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In this paper, we propose a deep learning framework for solving high-dimensional partial integro-differential equations (PIDEs) based on the temporal difference learning. We introduce a set of Lévy processes and construct a corresponding reinforcement learning model. To simulate the entire process, we use deep neural networks to represent the solutions and nonlocal terms of the equations. Subsequently, we train the networks using the temporal difference error, the termination condition, and properties of the nonlocal terms as the loss function. The relative error of the method reaches O (10- 3) in 100-dimensional experiments and O (10- 4) in one-dimensional pure jump problems. Additionally, our method demonstrates the advantages of low computational cost and robustness, making it well-suited for addressing problems with different forms and intensities of jumps. [ABSTRACT FROM AUTHOR]

The recently introduced coati optimization algorithm suffers from drawbacks such as slow search velocity and weak optimization precision. An enhanced coati optimization algorithm called CMRLCCOA is proposed. Firstly, the Sine chaotic mapping function is used to initialize the CMRLCCOA as a way to obtain better-quality coati populations and increase the diversity of the population. Secondly, the generated candidate solutions are updated again using the convex lens imaging reverse learning strategy to expand the search range. Thirdly, the Lévy flight strategy increases the search step size, expands the search range, and avoids the phenomenon of convergence too early. Finally, utilizing the crossover strategy can effectively reduce the search blind spots, making the search particles constantly close to the global optimum solution. The four strategies work together to enhance the efficiency of COA and to boost the precision and steadiness. The performance of CMRLCCOA is evaluated on CEC2017 and CEC2019. The superiority of CMRLCCOA is comprehensively demonstrated by comparing the output of CMRLCCOA with the previously submitted algorithms. Besides the results of iterative convergence curves, boxplots and a nonparametric statistical analysis illustrate that the CMRLCCOA is competitive, significantly improves the convergence accuracy, and well avoids local optimal solutions. Finally, the performance and usefulness of CMRLCCOA are proven through three engineering application problems. A mathematical model of the hypersonic vehicle cruise trajectory optimization problem is developed. The result of CMRLCCOA is less than other comparative algorithms and the shortest path length for this problem is obtained. [ABSTRACT FROM AUTHOR]

Due to the intermittent and unstable nature of PV systems, maximum power point tracking (MPPT) of PV systems is necessary in practice. At the same time, PV cells are also shaded by objects such as trees and houses in the environment causing local shadows, so the Lévy flight particle swarm optimisation (LFPSO) is proposed in this paper. The random walk process of Lévy flight is added to the particle swarm optimisation (PSO) to increase the diversity of the search, which can avoid falling into local optimal solutions. The experimental simulation results show that the algorithm proposed in this paper can still accurately track the global maximum power in the case of partial shading, which well avoids falling into the local maximum power. The performance is much higher than the traditional maximum power point tracking algorithm, which improves the efficiency of the PV system. [ABSTRACT FROM AUTHOR]

Multispectral thermometry stands as a prevalent non-contact method utilized for temperature measurement across various applications. To solve the problem that multispectral thermometry cannot obtain accurate temperature of the target under the unknown spectral emissivity, many scholars have proposed various optimization algorithms. However, there are still problems such as the large emissivity search range, uncertain initial solution and long solution time. To solve the above problems, a new objective function and constraint conditions are established. A self-adaptive cuckoo algorithm is proposed. Real number coding is used to improve the convergence ability and robustness of the algorithm. The adaptive function is used to evaluate the quality of the solution, and the result is avoided to fall into the local optimal solution by the random walk mechanism of Lévy flight. The validity of the proposed self-adaptive cuckoo algorithm is verified by inversion calculation of 6 different emissivity models and zirconia samples. The maximum relative error of self-adaptive cuckoo algorithm is 0.41% in the case of no noise interference, and 0.91% in the case of noise interference. The self-adaptive cuckoo algorithm can still inversion the temperature well with a low signal-to-noise ratio. The experimental results show that the inversion temperature error is less than 0.25%. This method provides a new idea for multispectral temperature measurement. [ABSTRACT FROM AUTHOR]

Let d be a positive integer, and let A be a set in Z d that contains finitely many points with integer coordinates. We consider a standard random walk X perturbed on the set A. This means that X is a Markov chain whose transition probabilities from the points outside A coincide with those of a standard random walk on Z d , whereas the transition probabilities from the points inside A are different. We investigate the impact of the perturbation on a scaling limit of X. It turns out that if d ⩾ 2, then in a typical situation the scaling limit of X coincides with that of the underlying standard random walk. This is unlike the case d = 1, in which the scaling limit of X is usually a skew Brownian motion, a skew stable Lévy process, or some other "skew" process. The distinction between the one-dimensional and multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in Z d for d ⩾ 3. More interestingly, in the situation where the standard random walk in Z 2 is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of X to the set A is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum, we have the situation in which the standard random walk admits a Donsker's scaling limit, whereas its locally perturbed version does not because of huge jumps from the set A, which occur early enough. [ABSTRACT FROM AUTHOR]

We explicitly calculate the Hausdorff dimension of the graph and range of an isotropic stable Lévy process X plus deterministic drift function f. For that purpose we use a restricted version of the genuine Hausdorff dimension, which is called the parabolic Hausdorff dimension. It turns out that covers by parabolic cylinders are optimal for treating self-similar processes, since their distinct non-linear scaling between time and space geometrically matches the self-similarity of the processes. We provide explicit formulas for the Hausdorff dimension of the graph and the range of X + f. In sum, the parabolic Hausdorff dimension of the drift term f alone contributes to the Hausdorff dimension of X + f. Furthermore, we derive some formulas and bounds for the parabolic Hausdorff dimension. [ABSTRACT FROM AUTHOR]

This study proposes a new prairie dog optimization algorithm version called EPDO. This new version aims to address the issues of premature convergence and slow convergence that were observed in the original PDO algorithm. To improve performance, several modifications are introduced in EPDO. First, a dynamic opposite learning strategy is employed to increase the diversity of the population and prevent premature convergence. This strategy helps the algorithm avoid falling into local optima and promotes global optimization. Additionally, the Lévy dynamic random walk technique is utilized in EPDO. This modified Lévy flight with random walk reduces the algorithm's running time for the test function's ideal value, accelerating its convergence. The proposed approach is evaluated using 33 benchmark problems from CEC 2017 and compared against seven other comparative techniques: GWO, MFO, ALO, WOA, DA, SCA, and RSA. Numerical results demonstrate that EPDO produces good outcomes and performs well in solving benchmark problems. To further validate the results and assess reliability, the authors employ average rank tests, the measurement of alternatives, and ranking according to the compromise solution (MARCOS) method, as well as a convergence report of EPDO and other algorithms. Furthermore, the effectiveness of the EPDO algorithm is demonstrated by applying it to five design problems. The results indicate that EPDO achieves impressive outcomes and proves its capability to address practical issues. The algorithm performs well in solving benchmark and practical design problems, as supported by the numerical results and validation methods used in the study. [ABSTRACT FROM AUTHOR]

This article studies the problem of optimal control with state constraints for mean‐field type stochastic systems, which is governed by a fully coupled forward‐backward stochastic differential equation with Teugels martingales. In this system, the coefficients contain not only the state processes but also its expectation value, and the cost function is of mean‐field type as well. We use an equivalent backward formulation to deal with the terminal state constraint, and then we obtain a stochastic maximum principle by Ekeland's variational principle. In addition, we discuss a stochastic linear‐quadratic control problem with state constraints. [ABSTRACT FROM AUTHOR]

A new test to detect the change-point in the quasi-autocorrelation coefficient (QAC) structure of a simple linear model with heavy-tailed series was developed. It is more general than previous approaches to the change-point problem in that it allows for the process with innovations in the domain of the attraction of a stable law with index κ (0 < κ < 2). Since the existing methods for QAC change detection are not satisfactory, we converted QAC change to mean change through the moving window method, which greatly improved the efficiency. Thus, the aim of this paper was to construct a ratio-typed test based on M-estimation for the testing of mean change. Under regular conditions, the asymptotic distribution under the no change null hypothesis was functional of a Wiener process, not that of a Lévy stable process. The divergent rate under the alternative hypothesis was also given. The simulation results demonstrate that the performances of our proposed tests were outstanding. Finally, the theoretical results were applied to an analysis of daily USD/CNY exchange rates with respect to QAC change. [ABSTRACT FROM AUTHOR]

The vibration signals of rolling bearings exhibit nonlinear and non-stationary characteristics under the influence of noise. In intelligent fault diagnosis, unprocessed signals will lead to weak fault characteristics and low diagnostic accuracy. To solve the above problem, a fault diagnosis method based on parameter optimization feature mode decomposition and improved deep belief networks is proposed. The feature mode decomposition is used to decompose the vibration signals. The parameter adaptation of feature mode decomposition is implemented by improved whale optimization algorithm including Levy flight strategy and adaptive weight. The selection of activation function and parameters is crucial in the application of deep belief networks. The improved deep belief networks are proposed based on Hard swish activation function and the improved whale optimization algorithm. The optimized diagnosis model is applied to weak fault diagnosis of bearings. The experimental results show that this method has faster convergence speed and anti-noise performance under the interference of −5 db noise, and effectively improves the adaptive feature extraction ability of the model and the accuracy of fault diagnosis. Under different conditions, the average accuracy reached 97.8% and 97.6%, with good generalization performance. [ABSTRACT FROM AUTHOR]

This paper introduces a novel metaheuristic algorithm named the opposition-based cooperation search algorithm with Nelder–Mead (OCSANM). This enhanced algorithm builds upon the cooperation search algorithm (CSA) by incorporating opposition-based learning (OBL) and the Nelder–Mead simplex search method. The primary application of this algorithm is the design of a fractional-order proportional–integral–derivative (FOPID) controller for a buck converter system. A comprehensive evaluation is conducted using statistical boxplot analysis, nonparametric statistical tests and convergence response comparisons to assess the algorithm's performance and confirm its superiority over CSA. Furthermore, the FOPID-controlled buck converter system based on OCSANM is compared with two top-performing algorithms: one using a hybridized approach of Lévy flight distribution with simulated annealing (LFDSA) and the other employing the improved hunger games search (IHGS) algorithm. This comparison encompasses transient and frequency responses, performance indices and robustness analysis. The results reveal the notable advantages of the proposed OCSANM-based system, including 25.8% and 8.7% faster rise times, 26% and 8.8% faster settling times compared with the best-performing approaches, namely LFDSA and IHGS, respectively. In addition, the OCSANM-based system exhibits a 34.7% and 9.6% wider bandwidth than the existing approaches-based systems. Incorporating voltage and current responses of the buck converter's switched circuit with the OCSANM-based FOPID controller further underscores the algorithm's effectiveness. To provide a comprehensive assessment, the paper also compares the proposed approach's time and frequency domain responses with those of 17 other state-of-the-art approaches attempting to control buck converter systems similarly. These findings affirm the effectiveness of the OCSANM in designing FOPID controllers for buck converter systems. [ABSTRACT FROM AUTHOR]

To address the anti-swing issue of the payload in bridge cranes, Proportional–Integral–Derivative (PID) control is a commonly used method. However, parameter tuning of the PID controller relies on empirical knowledge and often leads to system overshoot. This paper proposes an Improved Sparrow Search Algorithm (ISSA) to optimize the gains of PID controllers, alleviating adverse effects on payload oscillation and trolley positioning during the operation of overhead cranes. First, tent map chaos mapping is introduced to initialize the sparrow population, enhancing the algorithm's global search capability. Then, by integrating sine and cosine concepts along with nonlinear learning factors, the updating mechanism of discoverer positions is dynamically adjusted, expediting the solving process. Finally, the Lévy flight strategy is employed to update follower positions, thereby enhancing the algorithm's local escape capability. Additionally, a fitness function containing overshoot penalties is proposed to address overshoot issues. Simulation results indicate that the overshoot rates of all algorithms remain less than 3%. Moreover, compared with the Sparrow Search Algorithm (SSA), Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Whale optimization Algorithm (WOA), the optimized PID control system with the ISSA algorithm exhibits superior control performance and possesses certain robustness and adaptability. [ABSTRACT FROM AUTHOR]

We developed a new method to price double barrier options under pure non-Gaussian Lévy processes admitting jumps of unbounded variation. In our approach, we represent the underlying process as a sequence of spectrally positive, negative, and again positive jumps, with spectrally positive jumps corresponding to half of the time moment. This rule is applied to the increments of the Lévy process at exponentially distributed time points. The justification for the convergence of the method in time is carried out using the interpretation of time randomization as the numerical Laplace transform inversion with the Post-Widder formula. The main algorithm consists of the recurrent calculation of the sufficient simple expectations of the intermediate price function depending on the position of the extremum of spectrally positive or negative part of the underying Lévy process at a randomized time moment. It corresponds to a sequence of problems for integro-differential equations on an interval, each of which is solved semi-explicitly by the Wiener–Hopf method. We decompose the characteristic function of a randomized spectrally positive (negative) process as the product of two Wiener–Hopf factors, the first of which corresponds to the exponential distribution and the second is calculated as a ratio. The parameter of the exponential distribution is numerically found as the only root of the factorizing operator symbol using Newton's method. Thus, by combining the sequential application of the Wiener–Hopf operators in explicit form with the characteristic function of our interval, we obtain the solution to each auxiliary problem. The main advantage of the suggested methods is that being very simple for programming it makes it possible to avoid dealing with Wiener–Hopf matrix factorization or solving systems of coupled nontrivial Wiener–Hopf equations that require application of tricky approximate factorization formulas. [ABSTRACT FROM AUTHOR]

Path planning is an important research direction in the field of robotics; however, with the advancement of modern science and technology, the study of efficient, stable, and safe path-planning technology has become a realistic need in the field of robotics research. This paper introduces an improved sparrow search algorithm (ISSA) with a fusion strategy to further improve the ability to solve challenging tasks. First, the sparrow population is initialized using circle chaotic mapping to enhance diversity. Second, the location update formula of the northern goshawk is used in the exploration phase to replace the sparrow search algorithm's location update formula in the security situation. This improves the discoverer model's search breadth in the solution space and optimizes the problem-solving efficiency. Third, the algorithm adopts the Lévy flight strategy to improve the global optimization ability, so that the sparrow jumps out of the local optimum in the later stage of iteration. Finally, the adaptive T-distribution mutation strategy enhances the local exploration ability in late iterations, thus improving the sparrow search algorithm's convergence speed. This was applied to the CEC2021 function set and compared with other standard intelligent optimization algorithms to test its performance. In addition, the ISSA was implemented in the path-planning problem of mobile robots. The comparative study shows that the proposed algorithm is superior to the SSA in terms of path length, running time, path optimality, and stability. The results show that the proposed method is more effective, robust, and feasible in mobile robot path planning. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of stochastic Riemann-Liouville type fractional differential equations driven by Lévy noise. By using Itô formula for the considered equation, we attempt to explore the non-confluence property of solution for the considered equation under some appropriate conditions. Our approach is to construct some suitable Lyapunov functions which is novel in exploring the non-confluence property of differential equations. [ABSTRACT FROM AUTHOR]

This work investigates the emergence of structural super-diffusion in finite multiplex networks, focusing on situations where long-range interactions (LRIs) are present in at least one of the layers. Employing the Lévy random walk model, we explore how the likelihood of observing LRIs and the strength of the coupling among layers, specifically through the inter-layer diffusion coefficient, affect the relaxation time of the entire multiplex network. Our aim is to determine if this collective relaxation time is shorter compared to that in isolated layer configurations. We quantify the relaxation times through algebraic connectivity, the second-smallest eigenvalue of the network's (Supra-)Laplacian matrix. The formalism is adapted to scenarios where long-range jumps may exist by considering Mellin or Laplace transforms. We study samples of multiplexes whose layers are obtained from the models by Erdös-Rény, Watts-Strogatz and Barabási-Albert, and discuss in great detail the results for multiplexes with two layers. Besides the rather common enhancement of multiplex diffusion in comparison to the diffusion observed in their isolated layers, our results show that the timescale of layers with LRIs may be still faster than that due only to the usual multiplex network, speeding-up the whole system's diffusion. In addition, we also provide enough evidences that a novel linear diffusion regime emerges with strong inter-layer coupling and the presence of LRIs in one or more layers. Graphical Abstract highlights the main focus of the work, the occurrence of superdiffusion in a multiplex with M = 2 layers. The connectivity ratio η = Λ 2 / max (λ 2 1 , λ 2 2) (red line) crosses the dashed line η = 1 at a well defined value of D x , and remains in the super-diffusion region highlighted in green [ABSTRACT FROM AUTHOR]

In this paper, we study McKean–Vlasov stochastic differential equations driven by Lévy processes. Firstly, under the non-Lipschitz condition which include classical Lipschitz conditions as special cases, we establish the existence and uniqueness for solutions of McKean–Vlasov stochastic differential equations using Carathéodory approximation. Then under certain averaging conditions, we establish a stochastic averaging principle for McKean–Vlasov stochastic differential equations driven by Lévy processes. We find that the solutions to stochastic systems concerned with Lévy noise can be approximated by solutions to averaged McKean–Vlasov stochastic differential equations driven by Lévy processes in the sense of convergence in p th moment. [ABSTRACT FROM AUTHOR]

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting that goes back to Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional CLT by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a linearization trick and the univariate second-order Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality. [ABSTRACT FROM AUTHOR]

We construct an additive functional playing the role of the local time—at a fixed point x—for Markov processes indexed by Lévy trees. We start by proving that Markov processes indexed by Lévy trees satisfy a special Markov property which can be thought as a spatial version of the classical Markov property. Then, we construct our additive functional by an approximation procedure and we characterize the support of its Lebesgue-Stieltjes measure. We also give an equivalent construction in terms of a special family of exit local times. Finally, combining these results, we show that the points at which the Markov process takes the value x encode a new Lévy tree and we construct explicitly its height process. In particular, we recover a recent result of Le Gall concerning the subordinate tree of the Brownian tree where the subordination function is given by the past maximum process of Brownian motion indexed by the Brownian tree. [ABSTRACT FROM AUTHOR]

This work proposes a novel stochastic model describing the propagation dynamics of the hepatitis B virus. The model takes into account numerous disease characteristics, and environmental disturbances were collected using Lévy jumps and the conventional Brownian motions. Initially, the deterministic model is developed, and the asymptotic behavior of the model's solution near the equilibria is examined. The deterministic model is transformed into a stochastic model while retaining the Lévy jumps and conventional Brownian motions. Under specific assumptions, the stochastic system is shown to have a unique solution. The study further investigates the conditions that ensure the extinction and persistence of the infection. The numerical solutions to both stochastic and deterministic systems were obtained using the well-known Milstein and RK4 techniques, and the analytical findings are theoretically confirmed. The simulation suggests that the noise intensities have a direct relationship with the amplitudes of the stochastic curves around the equilibria of the deterministic system. Smaller values of the intensities imply negligible fluctuations of trajectories around the equilibria and, hence, better describe the extinction and persistence of the infection. It has also been found that both Brownian motions and the Lévy jump had a significant influence on the oscillations of these curves. A discussion of the findings of the study reveals other important aspects as well as some future research guidelines. In short, this study proposes a novel stochastic model to describe the propagation dynamics of the hepatitis B virus. [ABSTRACT FROM AUTHOR]

We study the weak convergence of the extremes of supercritical branching Lévy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are Lévy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$. [ABSTRACT FROM AUTHOR]

Temperature and humidity as the key factors affecting the storage and ripening of fruits and vegetables directly determine the quality of fruits and vegetables. In this paper, a temperature and humidity model was constructed based on an integrated device of controlled atmosphere storage and ripening. Aiming at the shortcomings of the temperature and humidity model such as large inertia, nonlinearity, and model uncertainty, a fuzzy‐PID controller based on the improved Lévy flight particle swarm algorithm (LFPSO) is proposed. Initially, a mathematical model of temperature and humidity is developed through mechanism research and parameter estimation. Subsequently, a temperature and humidity fuzzy‐PID control strategy is proposed for regulating temperature and humidity. An improved LFPSO is then introduced to optimize the key parameters of the fuzzy‐PID controller, such as the quantization factor and the scale factor. The superiority of the improved LFPSO algorithm is verified by comparing the test functions. Finally, the improved LFPSO‐fuzzy‐PID controller, fuzzy‐PID controller, and Smith‐PID controller are applied to the simulation model for comparison using the MATLAB Simulink simulation platform. The results show that the improved LFPSO‐fuzzy‐PID control algorithm has a good control effect in the temperature and humidity simulation system of controlled atmosphere (CA) ripening container, which provides a reference for solving the optimization problem of actual engineering design. Practical applications: In order to reduce the postharvest cold chain losses of fruits and vegetables, this paper proposes a containerized style device that combines fruits and vegetables storage and ripening for simplifying the losses caused by transshipment at cold chain nodes. Since temperature and humidity play a key role in fruits and vegetables storage and ripening, which directly affect the product quality, this paper establishes a complete mathematical model of CA ripening temperature and humidity system by combining mechanism modeling and parameter identification. In order to keep the temperature and humidity within the ideal range, Smith‐PID controller, fuzzy‐PID controller, and fuzzy‐PID control method based on improved Lévy flight particle swarm optimization are proposed to regulate the air conditioner and humidifier. The performance of three different types of controllers was tested on the MATLAB. The simulation results demonstrate that the improved LFPSO‐fuzzy‐PID controller is superior and more effective than Smith‐PID and fuzzy‐PID controllers. [ABSTRACT FROM AUTHOR]

The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes. [ABSTRACT FROM AUTHOR]

Dispersion curve inversion is one of the core contents of Rayleigh wave data processing. However, the dispersion curve inversion has the characteristics of multi‐parameter, multi‐extremum as well as nonlinearity. In the face of Rayleigh wave data processing under complex seismic‐geological conditions, it is difficult to reconstruct an underground structure quickly and accurately apply a single global‐searching non‐linear inversion algorithm. For this reason, we proposed a strategy to invert multi‐order mode Rayleigh wave dispersion curves by combining with grey wolf optimization (GWO) and cuckoo search (CS) algorithms. On the basis of introducing the mechanism of iterative chaotic map with infinite collapses (ICMIC) and the strategy of dimension learning–based hunting (DLH), an improved GWO was developed that was called IDGWO (ICMIC and DLH GWO). After searching the near‐optimal region through IDGWO, the CS with a variable step‐size Lévy flight search mechanism was switched adaptively to complete the final inversion. The correctness of our method was verified by the multi‐order mode dispersion curve inversion of a six‐layer high‐velocity interlayer model. Then it was further applied to the processing of real seismic datasets. The research results show that our method fully utilizes the advantages of each of the two global‐searching non‐linear algorithms after integrating IDGWO and CS, while effectively balancing the ability between global search and local exploitation, further improving the convergence speed and inversion accuracy and having good anti‐noise performance. [ABSTRACT FROM AUTHOR]

Introduction: Eye movement is one of the cues used in human--machine interface technologies for predicting the intention of users. The developing application in eye movement event detection is the creation of assistive technologies for paralyzed patients. However, developing an effective classifier is one of the main issues in eye movement event detection. Methods: In this paper, bidirectional long short-term memory (BILSTM) is proposed along with hyperparameter tuning for achieving effective eye movement event classification. The Lévy flight and interactive crossoverbased reptile search algorithm (LICRSA) is used for optimizing the hyperparameters of BILSTM. The issues related to overfitting are avoided by using fuzzy data augmentation (FDA), and a deep neural network, namely, VGG- 19, is used for extracting features from eye movements. Therefore, the optimization of hyperparameters using LICRSA enhances the classification of eye movement events using BILSTM. Results and Discussion: The proposed BILSTM--LICRSA is evaluated by using accuracy, precision, sensitivity, F1-score, area under the receiver operating characteristic (AUROC) curve measure, and area under the precision--recall curve (AUPRC) measure for four datasets, namely, Lund2013, collected dataset, GazeBaseR, and UTMultiView. The gazeNet, human manual classification (HMC), and multi-source information-embedded approach (MSIEA) are used for comparison with the BILSTM--LICRSA. The F1-score of BILSTM--LICRSA for the GazeBaseR dataset is 98.99%, which is higher than that of the MSIEA. [ABSTRACT FROM AUTHOR]

The escalation of cloud services, driven by their accessibility, improved performance, and cost-effectiveness, has led cloud service providers to consistently seek methods to expedite job completion, thereby boosting profits and reducing energy consumption expenses. Despite developing numerous scheduling algorithms, many of these techniques address only a specific objective within the scheduling process. To efficiently achieve better optimization results for the cloud task scheduling problem, a novel approach, the Enhanced Whale Optimization Algorithm (EWOA), is proposed. EWOA integrates the WOA with the Lévy flight. The incorporation of Lévy flight is tailored to broaden the search space of WOA, expediting convergence with adaptive crossover. The EWOA model is simulated using the Cloudsim tool and evaluated under diverse test conditions. The effectiveness of EWOA is assessed by employing various parameters and comparing them with existing algorithms. The results demonstrate that EWOA outperforms other algorithms in resource utilization, energy consumption, and execution cost, establishing its superiority in addressing the complexities of multi-objective cloud task scheduling. [ABSTRACT FROM AUTHOR]

Underground coal seam mining significantly reduces the stability of slopes, especially soil slopes, and an accurate evaluation of the stability of soil slopes under underground mining conditions is crucial for mining safety. In this study, the impact of coal seam mining is considered as the additional horizontal and vertical stresses acting on the slope, and an equation for calculating the safety factor of soil slopes under underground mining conditions is derived based on the rigorous Janbu method. Then, the Improved Radial Movement Optimization (IRMO) algorithm is introduced and combined with Lévy flight optimization to conduct global optimization searches, obtaining the critical sliding surface and corresponding safety factor of the soil slope under underground coal seam mining. Through comparisons with the numerical simulation results in three different case studies, the feasibility of applying the IRMO algorithm with Lévy flight to analyze the stability of soil slopes under underground mining is demonstrated. This ensures the accuracy and stability of the calculation results while maintaining a high convergence efficiency. Furthermore, the effects of the mining thickness and mining direction on slope stability are analyzed, and the results indicate that a smaller mining thickness and mining along the slope are advantageous for slope stability. The method proposed in this study provides valuable insights for preventing the slope instability hazards caused by underground coal seam mining. [ABSTRACT FROM AUTHOR]

The precise mathematical description of gaze patterns remains a topic of ongoing debate, impacting the practical analysis of eye-tracking data. In this context, we present evidence supporting the appropriateness of a Lévy flight description for eye-gaze trajectories, emphasizing its beneficial scale-invariant properties. Our study focuses on utilizing these properties to aid in diagnosing Attention-Deficit and Hyperactivity Disorder (ADHD) in children, in conjunction with standard cognitive tests. Using this method, we found that the distribution of the characteristic exponent of Lévy flights statistically is different in children with ADHD. Furthermore, we observed that these children deviate from a strategy that is considered optimal for searching processes, in contrast to non-ADHD children. We focused on the case where both eye-tracking data and data from a cognitive test are present and show that the study of gaze patterns in children with ADHD can help in identifying this condition. Since eye-tracking data can be gathered during cognitive tests without needing extra time-consuming specific tasks, we argue that it is in a prime position to provide assistance in the arduous task of diagnosing ADHD. [ABSTRACT FROM AUTHOR]

Patu digua search (BKS) optimization algorithm, Quasi-Oppositional based Teacher Actions in the Society - inspired Optimization (SGS) Algorithm, Quantum Monastery chief selection Inspired Optimization (QRMCSO) Algorithm are designed and applied to solve the Electrical loss reduction problem. Patu digua search optimization algorithm is designed based on the stalking deeds of Patu digua. The stalking approaches are deliberated by oppression, examination, and hurdling on the prey. Lévy flights are engaged as an ancillary of persistent irrational engagements to emulate the fudging performance. Chaotic sequences are integrated in Patu digua search optimization algorithm. In Quasi-Oppositional based Teacher Actions in the Society - inspired Optimization Algorithm, exploration region is envisaged as a synchronize organization with a quantity of axes equivalent to the decision parameters. Quasi-Opposition approach integrated into the approach to enhance the search and quality of the solution. In QRMCSO, before picking a successor Head of the monastery will do various analyses and these standards are imitated in the design through Chebyshev serviceable connexion system. [ABSTRACT FROM AUTHOR]

We develop a stochastic volatility framework for modeling multiple currencies based on CBI-time-changed Lévy processes. The proposed framework captures the typical risk characteristics of FX markets and is coherent with the symmetries of FX rates. Moreover, due to the self-exciting behavior of CBI processes, the volatilities of FX rates exhibit self-exciting dynamics. By relying on the theory of affine processes, we show that our approach is analytically tractable and that the model structure is invariant under a suitable class of risk-neutral measures. A semi-closed pricing formula for currency options is obtained by Fourier methods. We propose two calibration methods, also by relying on deep-learning techniques, and show that a simple specification of the model can achieve a good fit to market data on a currency triangle. [ABSTRACT FROM AUTHOR]

We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional $ L^p $ L p -Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes. [ABSTRACT FROM AUTHOR]

The beetle antennae search algorithm (BAS) is inspired by the search behavior of long-horned beetles and is widely used in optimization problems. However, in the typical BAS algorithm, there are problems such as the insufficient optimization ability of beetle individuals and the lack of consideration of historical dynamic information in the algorithm search process, resulting in sufferings from local optimum, low diversity, and imbalanced utilization, etc. To solve these problems, an improved beetle swarm antennae search algorithm based on multiple operators, dubbed MO-BSAS, is proposed. First, the elite opposition-based learning is used to initialize the beetle population, which improves the population diversity and optimization ability of the algorithm; then, the Lévy Flight strategy is used to improve the traditional beetle moving operator. In addition, a multi-operator search strategy determined by the damped sinusoidal probability factor is designed to balance the exploitation and exploration capabilities of the algorithm and speed up its convergence. The MO-BSAS algorithm is compared with BSA and 8 popular or state-of-the-art smart algorithms under 14 different benchmark functions, and the experimental results show that the MO-BSAS algorithm has a shorter convergence time and better convergence accuracy than the algorithms compared, which verifies the effectiveness of the improved algorithm. [ABSTRACT FROM AUTHOR]

Complex-valued random variables that satisfy usual stability condition for a complex parameter α such that |α − 1/2| < 1/2, are constructed. The characteristic function of the obtained random variables is found and limit theorems for sums of independent identically distributed random variables are proved. The corresponding Lévy processes and semigroups of operators corresponding to these processes are constructed. [ABSTRACT FROM AUTHOR]

A class of Lévy processes including symmetric α-stable processes for α ∈ (1, 2) is considered. An analog of local time for this class of processes is constructed and its properties are studied. It is shown that the constructed analog can be considered as a generalization of local time for a symmetric stable process. [ABSTRACT FROM AUTHOR]

First hitting times (FHTs) describe the time it takes a random "searcher" to find a "target" and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order s ε (0, 1) (describing a (2s)-stable Lévy flight whose squared displacement scales as t1/s in time t) and targets of radius \varepsilon \ll 1, we show that the MFHT is order one for s ε (1/2, 1) and diverges as log(1/\varepsilon) for s = 1/2 and \varepsilon 2s 1 for s ε (0, 1/2). We then use our asymptotic results to identify the value of s ε (0, 1] which minimizes the average MFHT and find that (a) this optimal value of s vanishes for sparse targets and (b) the value s = 1/2 (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations. [ABSTRACT FROM AUTHOR]

A system of n$$ n $$ components is here considered, with component deterioration modeled by non decreasing time‐scaled Lévy processes. When a component fails, a sudden change in the time‐scaling functions of the surviving components is induced, which makes the components stochastically dependent. We compute the reliability function of coherent systems under this new dependence model. We next study the distribution of the ordered failure times, and establish some positive dependence properties. We also provide stochastic comparison results in the usual multivariate stochastic order between failure times of two dependence models with different parameters. Finally, some numerical experiments illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Univariate Lévy processes have become quite common in the reliability literature for modeling accumulative deterioration. In case of correlated deterioration indicators, several possibilities have been suggested for modeling their dependence. The point of this article is the study of three different dependence models: use of a regular copula, superposition of independent univariate Lévy processes and use of a Lévy copula. The three methods are first presented and analyzed. In this way, it is shown that the multivariate process constructed through an ordinary copula cannot have independent increments in general, that is, it is not a Lévy process. The impact of a wrong choice for the model is next explored, based on data simulated from one model and next adjusted to all three models. It is shown that a wrong model can lead to either overestimate or underestimate the reliability function, which could be problematic in an application context. [ABSTRACT FROM AUTHOR]

We propose a doubly subordinated Lévy process, the normal double inverse Gaussian (NDIG), to model the time series properties of the cryptocurrency bitcoin. By using two subordinated processes, NDIG captures both the skew and fat-tailed properties of, as well as the intrinsic time driving, bitcoin returns and gives rise to an arbitrage-free option pricing model. In this framework, we derive two bitcoin volatility measures. The first combines NDIG option pricing with the Chicago Board Options Exchange VIX model to compute an implied volatility; the second uses the volatility of the unit time increment of the NDIG model. Both volatility measures are compared to the volatility based on the historical standard deviation. With appropriate linear scaling, the NDIG process perfectly captures the observed in-sample volatility. [ABSTRACT FROM AUTHOR]

In order to overcome the difficulty of parameter tuning caused by the large lag and time-varying nonlinearity of the tank for lysozyme fermentation, a temperature control method based on LRWOA-LSTM-PID is proposed in this paper. Firstly, according to the intrinsic mechanism of the fermenter, a temperature mechanism model based on a dynamic equation is designed, which can better reflect the temperature changes in the fermenter. Secondly, a Proportional Integral Derivative (PID) parameter tuning method based on a Long-Short Term Memory Network (LSTM) is proposed, which takes advantage of the ability of LSTM to learn time sequence information and obtains the variation trend between error sequences under continuous time sampling, thereby adjusting network weights more reasonably and accelerating PID parameter tuning. Finally, a Whale Optimization Algorithm (WOA) based on the Lévy flight and random walk strategy (LRWOA) is proposed for the initialization of LSTM parameters; this algorithm has excellent optimization capabilities and overcomes the problem of LSTM falling into local optimal solutions prematurely during parameter randomization. The results show that the method proposed in this paper can achieve rapid tuning of PID parameters, thereby improving the convergence speed of the system and reducing system overshoot. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic two-species Schoener's competitive model with Lévy jumps by the mean-reverting Ornstein-Uhlenbeck process. First, the biological implication of introducing the Ornstein-Uhlenbeck process is illustrated. After that, we show the existence and uniqueness of the global solution. Moment estimates for the global solution of the stochastic model are then given. Moreover, by constructing the Lyapunov function and applying Itô's formula and Chebyshev's inequality, it is found that the model is stochastic and ultimately bounded. In addition, we give sufficient conditions for the extinction of species. Finally, numerical simulations are employed to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

The route and speed of migration into Sahul by Homo sapiens remain a major research question in archaeology. Here, we introduce an approach which models the impact of the physical environment on human mobility by combining time-evolving landscapes with Lévy walk foraging patterns, this latter accounting for a combination of short-distance steps and occasional longer moves that hunter-gatherers likely utilised for efficient exploration of new environments. Our results suggest a wave of dispersal radiating across Sahul following riverine corridors and coastlines. Estimated migration speeds, based on archaeological sites and predicted travelled distances, fall within previously reported range from Sahul and other regions. From our mechanistic movement simulations, we then analyse the likelihood of archaeological sites and highlight areas in Australia that hold archaeological potential. Our approach complements existing methods and provides interesting perspectives on the Pleistocene archaeology of Sahul that could be applied to other regions around the world. The speed and route by which Homo sapiens colonised Sahul is an ongoing topic of research. Here, the authors model the physical environment as it changes through time in combination with Lévy walk foraging patterns to suggest a wave of dispersal following coastlines and rivers. [ABSTRACT FROM AUTHOR]

Globally, traffic accidents on the highway network contribute significantly to a high fatality rate, drawing considerable attention from health institutions. The efficiency of transportation plays a vital role in mitigating the severe consequences of these incidents. This study delves into the issues of emergency vehicles experiencing delays despite having priority. Therefore, we construct mixed-integer linear programming with semi-soft time windows (MIPSSTW) model for optimizing emergency vehicle routing in highway incidents. We analyze the time-varying and complex traffic situations and respectively propose corresponding estimation approaches for the travel time of road segments, intersections on the urban road network, and ramp-weave sections on the highway network. Furthermore, we developed a modified cuckoo search(MCS) algorithm to solve this combinatorial problem. Optimization strategies of Lévy flight and dynamic inertial weight strategy are introduced to strengthen the exploration capability and the diversity of solution space of the CS algorithm. Computational experiments based on the Chinese emergency medical system data are designed to validate the efficacy and effectiveness of the MIPSSTW model and MCS algorithm. The results show that our works succeed in searching for high-quality solutions for emergency vehicle routing problems and enhance the efficacy of strategic decision-making processes in the realm of incident management and emergency response systems. [ABSTRACT FROM AUTHOR]

This paper presents an enhanced whale optimization algorithm based on virtual force to optimize coverage and address the problem of uneven coverage during the deployment of underwater sensor networks. This method is guided by the Lévy flight and virtual force algorithms and adopts a differential mutation strategy based on random walks. This improves the fitness of the initial population and population richness of the algorithm. The effectiveness of this method was verified through simulation. [ABSTRACT FROM AUTHOR]