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With the accelerated construction of 5G and IoT, more and more 5G base stations are erected. However, with the increase of 5G base stations, the power management of 5G base stations becomes progressively a bottleneck. In this paper, we solve the problem of 5G base station power management by designing a 5G base station lithium battery cloud monitoring system. In this paper, first, the lithium battery acquisition hardware is designed. Second, a new communication protocol is established based on Modbus. Third, the windows desktop upper computer software and the cloud-based monitoring system are designed. Finally, this paper designs the improved ResLSTM algorithm which is fused with ResNet algorithm based on Stacked LSTM. The algorithm designed in this paper is tested in comparison with SVM and LSTM. The performance of the algorithm designed in this paper is better than SVM and LSTM. Furthermore, the communication test, as well as the training and testing of the ResLSTM algorithm are outstanding. The 5G base station lithium-ion battery cloud monitoring system designed in this paper can meet the requirements. It has great significance for engineering promotion. More importantly, the ResLSTM algorithm designed in this paper can better guide the method of lithium-ion battery SOC estimation. [ABSTRACT FROM AUTHOR]

In order to solve the challenges brought by multi-source and cross-domain scenarios to online music education, this paper designs an online music education system based on advanced artificial intelligence technology, which can provide personalized learning course resource recommendations for music online learners. The system includes four layers, consisting of user interface layer, application module layer, function module layer and data storage layer. At the application module level, this paper proposes a music recommendation algorithm based on a personalized multimodal network model. The recommendation algorithm performs music information retrieval (MIR) based on the similarity judgment of the contour of music pitch and the overall change, and constructs a multimodal network model based on the user's preference for resources to achieve personalized music recommendation. This paper crawls more than one million music score data from a well-known music platform database in China to establish a dataset to evaluate the performance of this method. The comparison results with three existing works show that the method proposed in this paper has good performance and can provide users with suitable music recommendations. The artificial intelligence technology-driven online music education mechanism proposed in this paper has good prospects. [ABSTRACT FROM AUTHOR]

Metaverse empowers learning and spawns learning metaverse. Learning Metaverse is a new type of Internet educational application and digital social ecosystem. Learning Metaverse Empowering Education will help promote the construction of new educational infrastructure and provide new opportunities for building a high-quality education support system. This paper aims at the online English teaching scenario in the multi-source and cross-domain environment, and constructs a new teaching scheme by using the metaverse technology. In order to improve the quality of English teaching and the interest of English teaching classroom, this paper proposes an immersive English learning visualization system with diversified interactive methods. Therefore, this paper designs an immersive interaction management method based on visual scene and a visual network management method based on visual algorithm tree, which can systematically manage user interactions and provide full-process support for visual operations in an immersive environment. In order to verify the effectiveness and naturalness of the interactive system, this paper conducts a comparative experiment between the two-dimensional desktop system and the immersive virtual reality system. The user experiment results show that the system can meet the needs of multi-source and cross-domain environment. The new online English teaching scheme proposed in this paper can provide users with efficient and natural interaction in the immersive visual English learning scenario. [ABSTRACT FROM AUTHOR]

The cultivation of creativity is closely related to language learning. How to design the creativity promotion mechanism of English teaching in the public environment is the challenge faced by English teachers. With the advent of the era of big data, English teachers can apply the latest research results to classroom teaching. For example, the rational use of social media helps students to learn and communicate in the language, cultivate students' creativity in learning English, and improves the quality of teaching. Corporate social media has become the most important way for corporate employees to record their lives, express opinions, share and communicate, and it is also one of the reliable and real-time sources of big data that reflects the true state of English learners. Real, accurate, and timely enterprise social media big data samples contain an enormous educational value, providing more possibilities for educational research. From the perspective of value, through sentiment analysis, topic mining, social network analysis, etc. on social media big data, learner portraits can be realized, thereby providing decision-making reference and support for stakeholders. This paper first builds a learning interest classification model based on TCNN-GRU deep learning, collects experimental data sets from an online English learner's social media platform and performs learning interest classification and labeling, and then uses the TCNN-GRU model to determine the user's learning interest tendency. On this basis, the concept of learning interest index is further proposed, and a neural network-based corporate social platform English learner portrait model is established. The experimental results show that, compared with the traditional machine learning model, convolutional neural network model, and recurrent neural network model, the TCNN-GRU model can obtain better results in learning interest classification. [ABSTRACT FROM AUTHOR]

The Industrial Internet is based on the network, the platform is the core, and the security is the guarantee. The Industrial Internet connects all industry elements and the entire industry chain through the large-scale network infrastructure, collects and analyzes industry data in real time, and forms a new application model for a new generation of information communication. With the rapid development of industrial Internet technology, the scale of industrial Internet data will become larger and larger, and the data dimension will become higher and higher. How to efficiently use cluster analysis for industrial Internet big data mining is an urgent problem that needs to be solved. This paper proposes an improved differential evolution particle swarm algorithm for industrial Internet big data clustering analysis. Differential Evolution (DE) strategy can improve the problem that the particle swarm optimization (PSO) algorithm tends to fall into local optimum in the later stage as the number of iterations increases. Considering the influence of the randomness of the arrangement order of the cluster center vectors among the individuals on the learning and updating among individuals, this paper designs a method of adaptively adjusting the arrangement order of the cluster center vectors to optimize the cluster center vector with maximum similarity among individuals. In order to effectively evaluate our method, both industrial and non-industrial datasets are selected. The experimental results verify the feasibility and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

A watercolor is an art form that uses water to express the painting process. Water is the specific boundary that distinguishes watercolor from other painting types. Fluidity is one of the essential factors constituting its distinctive artistic beauty. With vibrant hues and delicate brushstrokes, 3D watercolors enhance the image's content with more meaningful information. It has more visual impact than a watercolor that is flat. Appreciators can visualize the scenario in their minds, which leaves more room for imagination and is particularly useful for usage in animation. This paper uses 3D point cloud reconstruction to enhance the 3D effect of watercolor and proposes a 3D reconstruction method of the point cloud based on a generative adversarial network. First, a new mesh model is obtained by using a predictor to predict the offset of the mesh model edge. Then, the point cloud classifier in the discriminator is used to extract the high dimensional features of the original point cloud data and the surface sampling point set of the mesh model. Finally, the output data of the predictor and discriminator are associated with the adversarial training method, and the network model is optimized through several iterations to obtain a 3D mesh model satisfying the spatial characteristics of the point cloud. The experimental results demonstrate that the method suggested in this paper is superior to the three benchmarks in terms of watercolor's texture, color transition, highlight part, and iteration time, as well as its ability to enhance the 3D effect. [ABSTRACT FROM AUTHOR]

In the era of 5G/B5G, computing-intensive, delay-sensitive applications such as virtual reality inevitably bring huge amounts of data to the network. In order to meet the real-time requirements of applications, Mobile Edge Computing (MEC) pushes computing resources and data from the centralized cloud to the edge network, providing users with computing offload technology. However, the mismatch between the great computing requirements of computing-intensive tasks and the limited computing power of a single edge server poses a great challenge to computing offload technology. In this paper, a multi-agent cooperation mechanism for MEC and a routing mechanism based on deep reinforcement learning (DRL) are proposed. First of all, a multi-agent cooperation mechanism is proposed to realize the cooperative processing of computing-intensive and delay-sensitive applications, and the task unloading decision-making problem based on multi-agent cooperation is studied. Secondly, the cooperative processing of tasks by multi-agents involves data transmission. Considering the real-time requirements of tasks, this paper proposes an intelligent routing mechanism based on DRL to plan the optimal routing path. Finally, the simulation implementation and performance evaluation of the multi-agent cooperation mechanism and routing mechanism for MEC are carried out. The experimental results show that the intelligent routing mechanism based on DRL and graph neural network is superior to the comparison mechanism in terms of network average delay, throughput and maximum link bandwidth utilization. At the same time, the superiority of graph neural network in model generalization is verified on a new network topology National Science Foundation (NSF) Net. The results of route optimization are applied to the multi-agent cooperation mechanism, and the experimental results show that the mechanism is superior to the comparison scheme in terms of task success rate and average task response delay. The combination of these two mechanisms well solves the problem that it is difficult to deal with computing-intensive and delay-sensitive applications in mobile edge computing because of its limited resources. [ABSTRACT FROM AUTHOR]

Poisson equation is a partial differential equation with broad utility in theoretical physics. Dirichlet problem of Poisson Equations can be solved by Green's function. It is a very attractive problem to look for analogous results of the above problem in the fractal context. This paper studies the network set Higher-Dimensional Sierpinski Gaskets, solves Dirichlet problem of Poisson Equations on them by expressing Green's function explicitly. [ABSTRACT FROM AUTHOR]

The development of 5G technology has enabled the cloud-internet of things (IoT) to impact all areas of our lives. Sensors in cloud-IoT generate large-scale data, and the demand for massive data processing is also increasing. The performance of a single machine can no longer meet the needs of existing users. In contrast, a data center (DC) integrates computing power and storage resources through a specific network topology and satisfies the need to process massive data. Regarding large-scale heterogeneous traffic in DCs, differentiated traffic scheduling on demand reduces transmission latency and improves throughput. Therefore, this paper presents a traffic scheduling method based on deep Q-networks (DQN). This method collects network parameters, delivers them to the environment module, and completes the environment construction of network information and reinforcement learning elements through the environment module. Thus, the final transmission path of the elephant flow is converted based on the action given by DQN. The experimental results show that the method proposed in this paper effectively reduces the transmission latency and improves the link utilization and throughput to a certain extent. [ABSTRACT FROM AUTHOR]

Every enterprise in the supply chain will participate in managing the supply chain. The decisions made by each party will affect the future decisions of other members in the supply chain and themselves. There are trade-offs since the upstream and downstream of the multi-supply chain tackle problems from different perspectives based on their interests. With the increasing competition and cooperation among enterprises in multi-supply chains, game theory is widely used to analyze the competition and collaboration among enterprises. This study combines game theory and auction theory to obtain the relative optimal allocation through double auctions for multi-objective optimization. Nevertheless, there are a few issues, including the number of iterations and the potential for local monopoly in double auctions. To this end, the generalized genetic particle swarm optimization (GGPSO) algorithm is introduced to improve the double auctions mechanism, and finally, the global optimization of the supply chain is achieved. The simulation results show that the proposed method can efficiently complete the distribution and pricing among multi-supply chains and compensate the competing buyers with higher average quotations and the competing sellers with lower average quotes. Additionally, the GGPSO algorithm presented in this paper has a good performance in terms of the time needed to obtain the optimal solution, which is conducive to the global optimization of the supply chain. [ABSTRACT FROM AUTHOR]

With the deepening of hospital informatization construction, the electronic health record (EHR) system has been widely used in the clinical diagnosis and treatment process, resulting in a large amount of medical data. Electronic medical records contain a large amount of rich medical information, which is an important resource for disease prediction, personalized information recommendation, and drug mining. However, the medical information contained in electronic medical records cannot be automatically acquired, analyzed and utilized by computers. In this paper, we utilize machine learning algorithms for intelligent analysis of large-scale electronic medical records to explore and develop general methods and tools suitable for electronic medical record analysis in medical databases. This is of great value for summarizing the therapeutic effects of various diagnosis and treatment programs, disease diagnosis, treatment, and medical research. We propose an ECML-based intelligent analysis method for electronic medical records. First, we perform data preprocessing on the electronic medical record. Second, we design an intelligent analysis method for electronic medical records based on a deep learning model. Third, we design a model hyperparameter optimization method based on evolutionary algorithms. Finally, we compare and analyze the performance of the proposed model through experiments, and the experimental results show that the model proposed in this paper has good performance. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the thickness and quasisymmetric equivalence of a class of fractal sets based on the binary expansion of numbers. We give the sufficient and necessary conditions for the thickness of this kind of sets to be positive, and the necessary and sufficient condition for their quasisymmetric equivalence with the standard Cantor ternary set. This paper also studies the relationship between the thickness and the uniform completeness about the product sets of two sets defined by digit restrictions. [ABSTRACT FROM AUTHOR]

Due to the increasing growth of economy in the last few years, environmental problems have become a prominent difficulty in Chinese economic growth and social development. Therefore, in order to respond to the environmental protection policy of Anhui Province, all prefecture-level cities need to be evaluated for their environmental efficiency. A study of environmental efficiency in Anhui Province is presented in this paper. The time period is from 2015 to 2020. By exploiting MaxDEA software, the input–output system in different years is taken as the decision-making unit, and slacks-based model (SBM-DEA) which is based on an unexpected output is used to construct the index system by selecting two output indexes and four input indexes, so as to calculate the environmental efficiency of Anhui Province, including 16 prefecture-level cities, under the condition of constant non-radial and non-angle scale benefits. In addition, based on the persistence and anti-persistence theory of fractal theory, in this paper, we have analyzed the future trend of environmental efficiency of prefecture-level cities in Anhui Province by rescaled range analysis (R/S). The results show that Anhui province has 16 prefectures that have not achieved an optimal level of environmental performance and fluctuates downward from 2015 to 2020. The average value of comprehensive efficiency is 0.846. As for technical efficiency, the mean value is 0.954, and scale efficiency has an average value of 0.887. The technical progress of environmental efficiency development in Anhui Province is small and the room for improvement is large. Finally, according to the results of the study, some suggestions on environmental technology are put forward. [ABSTRACT FROM AUTHOR]

This paper employs the monotonicity analysis for non-negativity to derive a class of sequential fractional backward differences of Riemann–Liouville type a+1RL∇νaRL∇αu (t) based on a certain subspace in the parameter space (0, 1) × (0, 1). Auxiliary and restriction conditions are included in the monotonicity results obtained in this paper and they confirm the monotonicity of the function on {a + 2,a + 3,…}. A non-monotonicity result is also established based on the main conditions together with further dual conditions, and this confirms that the main theorem is almost sharp. Furthermore, we recast the dual conditions in a sing condition, and then we represent the sharpness result in a new corollary. Finally, numerical results via MATLAB software are used to illustrate the main mathematical results for some special cases. [ABSTRACT FROM AUTHOR]

This paper explores the truncated M-fractional three-component coupled nonlinear Schrödinger (tc-CNLS) equation, that regulates the behavior of optical pulses in optical fibers. These equations are utilized in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. The study of multi-component NLS equations has gained significant attention due to their ability to elucidate various complex physical phenomena and exhibit more dynamic structures of localized wave solutions. The freshly invented integration tools, known as the fractional modified Sardar subequation method (MSSEM) and fractional enhanced modified extended tanh-expansion method (eMETEM), are employed to ensure the solutions. The study focuses on extracting various types of optical solitons, including bright, dark, singular, bright-dark, complex, and combined solitons. Optical soliton propagation in optical fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. Furthermore, hyperbolic, periodic and exponential solutions are generated. The utilized methodology is effective in explaining fractional nonlinear partial differential equations (FNLPDEs) as it offers pre-existing solutions and additionally derives novel exact solutions by mixing outcomes from various procedures. Furthermore, we plot the visualizations of solutions by plotting 3D, 2D, and contour graphs with the corresponding parameter values. The findings of this paper can improve the understanding of the nonlinear dynamical behavior of a specific system and demonstrate the efficacy of the methodology used. We anticipate that our study will provide substantial benefits to a considerable group of engineering model experts. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures. [ABSTRACT FROM AUTHOR]

Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized M set with different kinds of exponent k (k − M set), properties are hard to understand when k is a complex number. In this paper, fractal property of generalized M set with complex exponent z is studied. First, a relation is constructed between generalized M set with complex and real exponent. Then, distribution of z − M set on complex plane is researched. Meanwhile, symmetry of generalized M set is proved. Finally, graphics, generated by escape time algorithm, are the validated results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to ρ = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function's differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math.76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts α after α -order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of f (x) does not increase after α -order Hadamard fractional integral D H − α f (x) , i.e. dim ¯ B Υ (D H − α f , I) ≤ dim ¯ B Υ (f , I). Particularly, while dim B Υ (f , I) = 1 , for example, a function is of unbounded variation and/or infinite length, it holds dim B Υ (D H − α f , I) = dim B Υ (f , I) = 1 , which answers completely this conjecture for one-dimensional fractal function. The published paper could verify Conjecture 1.1 under Riemann–Liouville or Wyel fractional integral, for only constructed functions with one-dimensional fractal [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. Li and W. Xiao, Fractal dimensions of Riemann–Liouville fractional integral of certain unbounded variational continuous function, Fractals25 (2017), doi:10.1142/S0218348X17500475; Y. S. Liang, Fractal dimension of Riemann–Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal.21 (2018) 1651–1658; Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sin.3 (2014) 517–524; J. Wang and K. Yao, Construction and analysis of a special one-dimensional continuous function, Fractals25 (2017), doi:10.1142/S0218348X17500207; X. Liu, J. Wang and H. L. Li, The classification of one-dimensional continuous functions and their fractional integral, Fractals26 (2018), doi:10.1142/S0218348X18500639]. [ABSTRACT FROM AUTHOR]

This paper aims to study the processing and repairing methods of blurred images, promote the development of partial differential equations in the field of image processing, and expand the application of stochastic fuzzy differential equations in the field of image processing and repair. This study starts with a typical blurred image repair method. First, a comparative analysis of several common blurred image repair methods including Wiener filtering restoration, inverse filtering restoration, and Lucy–Richardson (L-R) filtering restoration is performed. Second, based on the linear partial differential equation learning model (LPDE), the concept of fuzzy integral is introduced, and an improved stochastic fuzzy partial differential equation learning model (SFCPDE) is proposed. The effect of the learning model before and after improvement on blurred color image processing is compared and analyzed. Finally, based on the total variation (TV) blurred image repair algorithm, an improved TV blurred image repair algorithm is proposed. The comparison and analysis of the repair effects of several blurred image repair algorithms are performed. The results show that there are obvious differences in the repairing methods with or without noise. Inverse filtering works best when there is no noise. L-R filtering has the disadvantage of amplifying noise. Compared with LDPE, the training speed of SFCPDE is significantly improved, and the training error is less than LDPE. The SFCPDE learning model performs better in the processing of blurred color images. After 10 iterations, the improved TV algorithm is significantly better than the TV algorithm and the CDD algorithm in repairing blurred images. The PSNR value of the TV algorithm and the curvature-driven diffusion (CDD) algorithm after 10 iterations corresponds to about 60% of the PSNR value of the improved algorithm. The algorithms and models of stochastic fuzzy partial differential equations proposed in this paper have great application potential in the processing and repair of multiple blurred images. [ABSTRACT FROM AUTHOR]

Based on the improved BP neural network, this paper establishes an adaptive online controlling model and adopts the model to optimize the controlling accuracies in discrete nonlinear dynamic systems and inverted pendulum systems. To avoid the local minimum problem of the BP neural network's objective function in the training process, this paper proposes a neural network training method based on the quasi-Newton method (BFGS) optimization algorithm. Compared with other control methods, the neural network-based inverted pendulum control method proposed in this paper has higher control accuracy. Through the control simulation of its power system uses a discrete control method and the control of the inverted pendulum model system, this paper verifies the validity and good control has significantly improved the control method. [ABSTRACT FROM AUTHOR]

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion. [ABSTRACT FROM AUTHOR]

The research object of this paper is the mixed (κ , s) -Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed (κ , s) -Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under σ = (σ 1 , σ 2) order of the mixed integral is 3 − min { σ 1 κ , σ 2 κ } where  κ > 0. [ABSTRACT FROM AUTHOR]

Every fluid problem is greatly affected by its boundary conditions, especially the near-shore seabed could produce an irrevocable harm when a tsunami wave is approaching, and a real-life mathematical model could stave off the worst effect. This paper assumes that the unsmooth seabed is a fractal surface, and fractal-fractional governing equations are established according to physical laws in the fractal space. The geometrical potential theory is used to explain the force produced by the wave surface, and Kong-He friction law is applied to further figuring out the local and memory properties of the friction along the fractal boundary. This paper aims at studying tsunami waves in a fractal space, rendering a reliable mathematical model for both prediction of the tsunami motion and the coastal protection. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to deal with a system governed by a system of semilinear nonlocal fractional evolution inclusions with partial Clarke subdifferential and its optimal control. First, we establish an existence theorem of the mild solution for the presented control system by applying the measure of noncompactness, a fixed point theorem of a condensing multivalued map and some properties of partial Clarke subdifferential. Moreover, under some mild conditions, we obtain a result on the existence of an optimal control to the presented control system. Finally, an example is provided to demonstrate the main results. The results presented in this paper improve, extend and develop the corresponding results in the earlier and recent literature. [ABSTRACT FROM AUTHOR]

This paper establishes a linear relationship between the order of the Riemann–Liouville fractional calculus and the exponent of the Hölder condition, whether the Hölder condition is global, local, or at a single point. We propose and prove a control inequality between the Hölder derivative ( H f (x , α) as defined in Proposition 12) of a continuous function and the Hölder derivative of the Riemann–Liouville fractional calculus of this function. In addition, this paper provides a more accurate estimation of the Box dimension of the graph of the Riemann–Liouville fractional integral of an arbitrary continuous function. More specifically, it establishes the result that whenever there is a continuous function whose graph has the upper Box dimension s with 1 < s ≤ 2 , the graph of its Riemann–Liouville fractional integral of order ν , with 0 < ν < 1 , has the upper Box dimension not greater than s − (s − 1) ν. [ABSTRACT FROM AUTHOR]

The laws governing life should be as simple as possible; however, theoretical investigations into allometric laws have become increasingly complex, with the long-standing debate over the scaling exponent in allometric laws persisting. This paper re-examines the same biological phenomenon using two different scales. On a macroscopic scale, a cell surface appears smooth, but on a smaller scale, it exhibits a fractal-like porous structure. To elaborate, a few examples are given. Employing the two-scale fractal theory, we theoretically predict and experimentally verify the scaling exponent values for basal, active, and maximal metabolic rates. This paper concludes that Rubner's 2/3 law and Kleiber's 3/4 law are two facets of the same truth, manifested across different scale approximations. [ABSTRACT FROM AUTHOR]

Data-driven learning of the fractional discrete-time unified system is studied in this paper. A neural network method is suggested in the parameter estimation of fractional discrete-time chaotic systems. An optimization problem is obtained and the famous Adam algorithm is employed to train the neural network's weights and parameters. The parameter estimation result is compared with that of the stepwise response sensitivity approach (SRSA). This paper provides a high accuracy method for parameter inverse problems. The method also can be applied to data-driven learning of other fractional chaotic systems. [ABSTRACT FROM AUTHOR]

Scientists worldwide have always been interested in the study of networks that resemble trees and porous materials. Therefore, based on fractal theory, this paper systematically studies the heat transfer problem of the damaged tree-like networks under different saturations in multi-medium composite materials and derives their dimensionless thermal conductivity (DTC). The research has shown that the dimensionless thermal conductivity (DTC) decreases with an increase in damaged channels. The saturation significantly impacts the heat transfer of the damaged tree networks, and it exhibits linear changes with the ratio of the gas phase, liquid phase, and solid phase. Additionally, it can be observed that the fractal dimension of length distribution and diameter distribution, bifurcation numbers, bifurcation series, porosity, and other parameters in the networks are related to thermal conductivity. The comparison between the data derived from this model and experimental data shows that the proposed model can effectively deepen our understanding of the heat transfer mechanism of the damaged networks in composite materials under different saturation levels. Additionally, the model in this paper does not have an empirical constant, which avoids the influence of potential factors. [ABSTRACT FROM AUTHOR]

The Kozeny–Carman (KC) equation is a well-known semi-empirical formula, which is used to calculate the permeability of porous media in the seepage field. The KC constant is an empirical constant in the KC equation. In this paper, based on the fractal theory, the Fractal-Monte Carlo technique is used to simulate the KC constant of the roughened fibrous porous media (RFPM) with micro-scale effects. There is no empirical constants in the proposed model, and each parameter has its physical meaning. The KC constant model of RFPM can be expressed as a function of structural parameters, including relative roughness (), porosity (ϕ), pore area fractal dimension ( D f ), tortuosity fractal dimension ( D T ), capillary diameter (λ) and Knudsen number (Kn). The result shows that the KC constant increases with increases in ϕ , , D f and D T . On the contrary, with increases in λ and Kn , the KC constant decreases. In addition, the KC constant model constructed in the paper agrees well with the existing experimental data and the model. According to the proposed Fractal-Monte Carlo technique, it is possible to better clarify the transmission physical mechanism in RFPM with micro-scale effects. [ABSTRACT FROM AUTHOR]

The security of mobile wireless sensor networks has captivated extensive attention of researchers because of their wide range of applications and vulnerability to attacks caused by malware. In this paper, we investigate a delayed malware propagation model on mobile wireless sensor network incorporating nonlinear incidence rate, logistic growth rate and recovery rate. Local asymptotic stability of the endemic equilibrium and existence of Hopf bifurcation at crucial value of the time delay are analyzed. Then, properties of Hopf bifurcation are explored. Specifically, global exponential stability is investigated via linear matrix inequality. An example is presented finally to underline the effectiveness of findings in our paper numerically and graphically. [ABSTRACT FROM AUTHOR]

This paper studies the well-known mathematical model's analytical wave solutions (modified Benjamin–Bona–Mahony (BBM) equation), which demonstrates the propagation of long waves in the nonlinear dispersive media in a visual illusion. Six recent analytical and semi-analytical schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, sech–tanh expansion method, Adomian decomposition (ADD) method, El Kalla (EK) expansion method, variational iteration (VI) method) are applied to the considered model for constructing abundant analytical and semi-analytical novel solutions. This variety of solutions aims to investigate the analytical techniques' accuracy by calculating the absolute error between analytical and semi-analytical solutions that shows the matching between them. The analytical results are sketched through two-dimensional (2D), three-dimensional (3D), contour plot, spherical plot and polar plot. The stability characterization of the analytical solutions is investigated through the Hamiltonian system's features. The originality and novelty of this paper are discussed, along with previously published papers. [ABSTRACT FROM AUTHOR]

Diabetes is a chronic disease that poses a serious threat to health, and its early risk prediction has been a hot research topic in the field of medical artificial intelligence. Routine medical checkups are the most common way to monitor people's health status, and the data from medical checkups contain rich diagnostic information, which is valuable for diabetes risk prediction. Currently, most of the available studies on diabetes risk prediction are based on publicly available datasets, and the models and algorithms do not work well on real clinical datasets. Real routine checkup data are characterized by complex information, diverse features, high redundancy and poor balance, which pose great challenges for diabetes risk prediction. To address this problem, this paper proposes a multi-strategy data augmentation-based diabetes risk prediction method, after completing data pre-processing and feature selection, a counterfactual-based data balancing strategy is used to augment a minority class of instances, and a density clustering-based supplemental counterfactual data augmentation strategy is proposed to address the problem of insufficient representation of generated instances in the counterfactual method. Moreover, the uncertainty-weighted method is used in the model training phase. Based on the real checkup dataset, five machine learning methods including Logistic Regression (LR), SVM, Decision Tree, Random Forest and Gradient Boosting are used to model and use 5-fold cross-validation to carry out diabetes risk assessment and prediction. The experimental results showed that the sensitivity and precision of the models were significantly improved compared with the existing methods, and the sensitivity of the LR model for diabetes risk prediction on the real routine checkup dataset reached more than 90 % , which meet the requirements of clinical application. [ABSTRACT FROM AUTHOR]

Long short-term memory (LSTM) with significantly increased complexity and a large number of parameters have a bottleneck in computing power resulting from limited memory capacity. Hardware acceleration of LSTM using memristor circuit is an effective solution. This paper presents a complete design of memristive LSTM network system. Both the LSTM cell and the fully connected layer circuit are implemented through memristor crossbars, and the 1T1R design avoids the influence of the sneak current which helps to improve the accuracy of network calculation. To reduce the power consumption, the word embedding dimensionality was reduced using the GloVe model, and the number of features in the hidden layer was reduced. The effectiveness of the proposed scheme is verified by performing the text classification task on the IMDB dataset and the hardware training accuracy reached as high as 88.58%. [ABSTRACT FROM AUTHOR]

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of ℝ. We show that the Weyl–M derivative of order γ of a continuous function satisfying Hölder condition of order μ also satisfies Hölder condition of order μ − γ and the upper box dimension of the Weyl–M derivative increases at most linearly with the order γ. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order γ. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one. [ABSTRACT FROM AUTHOR]

In this paper, we calculate electrical resistance scaling exponents for analogues of "carpet" and "sponge" fractals in higher-dimensional embedding spaces. The calculation idealizes the voids that define such fractals as spherical, and exploits the elementary theory of dielectrics. Possible applications include models of elementary particles with "extra" dimensions, and new large-dimension methods for the theory of fractals. [ABSTRACT FROM AUTHOR]

This paper has investigated the predictability of the top 10 cryptocurrencies' price dynamics, ranked by their daily market capitalization and trade volume, via the information theory quantifiers. Our analysis considers the Complexity-entropy causality plane to study the temporal evolution of the price of these cryptocurrencies and their respective locations along this 2D map, bearing in mind after and during the Russia–Ukraine war. Moreover, we apply the permutation entropy and the Jensen–Shannon statistical complexity measure to rank these cryptocurrencies similarly to a complexity hierarchy. Our findings reflect that the Russian–Ukraine war affects the informational efficiency of cryptocurrency dynamics. Specifically, the cryptocurrencies notably showed a decrease in informational inefficiency (USD-coin, Binance-USD, BNB, Dogecoin, and XRP). At the same time, the cryptocurrencies with more expressiveness for the financial market, considering the volume traded and the capitalized market, were strongly impacted, presenting an increase in informational inefficiency (Tether, Cardano, Ethereum, and Bitcoin). It clarifies the potential of cryptocurrencies to mitigate exogenous shocks and their capability to use with portfolio selection, risk diversification and herding behavior. [ABSTRACT FROM AUTHOR]

This paper provides a new idea for interpolating complex curves and surfaces by fractal interpolation. A type of bivariate fractal interpolation functions with function vertical scaling factors for the special interpolation data set is presented. The interpolation algorithm is proposed for complex curves by contour lines of two types of fractal interpolation functions which are proposed by this paper and Navascués et al. [Construction of fractal surfaces, Fractals 28(2) (2020) 2050033], respectively. Meanwhile, some properties of the interpolation algorithm are introduced. Then, the two classes of fractal interpolation functions are extended to the cases of trivariate functions. We also introduce the interpolation algorithm for complex surfaces based on iso-surfaces of fractal interpolation functions constructed. Finally, several examples are provided. [ABSTRACT FROM AUTHOR]

An efficient high-order computational procedure is going to be created in this paper to determine the solution to the mobile–immobile advection–dispersion model (MIAD) of temporal fractional order 0 < α ≤ 1 , which can be employed to model the solute forwarding in watershed catchments and floods. To do it, the temporal-first derivative of MIAD is discretized by using the finite-difference technique's first-order precision and the linear interpolation's temporal-fractional derivative. On either side, the space derivative is simulated using a collocation approach based on the Legendre basis to generate the full-discrete method. The order of MIAD-convergence for the implicit numerical structure is explained. Additionally, a basic conceptual discussion of the temporal-discretized stability mechanism is included in this paper. Finally, two models are provided to show the reliability and excellence of the organized approach. [ABSTRACT FROM AUTHOR]

Uses of time-fractional diffusion wave model (TFDWM) in its singular case with damping-reaction terms are widely seen in classical physics applications, e.g. for the quantitative measurement of activity diagnoses light-mechanical waves resulting from many physical experiments. The goal and importance of this paper are to predict and build accurate and convincing numerical solutions for TFDWM in its singular version by employing the collective cubic uniform B-spline approach (CUBSA) and standard finite difference approach (SFDA). The fractional Caputo time derivative (FCTD) has been estimated and broken down using SFDT, whilst the standard splines will be utilized upon realizing spatial discretization. To study the prediction error of our approach, some convergence and bound results are given under certain constraints. We demonstrate applications of our collective algorithm to a couple of fractional singular-type models appearing in fluid dynamics and electromagnetics. Detailed analysis, delegate tables, and representative graphs are displayed and offered in different dimensions to handle the crossover meaning for several order values of FCTDs. Some conclusions, observations, recommendations, and future issues were briefly raised in the final section of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear mathematical model which addresses the transmission dynamics of COVID-19. The considered model consists of susceptible (S), exposed (E), infected (I), and recovered (R) individuals. For simplicity, the model is abbreviated as S E I R. Immigration rates of two kinds are involved in susceptible and infected individuals. First of all, the model is formulated. Then via classical analysis, we investigate its local and global stability by using the Jacobian matrix and Lyapunov function method. Further, the fundamental reproduction number ℛ 0 is computed for the said model. Then, we simulate the model through the Runge–Kutta method of order two abbreviated as RK2. Finally, we switch over to the fractional order model and investigate its numerical simulations corresponding to different fractional orders by using the fractional order version of the aforementioned numerical method. Finally, graphical presentations are given for the approximate solution of various compartments of the proposed model. Also, a comparison with real data has been shown. [ABSTRACT FROM AUTHOR]

The physical/probabilistic meanings of fractional derivatives or integrals are very basic and important problems in fractional calculus. In this paper, we investigate the probabilistic interpretations of the general fractional derivatives with memory effect and their connections with cut-off distributions. Some examples like the cut-off Rayleigh distribution and the Atangana–Gómez fractional derivative are especially discussed. This paper shows that the kernels of the general fractional derivatives with memory effect may be not the same as the density functions of the physical processes described. For example, the kernel of Atangana–Gómez fractional derivative is a Gaussian function but its corresponding probabilistic distribution is a cut-off Rayleigh distribution. [ABSTRACT FROM AUTHOR]

In this paper, two different types of plasma models with variable coefficients are described by using the fractal derivative. This paper aims at establishing fractal variational principles for two different types of discontinuous plasma physics by the fractal semi-inverse method, so that it can be applied to numerical analysis of the effect of the discontinuous time and discontinuous space on the solution properties. [ABSTRACT FROM AUTHOR]

This paper introduces evolving hierarchical networks based on hierarchical product. We discuss two important consensus behavior indexes in both un-weighted and weighted cases in those networked systems: (1) convergence speed in system without delay and noise; (2) delay robustness in system with communication time-delay. We analyze the approximative behaviors of these two consensus indexes by determining the second smallest and largest eigenvalue of Laplacian matrix, respectively, as well as how the scale of the weight factor affects the consensus performance. Moreover, the method applied in this paper is also applicable to many classic networks. Finally, two numerical simulations are given to verify the advantages of communication topology like hierarchical networks and the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the Hopf bifurcation of a fractional tri-neuron network with both leakage delay and communication delay under different fractional orders. By applying fractional Laplace transform, the stability theorem of linear autonomous system and Hopf bifurcation theorem, we obtain a class of asymptotic stability criterion of zero solution as well as delay-induced Hopf bifurcation conditions for the considered system. Simultaneously, the stability and Hopf bifurcation for tri-neuron network with single fractional order are also discussed as a special case of our proposed neural network model. Finally, a simulation example is given to illustrate the efficiency of the presented theoretical results in this paper. [ABSTRACT FROM AUTHOR]

With the acceleration development of urbanization, the infrastructure construction plays an increasingly crucial role in the expansion process of cities. However, the existing decision-making methods of urban infrastructure investment still stay at the static level, resulting in a huge waste of resources, which does not meet the needs of dynamic urban development and management. Based on geometric Brownian motion, option function and the theory of real option, this paper establishes a dynamic differential optimal investment opportunity decision model applied to urban infrastructure construction. An example was then conducted to test the operability and application of the proposed model. The results confirm that the proposed approach contributes to improving the scientificity of urban infrastructure planning and construction, and realizing the maximization of infrastructure's social value. Meanwhile, this paper contributes to promoting the development of city infrastructure investment decision-making methods from static to dynamic, and from partial to overall. [ABSTRACT FROM AUTHOR]

In view of the fractal evolution phenomenon in Internet information dissemination, this paper constructs a generalized fractal network by analyzing the characteristics of user dissemination behavior and according to the dissemination trend of actual information. On this basis, it reveals the fractal evolution mechanism in Internet information dissemination and its dissemination dynamic characteristics and rules. First, based on node similarity, a fractal propagation model of Internet information is constructed. Second, based on the scale-free network and according to the user characteristics, the information dissemination process is simulated and a generalized fractal network is evolved. Then, the key factors influencing the formation of fractal networks, the characteristics, and the rules of propagation dynamics are analyzed. Finally, the rationality and validity of the model are verified by crawling the real user forwarded data in the micro-blog. [ABSTRACT FROM AUTHOR]

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general system of fractal-fractional order derivative in Atangana–Baleanu–Caputo sense. On the application of fixed point approach, we establish sufficient conditions regarding existence and uniqueness of solution. The said requirements are obtained via using Krasnoselkii's and Banach fixed results. Further via nonlinear analysis, some interesting results for Hyers–Ulam (HU)-type stability are also derived. To compute numerical solution for the proposed nonlinear system, fractal-fractional order Adams–Bashforth method is used. To support our findings, we give some test problems. Also by Matlab, we also present their graphical interpretation. The analysis of this paper is in generalized format which can be applied to any real problem. Each equation is investigated separately for the said characteristics. [ABSTRACT FROM AUTHOR]

In this paper, the fractional evolutions are described by using the conformable derivative for the first time. We implement fractional functional variable method (FFVM) to obtain some new kinds of fractal soliton wave solutions for these fractional evolution equations. The simplicity and effectiveness of this proposed method are tested on the fractional Drinfeld–Sokolov system and fractional cubic Klein–Gordon equation. The FFVM provides a new perspective to construct exact fractal soliton wave solutions of complex fractional nonlinear evolution equations in mathematical physics. [ABSTRACT FROM AUTHOR]