One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a2,a3, and a4, and for three second-order and third-order Hankel determinants, H2,1X,H2,2X, and H3,1X. The optimality of each obtained estimate is given as well.
{"title":"Analytic Functions Related to a Balloon-Shaped Domain","authors":"Adeel Ahmad, Jianhua Gong, Isra Al-shbeil, A. Rasheed, Asad Ali, Saqib Hussain","doi":"10.3390/fractalfract7120865","DOIUrl":"https://doi.org/10.3390/fractalfract7120865","url":null,"abstract":"One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a2,a3, and a4, and for three second-order and third-order Hankel determinants, H2,1X,H2,2X, and H3,1X. The optimality of each obtained estimate is given as well.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"52 1","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138600757","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 : 2023-12-04DOI: 10.3390/fractalfract7120861
Amr Abou-Senna, Ghada AlNemer, Yongchun Zhou, Boping Tian
This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in Lp(p≥2) sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in Lp(0
{"title":"Convergence Rate of the Diffused Split-Step Truncated Euler–Maruyama Method for Stochastic Pantograph Models with Lévy Leaps","authors":"Amr Abou-Senna, Ghada AlNemer, Yongchun Zhou, Boping Tian","doi":"10.3390/fractalfract7120861","DOIUrl":"https://doi.org/10.3390/fractalfract7120861","url":null,"abstract":"This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in Lp(p≥2) sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in Lp(0<p<2) sense is addressed where all the three coefficients grow beyond linearly. Finally, theoretical findings are manifested via some numerical examples.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"32 50","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138601575","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 : 2023-12-04DOI: 10.3390/fractalfract7120862
Ke Sun, Zhiyao Ma, Guowei Dong, Ping Gong
In this work, an adaptive fuzzy backstepping fault-tolerant control (FTC) issue is tackled for uncertain fractional-order (FO) nonlinear systems with sensor and actuator faults. A fuzzy logic system is exploited to manage unknown nonlinearity. In addition, a novel FO nonlinear filter-based dynamic surface control (DSC) method is constructed, effectively avoiding the inherent complexity explosion problem in the backstepping recursive process, and in the light of the construction of auxiliary functions, compensating the coupling term introduced by faults. On account of certain assumptions, the stability criterion of the FO Lyapunov function is applied to guarantee the stability of the closed-loop system. Finally, the simulation example verifies the validity of the presented control strategy.
{"title":"Adaptive Fuzzy Fault-Tolerant Control of Uncertain Fractional-Order Nonlinear Systems with Sensor and Actuator Faults","authors":"Ke Sun, Zhiyao Ma, Guowei Dong, Ping Gong","doi":"10.3390/fractalfract7120862","DOIUrl":"https://doi.org/10.3390/fractalfract7120862","url":null,"abstract":"In this work, an adaptive fuzzy backstepping fault-tolerant control (FTC) issue is tackled for uncertain fractional-order (FO) nonlinear systems with sensor and actuator faults. A fuzzy logic system is exploited to manage unknown nonlinearity. In addition, a novel FO nonlinear filter-based dynamic surface control (DSC) method is constructed, effectively avoiding the inherent complexity explosion problem in the backstepping recursive process, and in the light of the construction of auxiliary functions, compensating the coupling term introduced by faults. On account of certain assumptions, the stability criterion of the FO Lyapunov function is applied to guarantee the stability of the closed-loop system. Finally, the simulation example verifies the validity of the presented control strategy.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"81 7","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138604609","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 : 2023-12-02DOI: 10.3390/fractalfract7120860
T. Chiheb, B. Meftah, Abdelkader Moumen, Mohamed Bouye
In this manuscript, by using a new identity, we establish some new Maclaurin-type inequalities for functions whose modulus of the first derivatives are GA-convex functions via Hadamard fractional integrals.
{"title":"Maclaurin-Type Integral Inequalities for GA-Convex Functions Involving Confluent Hypergeometric Function via Hadamard Fractional Integrals","authors":"T. Chiheb, B. Meftah, Abdelkader Moumen, Mohamed Bouye","doi":"10.3390/fractalfract7120860","DOIUrl":"https://doi.org/10.3390/fractalfract7120860","url":null,"abstract":"In this manuscript, by using a new identity, we establish some new Maclaurin-type inequalities for functions whose modulus of the first derivatives are GA-convex functions via Hadamard fractional integrals.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"29 9","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138606459","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 : 2023-11-14DOI: 10.3390/fractalfract7110820
Jiajie Cheng, Qiunan Chen, Xiaocheng Huang
The segmentation of crack detection and severity assessment in low-light environments presents a formidable challenge. To address this, we propose a novel dual encoder structure, denoted as DSD-Net, which integrates fast Fourier transform with a convolutional neural network. In this framework, we incorporate an information extraction module and an attention feature fusion module to effectively capture contextual global information and extract pertinent local features. Furthermore, we introduce a fractal dimension estimation method into the network, seamlessly integrated as an end-to-end task, augmenting the proficiency of professionals in detecting crack pathology within low-light settings. Subsequently, we curate a specialized dataset comprising instances of crack pathology in low-light conditions to facilitate the training and evaluation of the DSD-Net algorithm. Comparative experimentation attests to the commendable performance of DSD-Net in low-light environments, exhibiting superlative precision (88.5%), recall (85.3%), and F1 score (86.9%) in the detection task. Notably, DSD-Net exhibits a diminutive Model Size (35.3 MB) and elevated Frame Per Second (80.4 f/s), thereby endowing it with the potential to be seamlessly integrated into edge detection devices, thus amplifying its practical utility.
{"title":"An Algorithm for Crack Detection, Segmentation, and Fractal Dimension Estimation in Low-Light Environments by Fusing FFT and Convolutional Neural Network","authors":"Jiajie Cheng, Qiunan Chen, Xiaocheng Huang","doi":"10.3390/fractalfract7110820","DOIUrl":"https://doi.org/10.3390/fractalfract7110820","url":null,"abstract":"The segmentation of crack detection and severity assessment in low-light environments presents a formidable challenge. To address this, we propose a novel dual encoder structure, denoted as DSD-Net, which integrates fast Fourier transform with a convolutional neural network. In this framework, we incorporate an information extraction module and an attention feature fusion module to effectively capture contextual global information and extract pertinent local features. Furthermore, we introduce a fractal dimension estimation method into the network, seamlessly integrated as an end-to-end task, augmenting the proficiency of professionals in detecting crack pathology within low-light settings. Subsequently, we curate a specialized dataset comprising instances of crack pathology in low-light conditions to facilitate the training and evaluation of the DSD-Net algorithm. Comparative experimentation attests to the commendable performance of DSD-Net in low-light environments, exhibiting superlative precision (88.5%), recall (85.3%), and F1 score (86.9%) in the detection task. Notably, DSD-Net exhibits a diminutive Model Size (35.3 MB) and elevated Frame Per Second (80.4 f/s), thereby endowing it with the potential to be seamlessly integrated into edge detection devices, thus amplifying its practical utility.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"54 34","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902629","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 : 2023-11-13DOI: 10.3390/fractalfract7110819
Imtithal Alzughaibi, Mourad Ben Slimane, Obaid Algahtani
We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on Rd, with d≥2. Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the new spaces take into account the geometric disposition of the hyperbolic wavelet coefficients at each scale (j1,⋯,jd), and are therefore suitable for a multifractal analysis of rectangular regularity. We prove that hyperbolic oscillation spaces are closely related to hyperbolic variation spaces, and consequently do not almost depend on the chosen hyperbolic wavelet basis. Therefore, the so-called rectangular multifractal analysis, related to hyperbolic oscillation spaces, is somehow ‘robust’, i.e., does not change if the analyzing wavelets were changed. We also study optimal relationships between hyperbolic and mixed fractional lifting oscillation spaces and Besov spaces with dominating mixed smoothness. In particular, we show that, for some indices, hyperbolic and mixed fractional lifting oscillation spaces are not always sharply imbedded between Besov spaces or fractional Sobolev spaces with dominating mixed smoothness, and thus are new spaces of a really different nature.
{"title":"On Mixed Fractional Lifting Oscillation Spaces","authors":"Imtithal Alzughaibi, Mourad Ben Slimane, Obaid Algahtani","doi":"10.3390/fractalfract7110819","DOIUrl":"https://doi.org/10.3390/fractalfract7110819","url":null,"abstract":"We introduce hyperbolic oscillation spaces and mixed fractional lifting oscillation spaces expressed in terms of hyperbolic wavelet leaders of multivariate signals on Rd, with d≥2. Contrary to Besov spaces and fractional Sobolev spaces with dominating mixed smoothness, the new spaces take into account the geometric disposition of the hyperbolic wavelet coefficients at each scale (j1,⋯,jd), and are therefore suitable for a multifractal analysis of rectangular regularity. We prove that hyperbolic oscillation spaces are closely related to hyperbolic variation spaces, and consequently do not almost depend on the chosen hyperbolic wavelet basis. Therefore, the so-called rectangular multifractal analysis, related to hyperbolic oscillation spaces, is somehow ‘robust’, i.e., does not change if the analyzing wavelets were changed. We also study optimal relationships between hyperbolic and mixed fractional lifting oscillation spaces and Besov spaces with dominating mixed smoothness. In particular, we show that, for some indices, hyperbolic and mixed fractional lifting oscillation spaces are not always sharply imbedded between Besov spaces or fractional Sobolev spaces with dominating mixed smoothness, and thus are new spaces of a really different nature.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"140 41","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136352190","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 : 2023-11-13DOI: 10.3390/fractalfract7110818
Mengjiao Wang, Jiwei Peng, Shaobo He, Xinan Zhang, Herbert Ho-Ching Iu
Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D Hindmarsh–Rose (HR) neuron are coupled through a locally active memristor. The simulation results show that the network exhibits complex dynamic behavior and is different from the usual phase synchronization. More specifically, the membrane potential of the 2D HR neuron exhibits five stable firing modes as the coupling parameter k1 changes. In addition, it is found that in the local region of k1, the number of spikes in bursting firing increases with the increase in k1. More interestingly, the network gradually changes from synchronous to asynchronous during the increase in the coupling parameter k1 but suddenly becomes synchronous around the coupling parameter k1 = 1.96. As far as we know, this abnormal synchronization behavior is different from the existing findings. This research is inspired by the fact that the episodic synchronous abnormal firing of excitatory neurons in the hippocampus of the brain can lead to diseases such as epilepsy. This helps us further understand the mechanism of brain activity and build bionic systems. Finally, we design the simulation circuit of the network and implement it on an STM32 microcontroller.
{"title":"Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network","authors":"Mengjiao Wang, Jiwei Peng, Shaobo He, Xinan Zhang, Herbert Ho-Ching Iu","doi":"10.3390/fractalfract7110818","DOIUrl":"https://doi.org/10.3390/fractalfract7110818","url":null,"abstract":"Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D Hindmarsh–Rose (HR) neuron are coupled through a locally active memristor. The simulation results show that the network exhibits complex dynamic behavior and is different from the usual phase synchronization. More specifically, the membrane potential of the 2D HR neuron exhibits five stable firing modes as the coupling parameter k1 changes. In addition, it is found that in the local region of k1, the number of spikes in bursting firing increases with the increase in k1. More interestingly, the network gradually changes from synchronous to asynchronous during the increase in the coupling parameter k1 but suddenly becomes synchronous around the coupling parameter k1 = 1.96. As far as we know, this abnormal synchronization behavior is different from the existing findings. This research is inspired by the fact that the episodic synchronous abnormal firing of excitatory neurons in the hippocampus of the brain can lead to diseases such as epilepsy. This helps us further understand the mechanism of brain activity and build bionic systems. Finally, we design the simulation circuit of the network and implement it on an STM32 microcontroller.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136347177","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 : 2023-11-12DOI: 10.3390/fractalfract7110817
Afrah Ahmad Noman Abdou
This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results.
{"title":"Solving a Nonlinear Fractional Differential Equation Using Fixed Point Results in Orthogonal Metric Spaces","authors":"Afrah Ahmad Noman Abdou","doi":"10.3390/fractalfract7110817","DOIUrl":"https://doi.org/10.3390/fractalfract7110817","url":null,"abstract":"This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"3 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135038854","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 : 2023-11-11DOI: 10.3390/fractalfract7110816
Alexandru Tudorache, Rodica Luca
We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative.
{"title":"Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions","authors":"Alexandru Tudorache, Rodica Luca","doi":"10.3390/fractalfract7110816","DOIUrl":"https://doi.org/10.3390/fractalfract7110816","url":null,"abstract":"We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"37 24","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135086591","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 : 2023-11-11DOI: 10.3390/fractalfract7110815
Wensheng Wang
Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.
{"title":"Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient","authors":"Wensheng Wang","doi":"10.3390/fractalfract7110815","DOIUrl":"https://doi.org/10.3390/fractalfract7110815","url":null,"abstract":"Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension dimp(B). On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"37 21","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135086594","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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