.
{"title":"Octonionic wavelet transform and uncertainly principle","authors":"Guangbin Ren, Xin Zhao","doi":"10.1016/j.amc.2025.129449","DOIUrl":"10.1016/j.amc.2025.129449","url":null,"abstract":"<div><div>This article centers around the octonion wavelet transform, exploring its transformation function <span><math><msup><mrow><mi>ψ</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>S</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> derived from the admissible octonionic mother wavelet <em>ψ</em>, incorporating translation, rotation, and dilation components. We establish the inverse transform and the Plancherel formula, unveiling the inner product relationship of transformed functions. The Uncertainty Principle for the octonion wavelet transform reveals inherent bounds in wavelet analysis within the octonionic framework. However, it is essential to note that these discoveries are specific to the alternative properties of octonions and cannot be extended to general Cayley-Dickson algebras, where the sedenion wavelet transform lacks the isometry property observed in the octonionic setting.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129449"},"PeriodicalIF":3.5,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747354","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 : 2025-04-02DOI: 10.1016/j.amc.2025.129417
Aurelian Bejancu, Mohamed Dekhil
For , let , the Euler operator of the quadratic functional where D is the first derivative operator. Given arbitrary values to be interpolated at a finite knot-set, we prove the existence of a unique L-spline interpolant from the natural space of functions f, for which the functional is finite. The natural L-spline interpolant satisfies adjoint differential conditions outside and at the end points of the interval spanned by the knot-set, and it is in fact the unique minimizer of the functional, subject to the interpolation conditions. This extends the approach by Bejancu (2011) for , corresponding to Sobolev spline (or Matérn kernel) interpolation. For , which is the special case of tension splines, our natural L-spline interpolant with adjoint end conditions can be identified as an “-spline interpolant in ” (for , ), previously studied by Le Méhauté and Bouhamidi (1992) via reproducing kernel theory. Our L-spline error analysis, confirmed by numerical tests, is improving on previous convergence results for such tension splines.
{"title":"Univariate interpolation for a class of L-splines with adjoint natural end conditions","authors":"Aurelian Bejancu, Mohamed Dekhil","doi":"10.1016/j.amc.2025.129417","DOIUrl":"10.1016/j.amc.2025.129417","url":null,"abstract":"<div><div>For <span><math><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mi>β</mi></math></span>, let <span><math><mi>L</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, the Euler operator of the quadratic functional<span><span><span><math><munder><mo>∫</mo><mrow><mi>R</mi></mrow></munder><mrow><mo>{</mo><mo>|</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>|</mo><mi>D</mi><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>|</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></mrow><mi>d</mi><mi>t</mi><mo>,</mo></math></span></span></span> where <em>D</em> is the first derivative operator. Given arbitrary values to be interpolated at a finite knot-set, we prove the existence of a unique <em>L</em>-spline interpolant from the natural space of functions <em>f</em>, for which the functional is finite. The natural <em>L</em>-spline interpolant satisfies adjoint differential conditions outside and at the end points of the interval spanned by the knot-set, and it is in fact the unique minimizer of the functional, subject to the interpolation conditions. This extends the approach by Bejancu (2011) for <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo>=</mo><mi>β</mi></math></span>, corresponding to Sobolev spline (or Matérn kernel) interpolation. For <span><math><mn>0</mn><mo>=</mo><mi>α</mi><mo><</mo><mi>β</mi></math></span>, which is the special case of tension splines, our natural <em>L</em>-spline interpolant with adjoint end conditions can be identified as an “<span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>l</mi><mo>,</mo><mi>s</mi></mrow></msup></math></span>-spline interpolant in <span><math><mi>R</mi></math></span>” (for <span><math><mi>m</mi><mo>=</mo><mi>l</mi><mo>=</mo><mn>1</mn></math></span>, <span><math><mi>s</mi><mo>=</mo><mn>0</mn></math></span>), previously studied by Le Méhauté and Bouhamidi (1992) via reproducing kernel theory. Our <em>L</em>-spline error analysis, confirmed by numerical tests, is improving on previous convergence results for such tension splines.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129417"},"PeriodicalIF":3.5,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747273","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 : 2025-04-02DOI: 10.1016/j.amc.2025.129448
Wenwen Zhang, Pingrun Li
In this paper, our purpose is to obtain the general solutions of several kinds of Volterra singular integral equations (VSIEs) in the class of differentiable functions. By constructing some operators and using the properties of integral transforms and conformal mappings, we transform VSIEs in the class of differentiable functions into the Riemann-Hilbert problems with discontinuity on a circle. By means of the principle of analytic continuation and Sokhotski-Plemelj formula, we obtain solutions of Riemann-Hilbert problems in the case of non-normal type, and further discuss the asymptotic properties of the solutions at the nodes.
{"title":"Existence of solutions for Volterra singular integral equations in the class of differentiable functions","authors":"Wenwen Zhang, Pingrun Li","doi":"10.1016/j.amc.2025.129448","DOIUrl":"10.1016/j.amc.2025.129448","url":null,"abstract":"<div><div>In this paper, our purpose is to obtain the general solutions of several kinds of Volterra singular integral equations (VSIEs) in the class of differentiable functions. By constructing some operators and using the properties of integral transforms and conformal mappings, we transform VSIEs in the class of differentiable functions into the Riemann-Hilbert problems with discontinuity on a circle. By means of the principle of analytic continuation and Sokhotski-Plemelj formula, we obtain solutions of Riemann-Hilbert problems in the case of non-normal type, and further discuss the asymptotic properties of the solutions at the nodes.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129448"},"PeriodicalIF":3.5,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747463","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}
In this paper, we present a Nyström-type method for the numerical solution of a class of nonlinear highly oscillatory Volterra integral equations with a trigonometric kernel. The implementation of this method leads to a nonlinear system that involves oscillatory integrals, which is then addressed using a two-point generalized quadrature rule to construct a fully discretized scheme. The error analysis of the method, in terms of both frequency and step length, is also presented. It is demonstrated that the proposed method outperforms the one recently introduced in the literature. To validate the method, several numerical examples are provided, confirming its efficiency and accuracy.
{"title":"Convergence analysis of a Nyström-type method for a class of nonlinear integral equations with highly oscillatory kernels","authors":"Qusay Abdulraheem Kassid, Saeed Sohrabi, Hamid Ranjbar","doi":"10.1016/j.amc.2025.129450","DOIUrl":"10.1016/j.amc.2025.129450","url":null,"abstract":"<div><div>In this paper, we present a Nyström-type method for the numerical solution of a class of nonlinear highly oscillatory Volterra integral equations with a trigonometric kernel. The implementation of this method leads to a nonlinear system that involves oscillatory integrals, which is then addressed using a two-point generalized quadrature rule to construct a fully discretized scheme. The error analysis of the method, in terms of both frequency and step length, is also presented. It is demonstrated that the proposed method outperforms the one recently introduced in the literature. To validate the method, several numerical examples are provided, confirming its efficiency and accuracy.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129450"},"PeriodicalIF":3.5,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747462","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}
<div><div>The second smallest eigenvalue and the largest eigenvalue of the Laplacian matrix of a simple undirected connected graph <em>G</em> are called the algebraic connectivity <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the Laplacian spectral radius <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, respectively. For a first-order periodically sampled consensus protocol multi-agent system (MAS), whose interaction topology can be modeled as a graph <em>G</em>, a larger <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> results in a faster consensus convergence rate, while a smaller <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> contributes to a longer sampling period of the system. Adjusting the weights of the edges is an efficient approach to optimize the interaction topology of a MAS, which improves the consensus convergence rate and prolongs the sampling period. If <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> increases, then the weight of one edge <span><math><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>}</mo></math></span> increases, i.e., the increment <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>></mo><mn>0</mn></math></span>, and the entries of its eigenvector with respect to <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> are not equal. If <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> decreases, then the weight of one edge <span><math><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>}</mo></math></span> decreases, i.e., the increment <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo><</mo><mn>0</mn></math></span>, and the entries of its eigenvector with respect to <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> are not equal. Moreover, when considering adjusting the weights of edges, some necessary conditions for increasing <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and decreasing <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are also given respectively, both of which a
{"title":"Interaction topology optimization by adjustment of edge weights to improve the consensus convergence and prolong the sampling period for a multi-agent system","authors":"Tongyou Xu , Ying-Ying Tan , Shanshan Gao , Xuejuan Zhan","doi":"10.1016/j.amc.2025.129428","DOIUrl":"10.1016/j.amc.2025.129428","url":null,"abstract":"<div><div>The second smallest eigenvalue and the largest eigenvalue of the Laplacian matrix of a simple undirected connected graph <em>G</em> are called the algebraic connectivity <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the Laplacian spectral radius <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, respectively. For a first-order periodically sampled consensus protocol multi-agent system (MAS), whose interaction topology can be modeled as a graph <em>G</em>, a larger <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> results in a faster consensus convergence rate, while a smaller <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> contributes to a longer sampling period of the system. Adjusting the weights of the edges is an efficient approach to optimize the interaction topology of a MAS, which improves the consensus convergence rate and prolongs the sampling period. If <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> increases, then the weight of one edge <span><math><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>}</mo></math></span> increases, i.e., the increment <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>></mo><mn>0</mn></math></span>, and the entries of its eigenvector with respect to <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> are not equal. If <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> decreases, then the weight of one edge <span><math><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>}</mo></math></span> decreases, i.e., the increment <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo><</mo><mn>0</mn></math></span>, and the entries of its eigenvector with respect to <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> are not equal. Moreover, when considering adjusting the weights of edges, some necessary conditions for increasing <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and decreasing <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are also given respectively, both of which a","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129428"},"PeriodicalIF":3.5,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738709","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 : 2025-04-01DOI: 10.1016/j.amc.2025.129439
Jing Ye , Jiani Zou , Miaomiao Han
A graph G is called weakly f-degenerate with respect to a function f from to the non-negative integers, if every vertex of G can be successively removed through a series of valid Delete and DeleteSave operations. The weak degeneracy is defined as the smallest integer d for which G is weakly d-degenerate, where d is a constant function. It was demonstrated that one plus the weak degeneracy can act as an upper bound for list-chromatic number and DP-chromatic number. Let if , and if . In this paper, we prove that for every -minor free graph G, , which implies that is -choosable and -DP-colorable. This work generalizes the result obtained by Lih et al. in [Discrete Mathematics, 269 (2003), 303-309] and Hetherington et al. in [Discrete Mathematics, 308 (2008), 4037-4043].
{"title":"Weak degeneracy of the square of K4-minor free graphs","authors":"Jing Ye , Jiani Zou , Miaomiao Han","doi":"10.1016/j.amc.2025.129439","DOIUrl":"10.1016/j.amc.2025.129439","url":null,"abstract":"<div><div>A graph <em>G</em> is called weakly <em>f</em>-degenerate with respect to a function <em>f</em> from <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> to the non-negative integers, if every vertex of <em>G</em> can be successively removed through a series of valid Delete and DeleteSave operations. The weak degeneracy <span><math><mi>w</mi><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is defined as the smallest integer <em>d</em> for which <em>G</em> is weakly <em>d</em>-degenerate, where <em>d</em> is a constant function. It was demonstrated that one plus the weak degeneracy can act as an upper bound for list-chromatic number and DP-chromatic number. Let <span><math><mi>κ</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn></math></span> if <span><math><mn>2</mn><mo>≤</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>3</mn></math></span>, and <span><math><mi>κ</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>=</mo><mo>⌊</mo><mfrac><mrow><mn>3</mn><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></math></span> if <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>4</mn></math></span>. In this paper, we prove that for every <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-minor free graph <em>G</em>, <span><math><mi>w</mi><mi>d</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>≤</mo><mi>κ</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, which implies that <span><math><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is <span><math><mo>(</mo><mi>κ</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-choosable and <span><math><mo>(</mo><mi>κ</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-DP-colorable. This work generalizes the result obtained by Lih et al. in [Discrete Mathematics, 269 (2003), 303-309] and Hetherington et al. in [Discrete Mathematics, 308 (2008), 4037-4043].</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129439"},"PeriodicalIF":3.5,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738712","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}
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