Pub Date : 2023-08-18DOI: 10.1142/s0218348x23500950
Kang-le Wang
The Klein–Gordon–Zakharov equation is an important and interesting model in physics. A fractional Klein–Gordon–Zakharov model is described by employing beta-derivative. Some new solitary wave solutions are acquired by utilizing the fractional rational [Formula: see text]–[Formula: see text] method and fractional [Formula: see text] method. Some 3D graphs are depicted to elaborate these new solitary wave solutions. The work is very helpful to study other related types of fractional evolution equations.
{"title":"NOVEL APPROACHES TO FRACTIONAL KLEIN–GORDON–ZAKHAROV EQUATION","authors":"Kang-le Wang","doi":"10.1142/s0218348x23500950","DOIUrl":"https://doi.org/10.1142/s0218348x23500950","url":null,"abstract":"The Klein–Gordon–Zakharov equation is an important and interesting model in physics. A fractional Klein–Gordon–Zakharov model is described by employing beta-derivative. Some new solitary wave solutions are acquired by utilizing the fractional rational [Formula: see text]–[Formula: see text] method and fractional [Formula: see text] method. Some 3D graphs are depicted to elaborate these new solitary wave solutions. The work is very helpful to study other related types of fractional evolution equations.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47701816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-16DOI: 10.1142/s0218348x23020048
Boqi Xiao, Ali Akgül, D. Shou, Gongbo Long
{"title":"PREFACE: SPECIAL ISSUE ON “ANALYSIS AND MODELING OF HEAT AND MASS TRANSFER IN FRACTAL POROUS MEDIA”","authors":"Boqi Xiao, Ali Akgül, D. Shou, Gongbo Long","doi":"10.1142/s0218348x23020048","DOIUrl":"https://doi.org/10.1142/s0218348x23020048","url":null,"abstract":"","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46122270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-10DOI: 10.1142/s0218348x23500779
Yuke Huang, Cheng Zeng, Yumei Xue
This paper studies the average trapping time of honeypots on some evolving networks. We propose a simple algorithmic framework for generating networks with Sturmian structure. From the balance property and the recurrence property of Sturmian words, we estimate the average trapping time of our proposed networks with an asymptotic expression [Formula: see text], where [Formula: see text] is a bounded expression related to word [Formula: see text]. We next consider networks with multi-honeypots and generalize our basic models. Additionally, we give an symmetrical method to create a series of networks with the Sturmian structure, and the average trapping time satisfies [Formula: see text], which is independent of any word [Formula: see text]. The generalized methods may have some illuminating effects on the study of networks with randomness.
本文研究了一些进化网络中蜜罐的平均捕获时间。我们提出了一个简单的算法框架来生成具有Sturmian结构的网络。根据Sturmian词的平衡性质和递归性质，我们用渐近表达式[Formula: see text]估计我们所提出的网络的平均捕获时间，其中[Formula: see text]是与词[Formula: see text]相关的有界表达式。接下来，我们考虑具有多个蜜罐的网络，并推广我们的基本模型。此外，我们给出了一种对称的方法来创建一系列具有Sturmian结构的网络，并且平均捕获时间满足[公式:见文]，它独立于任何单词[公式:见文]。这些方法对随机网络的研究具有一定的启发性。
{"title":"TRAPPING PROBLEM OF HONEYPOTS ON FRACTAL NETWORKS WITH THE STURMIAN STRUCTURE","authors":"Yuke Huang, Cheng Zeng, Yumei Xue","doi":"10.1142/s0218348x23500779","DOIUrl":"https://doi.org/10.1142/s0218348x23500779","url":null,"abstract":"This paper studies the average trapping time of honeypots on some evolving networks. We propose a simple algorithmic framework for generating networks with Sturmian structure. From the balance property and the recurrence property of Sturmian words, we estimate the average trapping time of our proposed networks with an asymptotic expression [Formula: see text], where [Formula: see text] is a bounded expression related to word [Formula: see text]. We next consider networks with multi-honeypots and generalize our basic models. Additionally, we give an symmetrical method to create a series of networks with the Sturmian structure, and the average trapping time satisfies [Formula: see text], which is independent of any word [Formula: see text]. The generalized methods may have some illuminating effects on the study of networks with randomness.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43295483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-10DOI: 10.1142/s0218348x23500834
YA-JIE Li, Zhiqiang Wu, Yongtao Sun, Y. Hao, X. Zhang, Feng Wang, Heqing Shi
The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic [Formula: see text]-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.
{"title":"STOCHASTIC STABILITY AND PARAMETRIC CONTROL IN A GENERALIZED AND TRI-STABLE VAN DER POL SYSTEM WITH FRACTIONAL ELEMENT DRIVEN BY MULTIPLICATIVE NOISE","authors":"YA-JIE Li, Zhiqiang Wu, Yongtao Sun, Y. Hao, X. Zhang, Feng Wang, Heqing Shi","doi":"10.1142/s0218348x23500834","DOIUrl":"https://doi.org/10.1142/s0218348x23500834","url":null,"abstract":"The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic [Formula: see text]-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44021076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Electrical conductivity is an important physical property of porous media, and has great significance to rock physics and reservoir engineering. In this work, a conductivity model including pore water conductivity and surface conductivity is derived for water-saturated tree-like branching network. In addition, combined with Archie’s law, a general analytical formula for the formation factor is presented. Through the numerical simulation of the analytical formula above, we discuss the impact of some structural parameters ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in tree-like branching network on the resistance, conductivity and formation factor. The results show that the total resistance [Formula: see text] is proportional to [Formula: see text], [Formula: see text], and inversely proportional to [Formula: see text], [Formula: see text]. The relation between conductivity and porosity in this model is contrasted with previous models and experimental data, and the results show considerable consistency at lower porosity. It is worth noting that when [Formula: see text], the conductivity and porosity curve of this model overlap exactly with those plotted by the parallel model. The fractal conductance model proposed in this work reveals the operation of the current in the tree-like branching network more comprehensively.
{"title":"A FRACTAL ELECTRICAL CONDUCTIVITY MODEL FOR WATER-SATURATED TREE-LIKE BRANCHING NETWORK","authors":"Huaizhi Zhu, Boqi Xiao, Yidan Zhang, Huan Zhou, Shaofu Li, Yanbin Wang, Gongbo Long","doi":"10.1142/s0218348x23500755","DOIUrl":"https://doi.org/10.1142/s0218348x23500755","url":null,"abstract":"Electrical conductivity is an important physical property of porous media, and has great significance to rock physics and reservoir engineering. In this work, a conductivity model including pore water conductivity and surface conductivity is derived for water-saturated tree-like branching network. In addition, combined with Archie’s law, a general analytical formula for the formation factor is presented. Through the numerical simulation of the analytical formula above, we discuss the impact of some structural parameters ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in tree-like branching network on the resistance, conductivity and formation factor. The results show that the total resistance [Formula: see text] is proportional to [Formula: see text], [Formula: see text], and inversely proportional to [Formula: see text], [Formula: see text]. The relation between conductivity and porosity in this model is contrasted with previous models and experimental data, and the results show considerable consistency at lower porosity. It is worth noting that when [Formula: see text], the conductivity and porosity curve of this model overlap exactly with those plotted by the parallel model. The fractal conductance model proposed in this work reveals the operation of the current in the tree-like branching network more comprehensively.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49578824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-09DOI: 10.1142/s0218348x23500780
Zihan Yu, Z. Li, Yong Deng
Among all probability distributions, power law distribution is an intriguing one, which has been studied by many researchers. However, the derivation of power law distribution is still an inconclusive topic. For deriving a distribution, there are various methods, among which maximum entropy principle is a special one. Entropy of random permutation set (RPS), as an uncertainty measure of RPS, is a newly proposed entropy with special features. Deriving power law distribution with maximum entropy of RPS is a promising method. In this paper, certain constraints are given to constrain the entropy of RPS. Power law distribution is able to be finally derived with maximum entropy principle. Numerical experiments are done to show characters of proposed derivation.
{"title":"POWER LAW DISTRIBUTION BASED ON MAXIMUM ENTROPY OF RANDOM PERMUTATION SET","authors":"Zihan Yu, Z. Li, Yong Deng","doi":"10.1142/s0218348x23500780","DOIUrl":"https://doi.org/10.1142/s0218348x23500780","url":null,"abstract":"Among all probability distributions, power law distribution is an intriguing one, which has been studied by many researchers. However, the derivation of power law distribution is still an inconclusive topic. For deriving a distribution, there are various methods, among which maximum entropy principle is a special one. Entropy of random permutation set (RPS), as an uncertainty measure of RPS, is a newly proposed entropy with special features. Deriving power law distribution with maximum entropy of RPS is a promising method. In this paper, certain constraints are given to constrain the entropy of RPS. Power law distribution is able to be finally derived with maximum entropy principle. Numerical experiments are done to show characters of proposed derivation.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41677801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-09DOI: 10.1142/s0218348x23500767
Huan Pan, Zhengyu Liang, Jian Lu, Kai Tu, Ning Xie
Image denoising has been a fundamental problem in the field of image processing. In this paper, we tackle removing impulse noise by combining the fractal image coding and the nonlocal self-similarity priors to recover image. The model undergoes a two-stage process. In the first phase, the identification and labeling of pixels likely to be corrupted by salt-and-pepper noise are carried out. In the second phase, image denoising is performed by solving a constrained convex optimization problem that involves an objective functional composed of three terms: a data fidelity term to measure the similarity between the underlying and observed images, a regularization term to represent the low-rank property of a matrix formed by nonlocal patches of the underlying image, and a quadratic term to measure the closeness of the underlying image to a fractal image. To solve the resulting problem, a combination of proximity algorithms and the weighted singular value thresholding operator is utilized. The numerical results demonstrate an improvement in the structural similarity (SSIM) index and peak signal-to-noise ratio.
{"title":"NONLOCAL LOW RANK REGULARIZATION METHOD FOR FRACTAL IMAGE CODING UNDER SALT-AND-PEPPER NOISE","authors":"Huan Pan, Zhengyu Liang, Jian Lu, Kai Tu, Ning Xie","doi":"10.1142/s0218348x23500767","DOIUrl":"https://doi.org/10.1142/s0218348x23500767","url":null,"abstract":"Image denoising has been a fundamental problem in the field of image processing. In this paper, we tackle removing impulse noise by combining the fractal image coding and the nonlocal self-similarity priors to recover image. The model undergoes a two-stage process. In the first phase, the identification and labeling of pixels likely to be corrupted by salt-and-pepper noise are carried out. In the second phase, image denoising is performed by solving a constrained convex optimization problem that involves an objective functional composed of three terms: a data fidelity term to measure the similarity between the underlying and observed images, a regularization term to represent the low-rank property of a matrix formed by nonlocal patches of the underlying image, and a quadratic term to measure the closeness of the underlying image to a fractal image. To solve the resulting problem, a combination of proximity algorithms and the weighted singular value thresholding operator is utilized. The numerical results demonstrate an improvement in the structural similarity (SSIM) index and peak signal-to-noise ratio.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41674056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-09DOI: 10.1142/s0218348x23500846
Kangkang Wang, Peng Xu
A fractal modification of the modified KdV–Zakharov–Kuznetsov equation is suggested and its fractal generalized variational structure is established by means of the semi-inverse method. Furthermore, the obtained fractal generalized variational structure is discussed and verified through the two-scale transform from another dimension field in detail. The obtained fractal generalized variational structure reveals the conservation laws via the energy form in the fractal space and can be employed to study the fractal solitary wave properties.
{"title":"GENERALIZED VARIATIONAL STRUCTURE OF THE FRACTAL MODIFIED KDV–ZAKHAROV–KUZNETSOV EQUATION","authors":"Kangkang Wang, Peng Xu","doi":"10.1142/s0218348x23500846","DOIUrl":"https://doi.org/10.1142/s0218348x23500846","url":null,"abstract":"A fractal modification of the modified KdV–Zakharov–Kuznetsov equation is suggested and its fractal generalized variational structure is established by means of the semi-inverse method. Furthermore, the obtained fractal generalized variational structure is discussed and verified through the two-scale transform from another dimension field in detail. The obtained fractal generalized variational structure reveals the conservation laws via the energy form in the fractal space and can be employed to study the fractal solitary wave properties.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46542606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1142/s0218348x23500998
Jin Chen, Xinyu Wang
{"title":"Generalized Cantor-integers and interval density of homogeneous Cantor sets","authors":"Jin Chen, Xinyu Wang","doi":"10.1142/s0218348x23500998","DOIUrl":"https://doi.org/10.1142/s0218348x23500998","url":null,"abstract":"","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47847209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1142/s0218348x23501013
Keqin Cui, Wenjia Ma, Lifeng Xi
{"title":"Exact Formula of Distance Sums on Sierpinski Skeleton Networks","authors":"Keqin Cui, Wenjia Ma, Lifeng Xi","doi":"10.1142/s0218348x23501013","DOIUrl":"https://doi.org/10.1142/s0218348x23501013","url":null,"abstract":"","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":" ","pages":""},"PeriodicalIF":4.7,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45418219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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