Search Results (624)

This paper introduces a new nonparametric estimator for detecting the conditional mode in the functional input variable setting. The estimator integrates a local linear approach with an $L^1$-robust algorithm and treats the modal regression as the minimizer of the quantile derivative. As an asymptotic result, we derive the theoretical properties of the estimator by analyzing its convergence rate under the almost complete consistency framework. The result is stated under standard conditions, characterizing both the functional structure of the data and the local linear approximation properties of the model. Moreover, the expression of the convergence rate retains the usual form of the stochastic convergence rate in functional statistics. Simulations and real-data applications demonstrate the algorithm’s effectiveness, showing its advantage over existing methods in high-dimensional prediction tasks. Full article

This study presents a reinforcement learning–based approach to optimize replenishment policies in the presence of uncertainty, with the objective of minimizing total costs, including inventory holding, shortage, and ordering costs. The focus is on single-level assembly systems, where both component delivery lead times and finished product demand are subject to randomness. The problem is formulated as a Markov decision process (MDP), in which an agent determines optimal order quantities for each component by accounting for stochastic lead times and demand variability. The Deep Q-Network (DQN) algorithm is adapted and employed to learn optimal replenishment policies over a fixed planning horizon. To enhance learning performance, we develop a tailored simulation environment that captures multi-component interactions, random lead times, and variable demand, along with a modular and realistic cost structure. The environment enables dynamic state transitions, lead time sampling, and flexible order reception modeling, providing a high-fidelity training ground for the agent. To further improve convergence and policy quality, we incorporate local search mechanisms and multiple action space discretizations per component. Simulation results show that the proposed method converges to stable ordering policies after approximately 100 episodes. The agent achieves an average service level of 96.93%, and stockout events are reduced by over 100% relative to early training phases. The system maintains component inventories within operationally feasible ranges, and cost components—holding, shortage, and ordering—are consistently minimized across 500 training episodes. These findings highlight the potential of deep reinforcement learning as a data-driven and adaptive approach to inventory management in complex and uncertain supply chains. Full article

The prediction of ground deformation during ultra-shallow-buried pilot tunnel construction is critical for urban rail transit projects in complex geological settings, yet existing cross-section models often lack accuracy. This study proposes an enhanced non-uniform convergence model based on stochastic medium theory, which decomposes surface settlement into uniform soil shrinkage and non-uniform initial support deformation. A computational formula for horseshoe-shaped sections is derived and validated through field data from Kunming Rail Transit Phase I, demonstrating a 59% improvement in maximum settlement prediction accuracy (reducing error from 7.5 mm to 3.1 mm) compared to traditional methods. Its application to Beijing Metro Line 13 reveals two distinct deformation patterns: significant ground heave occurs at 2.5 times the tunnel width from the centerline, while maximum settlement concentrates above the excavation center and diminishes radially. To mitigate heave, early strengthening of the secondary lining is recommended to control initial horizontal deformation. These findings enhance prediction reliability and provide actionable insights for deformation control in similar urban tunneling projects, particularly under ultra-shallow burial conditions. Full article

In some problems, partial differential equations are reduced to ordinary differential equations. In special cases, when incorporating randomness, equations can be reduced to systems of stochastic differential Equations (SDEs). Stochastic averaging for a class of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise is considered. Stability criteria for systems of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise do not currently exist. Usually, studies on determining the sensitivity of solutions to the accuracy of setting the initial conditions are being conducted to explain the phenomenon of deterministic chaos. These studies show both convergence in mean square and convergence in probability to averaged systems of stochastic differential equations driven by fractional Brownian motion and Lévy process. The solutions to systems can be approximated by solutions to averaged stochastic differential equations by using the stochastic averaging. Full article

This study presents an innovative approach to quantifying uncertainty in Herschel–Bulkley (H-B) fluid flow through rectangular ducts, analyzing four scenarios: uncertain apparent viscosity (Case I), uncertain pressure gradient (Case II), uncertain boundary conditions (Case III) and uncertain apparent viscosity and pressure gradient (Case IV). Using the stochastic finite difference with homogeneous chaos (SFDHC) method, we produce probability density functions (PDFs) of fluid velocity with exceptional computational efficiency (243 times faster), matching the accuracy of Monte Carlo simulation (MCS). Key statistics and maximum velocity PDFs are tabulated and visualized for each case. Mean velocity shows minimal variation in Cases I, III, and IV, but maximum velocity fluctuates significantly in Case I (63.95–187.45% of mean), Case II (50.15–156.68%), and Case IV (63.70–185.53% of mean), vital for duct design and analysis. Examining the effects of different parameters, the SFDHC method’s rapid convergence reveals the fluid behavior index as the primary driver of maximum stochastic velocity, followed by aspect ratio and yield stress. These findings enhance applications in drilling fluid management, biomedical modeling (e.g., blood flow in vascular networks), and industrial processes involving non-Newtonian fluids, such as paints and slurries, providing a robust tool for advancing understanding and managing uncertainty in complex fluid dynamics. Full article

Markov chain perturbation theory is a rapidly developing subfield of the theory of stochastic processes. This review outlines emerging applications of this theory in the analysis of stochastic models of chemical reactions, with a particular focus on biochemistry and molecular biology. We begin by discussing the general problem of approximate modeling in stochastic chemical kinetics. We then briefly review some essential mathematical results pertaining to perturbation bounds for continuous-time Markov chains, emphasizing the relationship between robustness under perturbations and the rate of exponential convergence to the stationary distribution. We illustrate the use of these results to analyze stochastic models of biochemical reactions by providing concrete examples. Particular attention is given to fundamental problems related to approximation accuracy in model reduction. These include the partial thermodynamic limit, the irreversible-reaction limit, parametric uncertainty analysis, and model reduction for linear reaction networks. We conclude by discussing generalizations and future developments of these methodologies, such as the need for time-inhomogeneous Markov models. Full article

Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria ($W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$, e.g., $W^{*}\approx 1.40$ under typical parameters $\eta =0.5$, $\delta =0.3$). We further derive closed-form expressions for the steady-state variance under stochastic perturbations ($\mathrm{Var}\left(W_{ij}\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$) and demonstrate a less than 6% deviation between simulated and theoretical values when ${\sigma }_{\zeta }=0.1$. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for $\mu =0.5$. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research. Full article

The Crayfish Optimization Algorithm (COA) has limitations that affect its optimization performance seriously. The competition stage of the COA uses a simplified mathematical model that concentrates on relations of distance between crayfish only. It is deprived of a stochastic variable and is not able to generate an applicable balance between exploration and exploitation. Such a case causes the COA to have early convergence, to perform poorly in high-dimensional problems, and to be trapped by local minima. Moreover, the low activation probability of the summer resort stage decreases the exploration ability more and slows down the speed of convergence. In order to compensate these shortcomings, this study proposes an Improved Crayfish Optimization Algorithm (ICOA) that designs the competition stage with three modifications: (1) adaptive step length mechanism inversely proportional to the number of iterations, which enables exploration in early iterations and exploitation in later stages, (2) vector mapping that increases stochastic behavior and improves efficiency in high-dimensional spaces, (3) removing the X_shade parameter in order to abstain from early convergence. The proposed ICOA is compared to 12 recent meta-heuristic algorithms by using the CEC-2014 benchmark set (30 functions, 10 and 30 dimensions), five engineering design problems, and a real-world ROAS optimization case. Wilcoxon Signed-Rank Test, t-test, and Friedman rank indicate the high performance of the ICOA as it solves 24 of the 30 benchmark functions successfully. In engineering applications, the ICOA achieved an optimal weight (1.339965 kg) in cantilever beam design, a maximum load capacity (85,547.81 N) in rolling element bearing design, and the highest performance (144.601) in ROAS optimization. The superior performance of the ICOA compared to the COA is proven by the following quantitative data: 0.0007% weight reduction in cantilevers design (from 1.339974 kg to 1.339965 kg), 0.09% load capacity increase in bearing design (COA: 84,196.96 N, ICOA: 85,498.38 N average), 0.27% performance improvement in ROAS problem (COA: 144.072, ICOA: 144.601), and most importantly, there seems to be an overall performance improvement as the COA has a 4.13 average rank while the ICOA has 1.70 on CEC-2014 benchmark tests. Results indicate that the improved COA enhances exploration and successfully solves challenging problems, demonstrating its effectiveness in various optimization scenarios. Full article

This paper considers the adaptive fixed-time tracking control problem for stochastic systems subject to input saturation. Firstly, a smooth function approximation method is utilized to eliminate the effect of input saturation. Then, by combining the neural networks (NNs) approximation method with the backstepping-like technique, an adaptive fixed-time tracking control scheme is explicitly developed. The proposed scheme can ensure that the state variables are bounded in probability and the tracking error converges to a small region of the equilibrium point in a fixed time. Eventually, two numerical examples are given to indicate the performance and effectiveness of the presented strategy. Full article

This research proposes a fractional-order adaptive neural control scheme using an optimized backstepping (OB) approach to address strict-feedback nonlinear systems with uncertain control directions and predefined performance requirements. The OB framework integrates both fractional-order virtual and actual controllers to achieve global optimization, while a Nussbaum-type function is introduced to handle unknown control paths. To ensure convergence to desired accuracy within a prescribed time, a fractional-order dynamic-switching mechanism and a quartic-barrier Lyapunov function are employed. An input-to-state practically stable (ISpS) auxiliary signal is designed to mitigate unmodeled dynamics, leveraging classical lemmas adapted to fractional-order systems. The study further investigates a decentralized control scenario for large-scale stochastic nonlinear systems with uncertain dynamics, undefined control directions, and unmeasurable states. Fuzzy logic systems are employed to approximate unknown nonlinearities, while a fuzzy-phase observer is designed to estimate inaccessible states. The use of Nussbaum-type functions in decentralized architectures addresses uncertainties in control directions. A key novelty of this work lies in the combination of fractional-order adaptive control, fuzzy logic estimation, and Nussbaum-based decentralized backstepping to guarantee that all closed-loop signals remain bounded in probability. The proposed method ensures that system outputs converge to a small neighborhood around the origin, even under stochastic disturbances. The simulation results confirm the effectiveness and robustness of the proposed control strategy. Full article

The complementary dual of entropy has received significant attention in the literature. Due to the emergence of many generalizations and extensions of entropy, the need to generalize the complementary dual of uncertainty arose. This article develops the residual cumulative generalized fractional extropy as a generalization of the residual cumulative complementary dual of entropy. Many properties, including convergence, transformation, bounds, recurrence relations, and connections with other measures, are discussed. Moreover, the proposed measure’s order statistics and stochastic order are examined. Furthermore, the dynamic design of the measure, its properties, and its characterization are considered. Finally, nonparametric estimation via empirical residual cumulative generalized fractional extropy with an application to blood transfusion is performed. Full article

General aviation trajectory prediction plays a crucial role in enhancing safety and operational efficiency at non-towered airports. However, current research faces multiple challenges including variable weather conditions, complex aircraft interactions, and flight pattern constraints specified by general aviation regulations. This paper proposes a deep learning method based on stochastic processes aimed at addressing uncertainty issues in general aviation trajectory prediction. First, we design a probabilistic encoder–decoder structure enabling the model to output trajectory distributions rather than single paths, with regularization terms based on Lyapunov stability theory to ensure predicted trajectories maintain stable convergence while satisfying flight patterns. Second, we develop a multi-layer attention mechanism that accounts for weather factors, enhancing the model’s responsiveness to environmental changes. Validation using the TrajAir dataset from Pittsburgh-Butler Regional Airport (KBTP) not only advances deep learning applications in general aviation but also provides new insights for solving trajectory prediction problems. Full article

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results. Full article

Many dynamic systems experience unwanted actuation caused by an unknown exogenous input. Typically, when these exogenous inputs are stochastically bounded and a basis set cannot be identified, a Kalman-like estimator may suffice for state estimation, provided there is minimal uncertainty regarding the true system dynamics. However, such exogenous inputs can encompass environmental factors that constrain and influence system dynamics and overall performance. These environmental factors can modify the system’s internal interactions and constitutive constants. The proposed control scheme examines the case where the true system’s plant changes due to environmental or health factors while being actuated by stochastic variances. This approach updates the reference model by utilizing the input and output of the true system. Lyapunov stability analysis guarantees that both internal and external error states will converge to a neighborhood around zero asymptotically, provided the assumptions and constraints of the proof are satisfied. Full article

This paper introduces a functional framework for modeling cyclical and feedback-driven information flow using a generalized family of derangetropy operators. In contrast to scalar entropy measures such as Shannon entropy, these operators act directly on probability densities, providing a topographical representation of information across the support of the distribution. The proposed framework captures periodic and self-referential aspects of information evolution through functional transformations governed by nonlinear differential equations. When applied recursively, these operators induce a spectral diffusion process governed by the heat equation, with convergence toward a Gaussian characteristic function. This convergence result establishes an analytical foundation for describing the long-term dynamics of information under cyclic modulation. The framework thus offers new tools for analyzing the temporal evolution of information in systems characterized by periodic structure, stochastic feedback, and delayed interaction, with potential applications in artificial neural networks, communication theory, and non-equilibrium statistical mechanics. Full article