Mathematics doi: 10.3390/math14162879

Authors: Xinyan Zhang Xuebing Zhang

We study a reaction–diffusion model for population–toxin interactions on a two-dimensional torus, where avoidance is driven by a nonlocal average of toxin-related memory. The variables represent population density, toxin concentration, and a phenomenological information field generated by population–toxin encounters. This information diffuses, decays, and enters the taxis term through convolution with a perceptual kernel. We prove local well-posedness, global existence and uniform boundedness, and give sufficient conditions for exponential convergence to either the positive equilibrium or the toxin-only equilibrium. Numerical simulations with two perceptual radii yield stripe and spot patterns, illustrating that the sensing scale can alter spatial organization. These simulations do not establish an effect on population persistence and should be viewed as hypotheses for empirical study.

Mathematics doi: 10.3390/math14162878

Authors: Maha Mesfer Alghamdi

Multi-horizon forecasting of nonlinear dynamic time series with exogenous inputs is challenging when the target variable exhibits strong temporal persistence and the exogenous variables provide horizon-dependent corrective information. Direct forecasting models must learn both the carry-forward behavior of the target and the nonlinear deviations caused by changes in the process inputs. This study proposes a residual exogenous–autoregressive gated forecasting framework for nonlinear dynamic prediction. The proposed model decomposes the forecasting operator into a persistence component and a learnable residual correction term. Historical target dynamics and exogenous input dynamics are encoded using two dedicated CNN-LSTM branches, and their latent representations are combined through a sample-dependent sigmoid gating mechanism. The final prediction is obtained by adding the learned correction to the most recent target observation. The framework is evaluated on a benchmark sulfur recovery unit dataset for multi-horizon hydrogen sulfide H2S concentration forecasting using a leakage-aware nested blocked hyperparameter selection and evaluation protocol. Three forecasting horizons are considered: one-step, five-step, and ten-step ahead prediction. The proposed method achieved the lowest RMSE at the one-step and five-step horizons and remained highly competitive at the ten-step horizon, where its RMSE was nearly identical to the best PatchTST baseline. Across the three horizons, the proposed model obtained RMSE values of 0.0096±0.0020, 0.0436±0.0097, and 0.0521±0.0138, corresponding to RMSE reductions over the persistence baseline of 39.7%, 10.0%, and 13.7%, respectively. The model also maintained a compact parameter count and sub-millisecond inference latency, supporting its feasibility for online soft-sensing applications. Regression, time-series, error distribution, Taylor diagram, and SHAP analyses show that the residual gated formulation is particularly effective for short- and medium-horizon forecasting, while longer-horizon prediction remains more difficult because of increasing temporal uncertainty. The SHAP results indicate that historical H2S dominates short-horizon prediction, whereas airflow-related variables become more influential at the longer horizon. The results demonstrate that the proposed framework provides an interpretable and computationally compact learning approach for residual forecasting in persistent nonlinear dynamic systems.

Mathematics doi: 10.3390/math14162880

Authors: Ion Mihai Mohammed Mohammed Octavian Postavaru Andreea Olteanu

Kähler manifolds are the most studied complex manifolds. A Kähler manifold with constant holomorphic sectional curvature is said to be a complex space form. There are special classes of submanifolds in Kähler manifolds. Recently, Poyraz et al. defined partially totally real submanifolds in complex space forms. In the present paper, we study curvature invariants adapted to partially totally real submanifolds. First, we define a Riemannian invariant, denoted by δPTR(D) extending the definition of Chen’s CRδ-invariant and estimate it in terms of the squared mean curvature. Furthermore, we obtain Chen–Ricci inequalities that distinguish between tangent directions in the two distributions. Finally, we prove a Chen first inequality for totally real two-plane sections.

Mathematics doi: 10.3390/math14162877

Authors: Muhammed Resul Aydın Mehmet Emin Aydın

Facility location decisions directly affect operational costs, service quality, and customer allocation. However, minimising total cost may result in an imbalanced distribution of customers among open facilities, requiring both objectives to be considered simultaneously. This study proposes a novel non-dominated sorting adaptive binary artificial bee colony algorithm with adaptive operator selection, called NSABC, for the bi-objective uncapacitated facility location problem. The first objective minimises facility opening and customer assignment costs, while the second minimises customer allocation imbalance among open facilities. NSABC integrates Pareto-based archiving, smart initialisation, adaptive operator selection, and diversity-preservation mechanisms to generate high-quality and diverse trade-off solutions. Computational experiments on 15 OR-Library CAP benchmark instances evaluate the algorithms using Hypervolume and Inverted Generational Distance as complementary Pareto-front performance indicators, together with paired two-sided Wilcoxon signed-rank tests and Holm correction. NSABC achieves higher mean Hypervolume values on most instances and lower mean IGD values on 14 of the 15 instances. The statistical analysis significantly favours NSABC on 11 instances according to Hypervolume and on 9 instances according to IGD, whereas NSGA-III is significantly favoured on only one instance according to IGD. The performance advantages of NSABC were observed across benchmark instances of different sizes and scales, indicating its effectiveness under varying problem structures. These findings indicate that NSABC is a competitive and statistically supported alternative to NSGA-III for bi-objective facility location problems involving both economic efficiency and balanced customer distribution.

Mathematics doi: 10.3390/math14162876

Authors: Liuxin Chen Xing Wang

Digital and intelligent fresh-product supply chains increasingly rely on third-party logistics providers (TPLs) to record and disclose transport-process information. However, the TPL bears data-collection and digital-governance costs while capturing only part of the market value created by credible disclosure. This study develops a supplier-led Stackelberg game for a supplier–TPL–retailer supply chain. Contractual terms are negotiated before operation. Conditional on the negotiated contract, the supplier sets the wholesale price, the TPL selects the disclosure level, and the retailer determines the retail price. We derive decentralized equilibria under blockchain and non-blockchain regimes and compare cost-sharing and joint cost-sharing/revenue-sharing contracts. The results show that cost-sharing increases the TPL’s optimal disclosure level, but disclosure upgrades occur through discrete threshold jumps. Blockchain adoption depends jointly on fixed implementation costs and reliability improvements, and cost-sharing alone may not ensure both adoption and high-level disclosure. Introducing revenue-sharing allows the TPL to internalize part of the demand-side value generated by credible disclosure, leading to a Pareto-improving coordination interval for all supply-chain members. The findings provide a mathematical basis for designing incentive-compatible contracts for blockchain-enabled disclosure in digital fresh product supply chains.

Mathematics doi: 10.3390/math14162875

Authors: Jun Xian Ying Chen Lei Zhang Chengbin Xu

This paper investigates a nonlinear inverse problem in a time fractional diffusion-wave equation, in which a spatially varying potential coefficient is recovered from noisy final-time measurements. A local uniqueness result for the inverse problem is first established in a finite-dimensional admissible space. To support the reconstruction, the Fréchet derivative of the forward map and its adjoint representation are derived to provide the gradient information required in the inversion procedure. A Banach space regularization scheme with a combined L1 and L2 penalty is then proposed to stabilize the nonlinear inverse problem, and the resulting nonsmooth minimization problem is solved by a locally linearized split Bregman iterative scheme. Numerical experiments in one and two spatial dimensions demonstrate the accuracy and stability of the proposed method for smooth, corner-type, localized, and discontinuous coefficient profiles.

Mathematics doi: 10.3390/math14162874

Authors: Mustafa Canim

Text embeddings from retrieval-tuned (dual-encoder) models are increasingly used as context features in contextual bandits for recommendation, on the assumption that an embedding space optimized for inner-product similarity will speed up a linear exploration policy. This study tests that assumption with a controlled, shared-encoder design: the same BERT-base model in two forms, vanilla (mean-pooled) and retrieval-fine-tuned (MS MARCO dot-product), used as frozen bandit context. Experiments span three datasets (MIND, MovieLens-1M, Amazon CDs and Vinyl), two linear policies (LinUCB and linear Thompson sampling), and two dimensionality-reduction methods (PCA and random projection), over 20 seeds. Neither the training objective nor the reduction method determines performance on its own; cumulative regret is governed by their interaction, which is dataset-dependent. On MovieLens the reduction reverses the encoder ranking (PCA favors the retrieval-tuned encoder, random projection the vanilla one), with large effects in both directions, while on MIND the encoders are close. The reduction method can be the larger lever, moving up to 34% of the learnable margin. A ridge-regression probe on the candidate contexts screens the reduction choice offline, without running the bandit. For linear exploration policies, the encoder and the reduction should be treated as a joint choice rather than assuming retrieval-tuned embeddings are universally preferable.

Mathematics doi: 10.3390/math14162872

Authors: Yuwei Cai Zhihong Lin Zhongjian Gao Yao Li Wenxiu Dong

Traditional liquid-resistive mounts are widely used in vehicle powertrain vibration isolation because of their favorable low-frequency damping characteristics. However, they usually suffer from pronounced high-frequency dynamic stiffening, which significantly degrades their vibration-isolation performance in the high-frequency range and remains a critical limitation in passive mount design. To address this problem, this study presents the systematic modeling, comparative analysis, and structural optimization of liquid-resistive mounts with different internal configurations. Three representative mount structures, namely the decoupler-membrane type, the conventional bell-plate type, and a novel bell-plate configuration, are first described in terms of their structural characteristics and working mechanisms. Based on the lumped-parameter method, mathematical models of the three mounts are established, and their low- and high-frequency dynamic characteristics are comparatively investigated. The vibration isolation performance of the mounts is further evaluated under various excitation conditions to clarify the influence of structural modifications on the dynamic response and transmitted force. In addition, sensitivity analysis is performed using the ISIGHT software platform (ISIGHT 5.6 Design Gateway) to identify the key parameters governing high-frequency performance. Subsequently, structural optimization is conducted using nonlinear programming under the quadratic Lagrangian algorithm and the Six Sigma method. The results indicate that the introduction of a bell plate has little influence on the low-frequency dynamic characteristics, while it effectively suppresses high-frequency hardening and improves high-frequency vibration isolation. Moreover, the Six Sigma optimization method achieves better performance improvement than the NLPQL approach, providing a useful reference for the design and optimization of passive liquid-resistive mounts.

Mathematics doi: 10.3390/math14162873

Authors: Ru Xiao Jiyang Xu Jiabo Li

Accurate estimation of the state of health (SOH) of lithium ion batteries is a fundamental prerequisite for the safe and reliable operation of battery management systems. To address the issues of insufficient feature representativeness, manually dependent hyperparameter tuning, and limited generalization under small sample conditions in existing SOH estimation methods, this paper proposes a hybrid SOH estimation approach based on a convolutional neural network and bidirectional gated recurrent unit optimized by the RIME optimization algorithm. Unlike existing CNN GRU/LSTM models that rely on unidirectional recurrent structures and can only utilize forward temporal information, the proposed CNN-BiGRU architecture captures bidirectional contextual dependencies inherent in battery degradation, enabling more comprehensive characterization of aging dynamics from limited cycle data. Firstly, 13 health indicators (HIs) related to capacity degradation are extracted from the incremental capacity (IC) curves, and Spearman’s rank correlation coefficient is employed to select the optimal feature subset with the highest correlation. Secondly, a CNN BiGRU hybrid architecture is constructed, where CNN extracts local degradation features and BiGRU captures bidirectional temporal dependencies. More importantly, instead of relying on manual trial and error or grid search for hyperparameter determination, the RIME algorithm is introduced to automatically and globally optimize the core hyperparameters of the model. Finally, ablation and comparative experiments are conducted on the public NASA dataset. The results demonstrate that the proposed model significantly outperforms other comparison methods in terms of MAE, MAPE, and RMSE for three battery cells under both 80% and 60% training set ratios, confirming its comprehensive superiority in estimation accuracy, robustness, and generalization capability with limited samples.

Mathematics doi: 10.3390/math14162871

Authors: Mordecai Opoku Ohemeng Frederick T. Sheldon

Networked Cyber–Physical Systems (CPSs), like autonomous quadrotor swarms, tightly couple continuous physical kinematics, wireless information exchange, and discrete real-time task scheduling. While conventional consensus security architectures focus exclusively on data-layer falsification, they fundamentally decouple adversarial behavior from onboard computational resource state profiles. This paper addresses a core CPS vulnerability termed Computational Weaponization, the deliberate injection of complex computational workloads (adversarial LLM token parsing or cryptographic verification) to intentionally manipulate hardware execution delays. Through this exploit, strategic cyber–physical perturbations force resource-constrained embedded microcontrollers to saturate their task execution queues, inducing real-time scheduling starvation and physical tracking divergence. To mitigate this without optimization bottlenecks, we present a state-dependent, Self-Triggered Control (STC) and Prospect Theoretic Alignment (PTA) co-design framework. The proposed protocol models the hardware microprocessor’s execution delay as an endogenous dynamic state coupled directly to continuous tracking spaces. By mapping discrete topology reconfigurations and variable task delays to a switched impulsive time-delay system, we leverage an Input-to-State Stability (ISS) to derive sufficient linear matrix inequality conditions. We prove that the coupled cyber–physical–computational loop achieves asymptotic consensus and bounded trajectory containment under adversarial actions.

Mathematics doi: 10.3390/math14162870

Authors: Ayman A. Amin Fatimah E. Almuhayfith

A wide range of real-world multivariate time series encountered in practice exhibit two simultaneous and interacting seasonal cycles, for example hourly electricity demand, intraday financial prices, and sub-daily traffic volumes. Existing Bayesian frameworks for vector autoregressive (VAR) processes accommodate at most a single seasonal periodicity, leaving no established methodology for the double seasonal case commonly observed in high-frequency multivariate data. This paper bridges that gap by introducing the double seasonal VAR (DSVAR) models, which extend the univariate double seasonal literature to a coherent multivariate setting. These models are defined through a multiplicative triple autoregressive operator that naturally accommodates the second seasonal cycle. Under a Gaussian error assumption, we derive a comprehensive and analytically convenient Bayesian framework for both modeling and forecasting of DSVAR processes. We consider two prior families: a conjugate matrix normal-Wishart prior which yields exact closed-form inference, and a Jeffreys’ non-informative prior. Under each prior, we derive the marginal posterior distribution of the coefficient matrix as a matrix-t distribution and the marginal posterior of the precision matrix as a Wishart distribution. Moreover, we derive the predictive distribution of future observations as a multivariate-t with an exact analytic form, together with its highest predictive density regions. The methodology is validated through a Monte Carlo simulation experiment and applied to hourly electricity loads in Czech Republic and Germany, two physically interconnected markets with pronounced intraday and intraweek seasonal cycles. Benchmark comparisons against standard VAR, single-seasonal VAR, and univariate seasonal AR models confirm the substantial forecasting gains delivered by the proposed DSVAR framework at both short and long horizons.

Mathematics doi: 10.3390/math14162869

Authors: Ioannis G. Tsoulos Vasileios Charilogis Dimitrios Tsalikakis

A large collection of real-world classification and regression problems can be addressed using machine learning tools such as, for example, radial basis function networks (RBF networks). However, the techniques used for training RBF networks often exhibit various problems, such as getting trapped in the local minima of the error function, or even encountering numerical issues when solving systems of linear equations in order to estimate the parameters of the RBF network. This paper presents a multi-stage evolutionary technique based on genetic algorithms for the effective training of RBF networks. In the first stage, the value ranges of the RBF network parameters are estimated using the K-Means algorithm. In the second stage, the chromosomes of the genetic algorithm are initialized within the parameter ranges determined in the first stage, followed by the execution of the genetic algorithm. Each chromosome of the genetic algorithm is considered a candidate parameter vector for the machine learning model. The centers and variances of the RBF network are estimated by the genetic algorithm, while the network weights are determined by solving a system of linear equations. This method was applied to a large set of classification and data-fitting problems, yielding excellent results.

Mathematics doi: 10.3390/math14162868

Authors: Manal Badgaish

This paper investigates a class of generalized nonlinear variable-coefficient beam-type evolution equations motivated by the modeling of nonhomogeneous elastic structures with nonlinear effects. The invariant subspace method is employed to construct exact solutions by identifying appropriate finite-dimensional invariant subspaces associated with the nonlinear operators. This approach reduces the original partial differential equations to systems of nonlinear ordinary differential equations. Several representative nonlinear beam-type equations are studied, leading to polynomial, trigonometric, hyperbolic, and negative-power invariant subspaces together with their corresponding exact solutions. The results demonstrate the effectiveness of the invariant subspace method for constructing explicit exact solutions of nonlinear variable-coefficient fourth-order beam-type equations and highlight its applicability to a broad class of nonlinear beam models.

Mathematics doi: 10.3390/math14162867

Authors: Youn Su Kim Kwang Yoon Song Hoang Pham In Hong Chang

With the Fourth Industrial Revolution, software has evolved into complex large-scale systems. Because of the complexity of these systems, failures and defects in individual software components can propagate through the entire system, potentially causing substantial losses. Numerous studies have aimed to address this issue and improve software reliability. In this paper, we extend previous research by proposing NHPP-based software reliability models that consider dependent failures and initial faults. Furthermore, considering failures in both initial and test stages, we propose software reliability models suitable for real-world problems. The proposed models, incorporating these two aspects, were compared with 21 traditional software reliability models and three datasets using nine criteria. The results indicate that accounting for dependent failures and initial faults improves software reliability modelling in complex systems. The model considering initial failures at the start demonstrated the best performance on two datasets, whereas the model considering initial failures during the testing phase also showed strong results on one dataset. Furthermore, both proposed models demonstrated excellent performance across all three datasets. These results indicate that, in complex software systems, model performance depends on the treatment of initial faults and dependent failures.

Mathematics doi: 10.3390/math14162866

Authors: Ampol Duangpan Ratinan Boonklurb Phiraphat Sutthimat

The Hasse derivative image of a repeated-root cyclic code is typically non-cyclic. This prevents a direct ideal-theoretic calculation of derivative-twisted intersections and first leads to cyclic containers and two-sided bounds. This paper shows that, after expanding codewords in Hasse–Taylor coordinates at the roots of the underlying polynomial, the higher-order Hasse derivative operator decomposes into independent local blocks and its image becomes a coordinate subspace. This yields exact closed-form dimensions, expressed entirely in terms of the generator multiplicities and the base-prime digits of the derivative order, for the derivative image, for the derivative-twisted intersection of two codes and its self-intersection specialization, and for the smallest cyclic code containing the image together with its non-cyclic defect. As a consequence, for every positive derivative order in the admissible range, the derivative image is cyclic only when it is zero, and the multiplicity-drop and Lucas-refined containers developed here, as well as the rank of the derivative operator, are recovered as immediate relaxations or special cases. Applying the intersection-pair construction to a code paired with its derivative image produces entanglement-assisted quantum error-correcting codes whose dimension and entanglement cost are exact functions of the multiplicity digits and the derivative order, so that the derivative order tunes the entanglement consumption. For the minimum distance, a punctured matrix-product decomposition gives an exact formula for the derivative image in terms of the minimum distances of a finite family of simple-root constituents of length n0. Combining this formula with the corresponding constituent formula for the dual gives the complete distance parameter of the resulting EAQECCs without enumerating the non-cyclic derivative images.

Mathematics doi: 10.3390/math14162865

Authors: Cem Özkurt Bahadır Düzcan Semih Özenç Süleyman Uzun

Breast cancer histopathological image classification is an important task for computer-aided diagnosis, yet patch-level analysis remains challenging due to tissue heterogeneity, visual similarity between classes, and the risk of patient-level data leakage. This study evaluates classical and quantum-assisted deep learning configurations for binary invasive ductal carcinoma classification using histopathological image patches. Six models were compared using a common patient-disjoint split: a task-specific convolutional neural network, ResNet18 and DenseNet121 transfer learning models, a hybrid quantum convolutional neural network, and two quantum transfer learning models based on ResNet18 and DenseNet121 backbones. The models were assessed using accuracy, precision, recall, F1 score, ROC-AUC, and PR-AUC. The classical convolutional neural network achieved the best overall performance, with 88.00% accuracy, 90.00% recall, 0.9624 ROC-AUC, and 0.9677 PR-AUC. Among the quantum-assisted configurations, QTL-DenseNet121 achieved the strongest result, with 86.50% accuracy and 0.9315 ROC-AUC, while the hybrid quantum convolutional model achieved 81.00% accuracy using 144 trainable quantum parameters. The findings indicate that compact quantum-assisted classifier components can be feasibly integrated into medical image classification pipelines, although the results do not demonstrate quantum advantage and require further validation.

Mathematics doi: 10.3390/math14162864

Authors: Nurhan Adil Öztürk Vahit Çalışır

The variable-coefficient Korteweg–de Vries equation ut+f(t)uux+g(t)uxxx+h(t)ux+σ(t)u=0 is bilinearized by the Hirota method after gauge and Galilean reductions, and exact N-soliton solutions are shown to exist precisely under the integrability condition σ+ddtln(g/f)=0, equivalent to reducibility to the constant-coefficient equation. Closed-form laws are obtained for the soliton amplitude A=3k12g/f, width, velocity, and trajectory; the two-soliton collision is proved strictly elastic, with a coefficient-independent phase shift and no fusion or fission; the conservation laws are recast as exactly modulated invariants; and a constructive coefficient-programming design realizes prescribed-amplitude and shape-preserving solitons. The physically derived variable-depth equation satisfies the condition only at constant depth, so that a generic sloping bottom admits only the adiabatic single-soliton regime (A∝D−1), compared structurally with the classical shoaling laws. Every solution is verified by direct symbolic substitution.

Mathematics doi: 10.3390/math14162863

Authors: Zheming Zhang Hui Zhang

The Most Valuable Player Algorithm (MVPA) is a recently developed metaheuristic optimizer with a simple competition-based framework; however, its search capability is limited by insufficient information interaction and weak diversity maintenance. To address these issues, this study proposes a Multi-Strategy Enhanced Most Valuable Player Algorithm (MSEMVPA). Three complementary strategies are developed: an Adaptive Historical Differential Competition Strategy (AHDCS) that introduces historical search information and adaptive differential guidance to enhance exploration, an Adaptive Multi-Elite Reorganization Strategy (AMERS) that integrates diverse elite information to improve exploitation, and a Business Analysis Strategy (BAS) that reconstructs inferior individuals to maintain population diversity. The proposed MSEMVPA is evaluated on the CEC2017 benchmark suite through comparative experiments, ablation studies, convergence analysis, statistical tests, and computational complexity analysis. Furthermore, MSEMVPA is employed to optimize a multilayer perceptron model for enterprise bankruptcy prediction. Experimental results demonstrate that MSEMVPA achieves improved optimization accuracy and robustness compared with the original MVPA and several competitive algorithms. The results indicate that the proposed multi-strategy framework provides an effective approach for enhancing MVPA and solving complex optimization problems.

Mathematics doi: 10.3390/math14162862

Authors: Danni Yang Wenkang Zhang

This study examines how misinformation can persist when broadcast exposure and social feedback reinforce one another under stochastic platform conditions. Text classifiers and single-layer cascade models omit latent exposure, reply-driven amplification, random attention shocks, and uncertainty in intervention response. A stochastic duplex SEIR model is developed on heterogeneous networks, with an information exposure layer for broadcast and recommendation channels and a social feedback layer for replies, discussion, and amplification. The analysis combines degree-weighted mean-field equations, next-generation threshold calculations, Lyapunov stability arguments, Fokker–Planck linear noise approximation, Milstein simulation, and Wasserstein distributionally robust control. Theoretical results provide positivity, stochastic threshold conditions, extinction and persistence regimes, and sufficient conditions for stationary behavior and robust control stability. Numerical simulations show extinction–persistence transitions, cross-layer resonance, noise-induced threshold shifts, stationary bands, control cost–safety trade-offs, and sensitivity to unidentifiable stochastic parameters. A CoAID tweet–reply case study maps public interaction traces to observable duplex indicators, including tweet–reply densities, propagation elasticities, coupling proxies, and classifier features. Duplex observable features improve over a single-layer public data baseline, while model-assisted stochastic features add modest gains in the available public projection. Structural fitting of the stochastic duplex process would require time-stamped user-level multiplex trajectories, recommendation exposures, and intervention logs. The case study also clarifies the data granularity needed for future platform-level calibration and operational readiness. The framework supports data-informed platform governance by linking propagation thresholds, algorithmic down-ranking, reply thread moderation, intervention cost, and robustness bounds within a common threshold control language for practical settings.

Mathematics doi: 10.3390/math14162861

Authors: Aslı Güldürdek Ahu Açıkgöz

Given an ideal topological space (X,τ,I), we consider two opposite ways of idealizing the class of h-open sets. The first, the class of hI-open sets introduced by Açıkgöz and Noiri, pushes the ideal inward through the Janković–Hamlett local-function closure: a set A is hI-open if A⊆int(A∪cl∗(V)) for every nonempty proper open subset V of X. The second, introduced here as the class of Ih-open sets, pushes the ideal outward by tolerating a small defect: a set A is Ih-open if A∖int(A∪V)∈I for every nonempty proper open subset V of X. We show that h-openness implies both hI-openness and Ih-openness, while hI-openness and Ih-openness are mutually independent. We also show that the two idealizations collapse to h-openness at opposite ends of the ideal lattice: hI-openness coincides with h-openness when I=P(X), whereas Ih-openness coincides with h-openness when I={∅}. Replacing the ordinary interior by the α-, pre-, semi- and β-interior operators, we obtain the variants Ihα-, Ihp-, Ihs- and Ihβ-openness, and we describe how they are related. Finally, we study the family τIh of all Ih-open sets and the induced operator intIh. We prove that τIh is a topology whenever X is finite or, more generally, whenever I is closed under arbitrary unions of its members. We also give a counterexample showing that, without such a hypothesis, arbitrary unions of Ih-open sets need not be Ih-open.

Mathematics doi: 10.3390/math14152860

Authors: Faraj H. Alyami Sheeraz Iqbal Md Shafiullah Saleh Al Dawsari

In incentive-based demand response, an aggregator is paid for the gap between a customer’s metered load and the baseline load that would have occurred without a curtailment signal. This baseline is never recorded during the event, yet the settlement depends on it, so it must be reconstructed from the load observed before and after the curtailment window. At the portfolio level on which settlement is cleared, this amounts to filling a single contiguous gap, aligned with the daily peak, in an otherwise complete record. To estimate the portfolio-level customer baseline load (CBL), we propose AFCANet, which folds the one-dimensional CBL time series into a period-aligned two-dimensional tensor whose two axes describe different things. The intra-period axis traces the shape of a single daily cycle, which is locally smooth and strongly autocorrelated, while the inter-period axis links the same clock time across successive days, a longer-range and less locally smooth dependency. At the core of AFCANet is the Axis-Factorized Convolution–Attention (AFCA) block, which assigns a convolution to the intra-period axis, where its locality and weight-sharing suit the smooth daily shape, and self-attention to the inter-period axis, where its ability to link distant positions suits the cross-day dependency. Experiments use metered load from the Low Carbon London trial dataset with half-hourly resolution, evaluated under a control-group protocol in which the masked baseline is directly verifiable; AFCANet attains a MAE of 20.88 kWh, a MAPE of 1.50%, and a near-zero bias of 4.56 kWh, improving on averaging, regression, and learning-based imputation baselines. A controlled ablation that swaps the two operators confirms that the matched axis assignment is the source of the gain. Since the evaluation relies on synthetic curtailment windows in which no behavioral response is present, the reported accuracy should be read as an upper bound on the performance attainable in live demand-response events. The near-zero bias is of direct practical value to load aggregators, as a baseline free of systematic over- or under-estimation supports accurate curtailment measurement and fair financial settlement in incentive-based demand response.

Mathematics doi: 10.3390/math14152859

Authors: Wentao Yang Yibo Wang Yuhan Guo Runze Zhang

Converter-dominated power systems increasingly require flexible resources that can support frequency while preserving a physically interpretable energy trajectory. Thermal-energy-storage (TES)-based virtual energy storage systems (VESSs) can shift electrical demand within thermal-energy and comfort constraints, and have therefore attracted extensive interest. However, existing studies commonly coordinate requested or normalized power without fully connecting it to actuator execution and the electrical-to-thermal energy path. Command-level sharing cannot be directly equated with executed physical power, and controller storage cannot be combined with joule-valued hardware energy without dimensional separation. Therefore, this paper proposes a physically coupled and energy-traceable virtual asynchronous machine (VAM) control method for TES-based VESSs. First, a loss-resolved averaged model establishes the point-of-common-coupling (PCC)–converter–DC-link–actuator–TES physical chain and separates the hardware Hamiltonian from the dimensionless control Lyapunov function. Second, a neighbor-coupled marginal controller is embedded in a command–projection–execution chain so that frequency regulation and weighted sharing are evaluated using the executed service. Third, a constraint-handling mechanism combines directional headroom gating, actuator saturation and ramp limits, thermal comfort bounds, and request-inactive state reset to maintain executable trajectories under the declared constraints. Simulations under a sustained 120kW disturbance show that primary-only control retains a −0.08883Hz steady-state offset, whereas the proposed nominal case restores frequency. In the constrained case, the 30 s terminal trend remains above the prescribed limit, while both terminal windows of the 60 s run satisfy the restoration criterion; the final-window mean error and dimensionless eligible-unit sharing spread are −6.317×10−5Hz and 6.564×10−5, respectively. The model-internal electrical–thermal balance achieves a dimensionless relative RMS residual of 6.2361×10−8. Because the PCC voltage/current pair is reconstructed from the same power source, this residual quantifies model-internal consistency rather than independent measured closure. These results demonstrate constrained frequency restoration, executed-power coordination, and energy traceability within the averaged-model scope.

Mathematics doi: 10.3390/math14152856

Authors: Austine Efut Ofem Seithuti Philemon Moshokoa Malesela Clifford Kekana

This paper proposes a two-inertial forward–backward algorithm with adaptive line search for solving convex bilevel optimization problems in real Hilbert spaces. The lower-level problem is reformulated as a fixed point problem associated with the forward–backward operator, while the upper-level objective is incorporated through a viscosity approximation framework. The proposed method employs adaptive line search to avoid requiring prior knowledge of a global Lipschitz constant and incorporates two-inertial extrapolation terms to improve practical performance. Under mild assumptions, we establish the strong convergence of the generated sequence to the unique viscosity-selected solution. Numerical experiments on image restoration, sparse signal recovery, digital twin optimization, and aerospace topology optimization illustrate the effectiveness of the proposed method and its competitive performance when compared with several recent bilevel optimization algorithms.

Mathematics doi: 10.3390/math14152858

Authors: Xingxing Zhang Cankun Xie Shaobo Li

The recently proposed Animated Oat Optimization (AOO) algorithm exhibits competitive search behavior, but its fixed branching rules and limited use of inter-individual information may cause diversity loss and premature stagnation. This study proposes an Improved Animated Oat Optimization algorithm (IAOO) that integrates the original AOO operator, a hybrid DE/rand/1–DE/best/1 operator with a decreasing scale factor, and an elite neighborhood-directed local search within a feedback-driven framework. Strategy probabilities are updated according to normalized successful fitness gains, enabling search effort to adapt to the current optimization state. IAOO was evaluated through 30 independent runs on the CEC2017 (dim = 30/100), CEC2020, and CEC2022 suites and achieved Friedman mean ranks of 1.50, 1.37, 2.70, and 1.83, respectively, achieving competitive Friedman mean ranks among the compared algorithms and demonstrating statistically supported advantages on most benchmark suites. In three-dimensional UAV reference-path planning, IAOO reduced the mean path cost from 406.26 for AOO to 298.94, corresponding to a 26.4% reduction, while the standard deviation decreased from 67.01 to 40.73. Its runtime increased only from 26.99 s to 27.13 s. These results indicate that feedback-based operator cooperation improves solution quality and robustness with limited computational overhead. The current UAV model produces geometrically feasible and kinematically constrained reference paths; full six-degree-of-freedom tracking validation remains future work.

Mathematics doi: 10.3390/math14152857

Authors: Ze-Jin Tao Pyung-Hoi Koo

Demand uncertainty and supply imbalances pose persistent challenges for procurement decisions. In practice, firms often mitigate such mismatches through inventory swapping and spot market procurement. Motivated by this observation, this paper develops a procurement model that integrates long-term wholesale procurement, ex-post inventory swapping, and spot-market sourcing within a unified framework. While prior studies have largely examined spot-market procurement and inventory swapping separately, little attention has been paid to how these two mechanisms jointly mitigate demand uncertainty. The analysis yields three main findings. First, inventory swapping remains economically valuable even when spot-market procurement is available, indicating that the two mechanisms are not pure substitutes. Second, the economic value of inventory swapping increases with higher spot-market prices and greater demand volatility, as reallocating existing inventory becomes more cost-effective than emergency spot-market procurement. Third, the impact of the partner buyer’s order quantity depends critically on the swap transfer price, with lower transfer prices strengthening the value of inventory swapping. These findings contribute to the procurement and supply flexibility literature by clarifying how multiple procurement channels interact under demand uncertainty. The study also provides managerial implications for designing procurement systems that integrate inventory swapping with spot-market procurement.

Mathematics doi: 10.3390/math14152855

Authors: Ziyue Yuan Duo Li Xuekai Cen Zhongnan Ye Xinyu Yan

Post-disaster repair priorities can change when building demand and available supply recover at different rates. This study models a power-grid-building system, defines demand loss as cumulative unmet demand divided by cumulative demand, and uses a genetic algorithm (GA) with deterministic feasibility rules to select repair task order and repair mode. The two GA searches used the same settings, 20 runs for each objective, and 36,200 schedules evaluated per run. In the baseline case, the lowest demand loss found was 0.3925 for the demand-targeted search and 0.3995 for the supply-targeted search. The demand-targeted result was 1.7464% lower and reduced cumulative unmet demand by 238 kW-day. Across the same 20 random seeds, the demand-targeted search produced lower demand loss in 16 runs and the supply-targeted search produced lower demand loss in four runs. In a separate comparison with different computational effort, the demand-targeted GA result had 20.3% lower demand loss than one deterministic greedy schedule. Additional five-run analyses show that the observed results depend on GA settings, crew availability, repair duration, and demand timing. The findings apply to the tested deterministic case study and support demand-aware repair scheduling when demand and supply recover at different rates.

Mathematics doi: 10.3390/math14152854

Authors: Md. Mutakabbir Khan Md. Jasim Uddin M. T. Alharthi Ibraheem M. Alsulami Najat A. Alghamdi

This research examines the behavioral shifts within a discrete-time predator–prey framework, constructed by applying the forward Euler discretization to a continuous model. The system incorporates Smith’s growth dynamics for the prey population alongside a Holling type III functional response to characterize predator behavior. Through bifurcation analysis, it is demonstrated that the interior fixed point undergoes stability loss via Neimark–Sacker and period-doubling transitions, leading to the emergence of quasiperiodic oscillations and chaos. Furthermore, the application of normal-form theory verifies the nondegeneracy of these bifurcations and establishes the direction of the resulting orbits. We use phase portraits, Lyapunov exponents, and bifurcation diagrams to confirm the model’s rich dynamics. These numerical tools demonstrate how the system moves from stable equilibria to more intricate behaviors. The application of partial rank correlation coefficients reveals the most influential parameters governing the system’s asymptotic population levels, providing a global perspective on parameter sensitivity. The Ott–Grebogi–Yorke (OGY) chaos control strategy is employed to suppress unwanted bifurcations and stabilize chaotic oscillations within the system. These results underscore the role of nonlinear interactions and discrete-time frameworks in precipitating unpredictable population fluctuations while simultaneously offering a suite of mechanisms for enhancing the stability of ecological networks.

Mathematics doi: 10.3390/math14152853

Authors: Siqi Liu Xuerong Meggie Wen Akim Adekpedjou Guangbao Guo

Laplace factor models (LFMs) provide a heavy-tailed alternative to Gaussian factor models by representing high-dimensional observations through a low-rank common component and Laplace-distributed idiosyncratic errors. This paper develops an assumption-consistent finite-sample analysis of matrix concentration, covariance estimation, and Monte Carlo integration under this model. We first formulate the model with explicit dimensional, independence, covariance, and identifiability conditions. Standard matrix Laplace-transform and matrix Bernstein inequalities are then recalled with their precise applicability conditions. Because untruncated Laplace variables are neither almost surely bounded nor strongly log-concave, these standard results cannot be applied directly in the forms commonly used for bounded or Gaussian-like observations. To address this issue, we analyze a coordinatewise truncated covariance estimator and derive an operator-norm bound that separates the stochastic estimation error from the truncation bias. The resulting rate depends on the effective rank and the logarithm of the ambient dimension and is therefore not dimension-free. For Monte Carlo integration, we replace strong-log-concavity arguments by a sub-exponential concentration analysis that is compatible with independent Laplace errors and yields non-asymptotic absolute- and relative-error bounds. Simulation studies compare empirical tails with the classical matrix Bernstein bound, evaluate ordinary, truncated, winsorized, PCA, POET-type, and Huberized covariance estimators, and we compare Laplace-based and Studentized confidence intervals. The results show that the classical Bernstein bound can be conservative, and truncation involves a substantial bias–variance trade-off. In a Wine chemical-analysis application, three factors explain 66.53% of the standardized variance, and POET-type covariance estimation attains a cross-validated balanced accuracy of 0.9901. These findings clarify both the scope and the limitations of finite-sample analysis for LFMs.

Mathematics doi: 10.3390/math14152852

Authors: Yanhong Zuo Kun Yang Shipeng Chen Cheng Jin Hao Zhang

High-quality information data refers to data meet the quality requirements of operational systems. Among these, the accuracy of information data measurement values serves as a key indicator for evaluating high-quality information data. Information data quality is reflected in two aspects: the measurement accuracy and signal transmission strength. The combined effect of these two issues results in significant measurement errors in the information detected by the information-detection system. Therefore, improving data quality essentially involves reducing the differences between information data to enhance measurement accuracy and increasing signal strength during information data transmission to compensate for energy loss and enhance anti-interference capability during long-distance transmission. In the field of signal detection and transmission, fractional calculus (FC) demonstrates dual advantages in data measurement: it can regulate signal strength across different frequencies and improve the algorithm’s generalization ability and data fusion accuracy, thereby significantly enhancing data quality. The spectral characteristics of the fractional differential operator indicate that data quality is closely related to frequency ω and fractional order v. In this study, the influence mechanism of parameters h and v on the information data quality of fractional-order differential operators constructs a mathematical model for information data fusion based on fractional-order operators, realizes high-precision detection technology for information data, and constructs a mathematical model for signal strength based on fractional-order differential operators to realize long-distance transmission of information data. By jointly solving these two models, high-precision, long-distance detection of information data under a given quality is achieved. Finally, application examples verify the feasibility of this method. In this case, we successfully met the quality requirement of a standard deviation Sg = 0.05 ± 0.05 for the information data transmitted over a distance of 3 km by the information detection system. The feasibility of the method was verified.

Mathematics doi: 10.3390/math14152851

Authors: Luiz M. B. C. Campos Manuel J. S. Silva

The original Bessel differential equation that describes, among many others cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalised Bessel differential equation that describes coupled acoustic–vortical waves. The solutions of the generalised Bessel differential equation can be obtained for all possible combinations of complex variable, order and degree by three alternative methods: (i) convergent power series of Frobenius–Fuchs type around the regular singularity at the origin; (ii) asymptotic expansions of Thomé normal integral type around the irregular singularity at infinity; and (iii) Laplace transform along suitable paths in the complex plane which are the focus of the present paper. This leads to the generalised Bessel, Neumann and Hankel functions of two kinds, for which are obtained: (i) power series; (ii) asymptotic expansions; and (iii/iv) representations as definite and contour integrals. For the generalised Bessel functions are obtained four integral representations: (i) as two definite integrals along the unit interval with branch-points at both ends; (ii) as integrals along tear-drop loops surrounding one branch-point each; and (iii) as an integral along a Pochhammer double-laced loop around both branch-points. For the modified generalised Neumann function are obtained two integral representations: (i) along the negative real axis joining infinity to the branch-point at origin and (ii) along a Hankel-type semi-infinite loop around the single branch-point. The generalised Hankel functions of two kinds are linear combinations of generalised Bessel and Neumann functions, and are used to describe cylindrical acoustic or vortical waves and their coupling. The acoustic–vortical waves in a cylindrical duct are illustrated as plots of generalised Bessel functions with different orders and degrees. The original Bessel functions describe purely acoustic or vortical cylindrical waves that, for large radius, are asymptotically decaying and oscillating, hence being stable. The generalised Bessel functions describe coupled acoustic–vortical waves that are asymptotically monotonic and unstable. This provides a physical interpretation for the different mathematical properties of the original and modified Bessel functions.

Mathematics doi: 10.3390/math14152850

Authors: Qin Li Min Wan

Residual neural ordinary differential equations (NODEs) of the form f^=f+hθ can attain small one-step prediction error yet diverge under autonomous long-horizon rollout. A recent diagnostic attributes this to the one-sided Lipschitz (OSL) constant—the supremum over visited states of the logarithmic norm μ2(Jf^)=λmax(12(Jf^+Jf^⊤))—which, when positive, signals local expansion and amplifies persistent approximation error. In this work we convert this post hoc diagnostic into a train-time method by augmenting the one-step objective with a contraction penalty λEx[(μ2(Jf^(x))−c)+], and we study when this improves robust forecasting across stable, expansive, marginal, and chaotic regimes under realistic sensor-corruption noise. We prove that, at the penalty’s minimizer, the empirical OSL constant is controlled on the training set. A sample-to-domain covering condition then yields, via a Gronwall-type comparison, a conditional uniform-in-time rollout-error bound, with a time-averaged variant that justifies penalizing the mean rather than the maximum log-norm. Empirically, on a six-system, four-noise benchmark the penalty reliably drives the OSL constant down by one-to-two orders of magnitude, but whether this helps long-horizon accuracy is strongly regime-dependent and λ-sensitive: a common default (λ=0.1) over-damps and degrades rollout, whereas a calibrated λ≈0.01 helps only for measurably expansive baselines. Under a seed-decoupled re-evaluation with 15 seeds, the benefit is robust on a near-unstable, rotation-dominated oscillator—a 2.3× lower 100-step rollout error (p=0.018) at matched one-step error—but the apparent 5-seed improvement on a six-dimensional chemical reaction network does not replicate: with model and dataset seeds decoupled it reverses to a significant degradation, identifying the original effect as a seed-coupling artifact. The method thus yields a single robust positive result, and degrades already-contractive, conservative, and chaotic systems; for chaotic systems this is unavoidable, because enforced contraction suppresses the positive Lyapunov exponents that define the attractor. Finally, while a contraction-aware spectral penalty matches the log-norm penalty, standard ∥J∥2≤1 spectral normalization fails on rotation-dominated dynamics (8× worse rollout, p=0.0005, 15 seeds), confirming that the rotation-invariance of μ2 is the operative property.

Mathematics doi: 10.3390/math14152849

Authors: Junghyeop Im Minsoo Kim Minkyu Jung Hyeonjun Im Duehee Lee

Wind power producers face severe financial risks from intermittent generation and volatile prices. In this paper, we develop a multivariate scenario-based optimization framework that integrates heavy-tailed Laplace forecast-error path generation, principal component (PC) score matching, and Frequency-Concentrated Projection (FCP) to determine day-ahead bidding curves that maximize expected settlements. To model uncertainty, 24-h scenarios are generated by sequentially accumulating heavy-tailed Laplace forecast-error increments. Trajectories of specific variables are integrated via PC-score matching to construct joint scenarios preserving inter-variable dependencies. A dense optimal response derived from these scenarios is compressed into a market-compatible 11-point bidding curve using FCP, which strategically allocates submission points to highly probable clearing intervals. Evaluation on 2021 NYISO West data demonstrates substantial improvements in both feasibility of scenarios and financial performance. The Laplace specification captures extreme price spikes, so it significantly reduces downside risk compared to a Gaussian baseline. PC-score matching ensures feasibility of structure by preserving daily trajectory shapes. Leveraging these robust scenarios, the FCP curve yields substantially higher realized settlements than the Uniform Support Baseline (USB), which uniformly places the limited submission points across the price range, recovering approximately 90% of the settlement gap between USB and the Dense Optimal Response (DOR), which serves as a non-submittable dense-grid upper-bound benchmark. Ultimately, this framework translates complex uncertainty models into actionable strategies, enabling producers to systematically maximize economic returns under rigid market constraints.

Mathematics doi: 10.3390/math14152848

Authors: Omar Mossa Alsalhi

We investigate the commutativity of Toeplitz operators with trigonometric polynomial symbols on the Beurling subspaces zkHs2 of the Hardy–Sobolev spaces. An explicit formula for the commutator of two Toeplitz operators is first derived in terms of weighted shift operators. As a consequence, every such commutator is shown to have finite rank. It is then proved that if the degrees of the symbols are at most k, then the corresponding Toeplitz operators commute on the Beurling subspace zkHs2. Moreover, this result is shown to be sharp by proving that, in general, the Beurling subspace zkHs2 cannot be replaced by zk−1Hs2. Finally, these results generalize the corresponding commutativity theorem for Toeplitz operators on the classical Hardy space.

Mathematics doi: 10.3390/math14152846

Authors: Muhammet Kamali Shigeyoshi Owa

Let Ap denote the class of p-valently functions f(z) in the open unit disc U with f(j)(0)=0(j=0,1,2,…,p−1). In general, we write f(z)∈Ap by f(z)=zp+∑k=1∞ap+kzp+k(p∈N). For f(z)∈Ap,p-valently α-convex functions of order β in U are introduced. Some interesting properties for such functions can be seen. In this paper, we introduce fractional derivatives Dzj+λf(z) for f(z)∈Ap with j=0,1,2,…,p−1 and 0≤λ<1. Applying the fractional derivatives Dzj+λf(z), we would like to generalize p-valently α-convex functions of order β in U. Some results for such functions f(z)∈Ap are discussed with example functions. Further, a conjecture for our research in this paper is given with an example function.

Mathematics doi: 10.3390/math14152847

Authors: Yuxin Ye Xuhua Qiu Kunjie Zhu Miltos Ladikas

Financial tail-risk observations are non-exchangeable: serial dependence and regime shifts make their joint law depend on the time ordering, invalidating the exchangeability that standard conformal guarantees assume, and expected shortfall is not elicitable on its own, so a forecaster cannot be calibrated to it as a quantile is to its coverage. We ask whether a black-box value-at-risk and expected-shortfall forecaster can be calibrated under such dependence while retaining finite-sample guarantees. We tune a single inflation parameter by conformal risk control on a bounded monotone loss that couples value-at-risk breach frequency with breach magnitude normalised by the model’s predicted value-at-risk–expected-shortfall gap; the guarantee is thus for a tail-gap-normalised exceedance-severity surrogate, and its expected-shortfall reading depends on the predicted gap being a sound tail-gap estimate. Under exchangeability, the method gives finite-sample expected-risk control; for dependent data we invoke a non-exchangeable swap-distance bound and add, for separated calibration points, a regime-drift bound with an explicit cumulative β-mixing cost, plus a high-probability realised-path statement and a heavy-tail rate of order D(p−1)/p. Building regimes causally from previous-month FRED-MD vintages across eight exchange rates, a Bitcoin series, and the GIFT-Eval finance domain, the weighted controller attains a 2.51% violation rate and a Fissler–Ziegel score of 0.431 against 0.441 and 0.439 for the strongest conformal baselines—an incremental gain, not significant at the 5% level, that concentrates in turbulent regimes and at matched capital, supporting calibration of a joint frequency-and-normalised-severity budget rather than distribution-free control of the expected-shortfall forecast itself.

Mathematics doi: 10.3390/math14152845

Authors: Mihoub Bouderbala Souad Ayadi Meltem Erden Ege Ozgur Ege Mohammed Rabih

This paper establishes the existence, uniqueness, and Ulam–Hyers stability of solutions for a novel class of coupled sequential hybrid Hilfer fractional q-Duffing equations. By integrating Dhage’s hybrid structure with generalized Hilfer q-operators, we extend recent results on fractional quantum systems. Existence is proven via Dhage’s fixed point theorem in Banach algebras, while uniqueness follows from the Banach contraction principle with explicit verification of operator invariance. All auxiliary functions satisfy rigorous continuity, boundedness, and Lipschitz conditions, and the solution representation is derived with complete calculation of q-integration constants. The theoretical findings are rigorously validated through a detailed numerical example that explicitly verifies all contraction and stability constants.

Mathematics doi: 10.3390/math14152844

Authors: Maher Kachour Hassan S. Bakouch Fatimah E. Almuhayfith Jumanah Ahmed Darwish Talha Arslan

The geometric distribution is a fundamental model for count data and discrete-time lifetimes, but its memoryless property implies a constant hazard rate that is often too restrictive in practice. This paper introduces a new three-parameter discrete distribution, termed the generalized mixture of geometric (GMG) distribution, which extends the geometric model while preserving its tail behaviour. The proposed distribution allows controlled departures from memorylessness at early counts and admits clear parameter interpretations governing tail behaviour, shape, and perturbation intensity. Closed-form expressions for the cumulative distribution function, survival function, and hazard rate are fundamental for both theoretical analysis and reliability modelling. In addition to their practical usefulness, these quantities provide valuable insight into the structural differences between the GMG and geometric distributions. Shannon entropy generalizes the geometric baseline and its existence is established. Parameter estimation is addressed through a preliminary moment-based procedure and maximum likelihood estimation. The moment-based estimator is used to initialise numerical likelihood maximisation, while the inferential and numerical properties of the maximum likelihood estimator are investigated. Particular attention is paid to parameter configurations that may lead to weak numerical identification, and a stable optimisation strategy is discussed. A comprehensive Monte Carlo simulation study is conducted to assess the finite-sample performance of the maximum likelihood estimator. Finally, the practical usefulness of the GMG distribution is illustrated through four real-life count datasets, where its performance is compared with several established competing models.

Mathematics doi: 10.3390/math14152843

Authors: Taim Saker Mirko Tarulli George Venkov

We study the focusing nonlinear Schrödinger equation with time-dependent linear damping in the mass-supercritical and energy-subcritical regime. After a gauge renormalization, the problem becomes a focusing Schrödinger equation with a time-dependent coefficient in the nonlinearity. Under a natural dissipativity assumption on this coefficient, we prove forward scattering for radial H1 data below the ground-state threshold in dimensions d≥3. In contrast to previous scattering results based on a strictly positive averaged damping rate or on specially oscillatory initial data, we allow the averaged damping rate to vanish and require neither an oscillatory-data assumption nor a perturbative smallness condition. The proof combines variational trapping for the renormalized flow, localized coercivity, a localized Morawetz estimate, energy evacuation, and a radial scattering criterion adapted to the time-dependent setting.

Mathematics doi: 10.3390/math14152842

Authors: Kavitha Velusamy Sowmiya Ramasamy George Washington Samuelraj Chrysolite Mallika Arjunan Mani Seenith Sivasundaram

We develop a residual-adaptive preconditioned quantum pseudo-spectral method for generalised ψ-Caputo initial-value problems containing a discrete delay, weakly singular hereditary memory, and nonlinear reaction terms. A ψ-fractional Chebyshev basis yields closed-form operational matrices that are exact on the chosen finite spectral space. To make the hereditary term compatible with block encoding, the power-law kernel is approximated by a sum of exponentials and supplemented by an explicit local near-field correction, converting global memory into finitely many local auxiliary modes. A structure-preserving preconditioner controls the condition number, while a residual-adaptive multidomain strategy and damped Newton iteration treat layers and nonlinearities. We prove well-posedness in Mittag–Leffler weighted graph spaces, derive a combined spectral–kernel–residual error estimate, and state the quantum linear-system complexity with explicit block-encoding normalisations and right-hand-side preparation assumptions. Numerical tests show high accuracy for solutions smooth in the ψ-coordinate, improved robustness for singular and layered solutions, substantial condition-number reduction, and lower history cost under sum-of-exponentials compression. To evaluate performance, we compare against L1 product integration and Jacobi collocation, systematically quantifying their respective accuracy, computational cost, and conditioning characteristics.

Mathematics doi: 10.3390/math14152841

Authors: Nicola Fabiano Zouaoui Bekri Abdulaziz Khalid Alsharidi

We introduce a novel generalization of the classical Ćirić–Reich–Rus (CRR) contraction, termed the Asymmetric Cross-Iterate CRR (ACI-CRR). The contraction condition involves distinct iterate orders p≠q in an asymmetric cross-coupled form. We establish the existence and uniqueness of a fixed point under the assumption of continuity. We then analyze the symmetric case p=q, demonstrating that continuity is indispensable for p>1, and contrast this with the classical p=q=1 case where continuity is unnecessary. We conclude with a structural example and a conjecture regarding the necessity of continuity in higher-iterate asymmetric contractions.

Mathematics doi: 10.3390/math14152840

Authors: Abdelhamid Taieb Zaidi

We study a Caputo fractional three-point boundary value problem of order α∈(c−1,c] and establish three interrelated contributions, all resting on a single refined pointwise L2 estimate for the associated Green function Gr(t,s). First, we derive a tighter upper bound for sup0<t<1∫01Gr2(t,s)ds by retaining a sign-definite negative mixed term that the classical L1-based analysis of Shivanian discards. The resulting admissible Lipschitz constant κB is explicit in α, a, b, c and exceeds Shivanian’s constant κS under an explicit algebraic condition; a closed-form refinement κB∗≥κB follows by maximising the pointwise bound in closed form. On Shivanian’s benchmark, the admissible constant rises from 14.646 to 23.220 and then to 32.165. Second, the same contraction constant yields an explicit Ulam–Hyers stability theorem for this problem. While a stability estimate already follows from the classical L1 condition, the refined constant both enlarges the range of admissible Lipschitz constants for which stability is certified and yields a strictly smaller stability constant. Third, we establish quantitative continuous-dependence bounds with respect to the nonlinearity h and the boundary parameter a, and characterize the deterioration of the contraction-based boundary-parameter estimate as the problem approaches resonance. For a fixed Lipschitz constant, this estimate becomes singular as the perturbed contraction factor approaches one and ceases to apply once the contraction condition fails. A more accurate analysis of the Green function thus simultaneously sharpens solvability conditions, stability estimates, and sensitivity bounds for nonlinear Caputo fractional boundary value problems.

Mathematics doi: 10.3390/math14152839

Authors: Hao Wu Xiaoqing Zeng

Short section delays in metro corridors can spread quickly through transfer hubs and return to operations as longer dwell times, local crowding, and downstream delay. Existing disruption–recovery studies often optimize train regulation, station control, and passenger guidance in separate modules, which makes it difficult to assemble a linked response that is both operationally coherent and auditable. This study develops an LLM-driven multi-agent decision framework for abnormal response in urban rail transit. Role-specific agents generate train-, station-, transfer-, and information-side measures; a safety-review agent removes infeasible actions; and an arbitration agent produces a structured DecisionCard for simulation backfeeding. A corridor-level mesoscopic model is built for the Shanghai Metro Line 2 section between East Nanjing Road and Lujiazui, covering four key transfer hubs. The evaluation combines five online LLM cases with a broader 3-by-3 delay–demand matrix and transfer-capacity sensitivity tests. Relative to B4-SOP-Limited, Online-MA-Rolling reduces total delay by 3.5–48.8% and peak transfer queues by 2.0–90.3% in the five online cases. Across the nine delay–demand combinations, the multi-agent strategy reduces total delay by 58.9% on average and transfer-queue peaks by 32.6% on average compared with B4. When transfer service capacity is reduced by 25%, however, queue relief collapses and the delay advantage can reverse under high load. The main contribution of the framework is, therefore, not the use of LLMs alone, but the production of linked, reviewable, and simulation-testable response plans within a clearly bounded operating envelope.

Mathematics doi: 10.3390/math14152838

Authors: Danhua Chen Jie Hu Yuting Liu Kaige Shen Tie Xu Yuanyi Huang Pei Zhang

To solve the conflict between trajectory tracking and stability control for distributed drive electric vehicles under complex driving conditions, an integrated longitudinal and lateral coordinated control strategy based on a hierarchical architecture is proposed in this paper. First, a stability-boundary dataset is constructed using an improved sum-of-squares programming (ISOSP) algorithm. Then, a long short-term memory (LSTM) network is optimized with the sparrow search algorithm (SSA). Finally, a prediction model is established to identify the dynamic stability region in real time. A hierarchical architecture is adopted for the control strategy. The upper layer integrates longitudinal–lateral tracking and stability control to describe the desired motion states accurately. In the middle layer, a stability margin is defined based on the stability region, and a risk factor is introduced to reconstruct the optimization objective. Through this design, the coordinated control of trajectory tracking and vehicle stability is achieved. In the lower layer, the minimization of the tire-workload rate is taken as the objective, and the optimal allocation of four-wheel torque is realized through quadratic programming. Hardware-in-the-loop (HIL) tests based on an NI PXIe-1078 real-time simulator, a host computer, and a domain controller indicate that the proposed strategy achieves good control performance under both variable-speed high-adhesion and high-speed low-adhesion double lane change (DLC) conditions. Especially in the extreme condition of high speed and low adhesion, the root-mean-square errors (RMSEs) of lateral displacement and sideslip angle are controlled within 0.4125 m and 0.0365 rad, respectively. Consequently, the high-precision tracking capability and real-time stability maintenance of the coordinated control strategy under extreme conditions are successfully verified.

Mathematics doi: 10.3390/math14152837

Authors: Adolfo Pimienta Johnny Cuadro Oswaldo Dede Margarita Gary

We investigate the Hyers–Ulam stability of a functional equation motivated by the semi-parallelogram law, a three-variable identity that extends the classical quadratic equation. Our first result shows that every solution of this equation splits uniquely as the sum of an additive mapping and a quadratic mapping. Under natural growth conditions on the error term, we obtain stability estimates via two classical routes: the direct method and the fixed point alternative in generalized metric spaces. What makes the three-variable setting worthwhile is that it accommodates asymmetric control functions, for instance, φ(x1,x2,x3)=∥x1−x2∥p∥x3∥p, which cannot be captured by the standard two-variable quadratic equation. This extra freedom proves useful when perturbations depend on the relative position of the variables, rather than on each variable separately. We also examine power-type perturbations in detail. The analysis reveals a critical exponent p=2: when p<2, stability holds with explicit constants; at p=2, the contraction argument breaks down, pointing to an intrinsic limitation of the method. Several examples accompany the main results, among them a closer look at the role of the critical exponent and a visual discussion of how asymmetric controls arise in practice.

Mathematics doi: 10.3390/math14152836

Authors: Amira A. Allam Mahmoud Owais

This study addresses the Traffic Sensor Location Problem for complete link-flow observability under non-uniform sensor measurement uncertainty. The proposed framework minimizes the accumulated error propagated from observed link flows to inferred unobserved link flows while preserving the structural conditions required for complete network observability. Its methodological novelty lies in combining a structured new-link selection procedure with a fuzzy-enhanced Binary Particle Swarm Optimization (FBPSO) algorithm that adaptively balances exploration and exploitation. An ILU-preconditioned GMRES procedure is also incorporated to efficiently solve the sparse linear systems generated during the evaluation of candidate sensor configurations. The proposed framework is evaluated using the Fishbone and Sioux Falls benchmark networks and the large-scale Austin transportation network, which contains 7388 non-centroid nodes and 18,961 directed links. Its performance is compared with standard Binary Particle Swarm Optimization (BPSO) and the Binary Bat Algorithm (BBAT) under uniform and non-uniform measurement-error conditions. For the Fishbone network, all three methods reach the same minimum accumulated inference error of 89.21, indicating agreement on the best solution for this small test case. For the Sioux Falls network, FBPSO obtains an inference error of 634.46, compared with 641.78 for BPSO and 640.94 for BBAT. For the Austin network, FBPSO achieves the lowest final inference error and continues improving after the comparison methods reach prolonged plateaus. These findings demonstrate that the proposed framework provides an effective and scalable approach for uncertainty-aware traffic-sensor placement and reliable network-wide link-flow inference.

Mathematics doi: 10.3390/math14152834

Authors: Şeyma Beşir Nisa Özge Önal Tuğrul Ertuğrul Karaçuha Vasil Tabatadze

This study proposes a two-variable fractional calculus–based modeling framework for analyzing gross domestic product (GDP) growth rates using key macroeconomic indicators of G8 countries and Türkiye over the period of 1998–2022. The economic variables considered include exports, imports, inflation, foreign direct investment, and unemployment rates, with data obtained from the World Bank. Caputo-type fractional derivatives combined with the least squares method are employed to construct two-variable economic models, where GDP growth rate is treated as the dependent variable. Multiple combinations of economic factors are systematically examined to evaluate their modeling performance. The accuracy of the proposed models is assessed using the Mean Absolute Percentage Error (MAPE). The results indicate that the export–inflation combination yields the lowest modeling error for Japan, while the inflation–unemployment combination produces the highest error for Russia. Overall, the findings demonstrate that fractional calculus–based bivariate models provide an effective and flexible framework for capturing nonlinear and memory-dependent dynamics in economic growth analysis.

Mathematics doi: 10.3390/math14152835

Authors: Abd Ulazeez Alkouri Osama Ogilat Hasan Almutairi

We introduce and study bipolar complex intuitionistic fuzzy Lie algebras (BCIFLAs), a new algebraic framework merging bipolar fuzzy theory, complex-valued membership functions in Cartesian form, and intuitionistic fuzzy Lie algebra theory. Adopting the Cartesian coordinate formulation of bipolar complex intuitionistic fuzzy sets (BCIFSs), we define bipolar complex intuitionistic fuzzy Lie subalgebras (BCIFLSAs) and ideals (BCIFLIs) and establish their fundamental properties, including closure under arbitrary meets. We prove that images and preimages of BCIFLSAs and BCIFLIs are preserved under Lie algebra homomorphisms, characterize them via level sets, and show that the sum of two BCIFLIs is, again, a BCIFLI. Non-degenerate three-level examples on sl(2,R) and t(2,R) illustrate the gap between the subalgebra and ideal conditions. This framework strictly generalizes fuzzy, intuitionistic fuzzy, complex intuitionistic fuzzy, and bipolar fuzzy Lie algebras.

Mathematics doi: 10.3390/math14152833

Authors: Yanan Yu Zhihao Yin Hongfu Huang

Purpose: This paper investigates how product quality and green information transparency jointly influence customer trust in green products and examines competing retailers’ green disclosure and suppliers’ quality investment strategies under information asymmetry. Methodology: We develop a game-theoretic model of two competing retailers and two suppliers with asymmetric quality investment costs. We characterize equilibrium outcomes in sequential disclosure games and study the effects of the trust factor and transparency level on the equilibrium outcomes. Findings: Three main results emerge. First, a second-mover advantage exists in sequential disclosure, namely, the later retailer free-rides on the first mover’s information. Second, higher trust intensifies competition by homogenizing customer utility, whereas greater transparency softens competition by making valuations more heterogeneous. Third, customer trust can incentivize both retailers to disclose simultaneously. Novelty: First, this paper studies consumers’ heterogeneous green preferences and considers the information asymmetry of green attributes between supply and demand. Second, this paper determines the suppliers’ quality investment considering the retailers’ strategic green disclosure and constructs the endogenous trust function. Third, this paper analyzes the retailers’ green disclosure in equilibrium, further determines the market coverage, and points out the effects of key parameters on the market coverage and the product competition.

Mathematics doi: 10.3390/math14152832

Authors: Zhipeng Chen Qisheng Wang Yan Feng

For a nonnegative integer m, let um(n) denote the reduced numerator of ∑k=1nnk2n+kk2k2m+1. OEIS A357513 originally posed the finite-exception conjecture that um(p−1)≡0(modp4) for all but finitely many primes p. We prove the explicit uniform sufficient threshold p>2m+6. Thus, every exceptional prime satisfies p≤2m+6, while {0,1,…,2m+6} serves only as a finite witness in the ambient set N. The proof reduces the Apéry-type summand to two inverse-power sums in Z/p4Z, applies finite-field and pairing cancellations, and transfers the result to the reduced numerator through a common denominator prime to p. An immutable supplementary Lean 4 package was used to verify a theorem whose type directly exposes the threshold 2m+6<p.

Mathematics doi: 10.3390/math14152830

Authors: Abdulaziz Mutlaq Alotaibi Akbar Ali

The reciprocal mean square (RMS) index of a graph G with edge set E(G) is defined by RMS(G)=∑uv∈E(G)[(d(u))2+(d(v))2]−1, where d(u) and d(v) are the degrees of vertices u and v, respectively. We first establish several bounds for the RMS index in terms of standard graph parameters (including the size, minimum degree, and maximum degree) and related degree-based indices, such as the forgotten index, the Sombor index, the reciprocal hyper-Zagreb index, the inverse degree index, and the harmonic index. We then determine the extremal values of the RMS index over the classes of n-order trees and unicyclic graphs for n≥3. For n-order trees, the path Pn and the star Sn are the extremal graphs for the considered index, as expected. However, for unicyclic graphs of sufficiently large order, the graph minimizing the RMS index is, rather surprisingly, not the one containing a universal vertex. This indicates that the extremal behavior of the RMS index may differ from that of many existing degree-based indices.

Mathematics doi: 10.3390/math14152831

Authors: Mohammad Arif Wooseong Kim Adeel Iqbal Eun-Kyu Lee

Jamming introduces unintentional disruptions in the system to exploit the legitimate communicating equipment. Clustered jamming considers jammers that are present in multiple groups to disrupt the intended communication. Vehicle-to-everything (V2X) transmissions are critical for smart transportation. This research considers full-duplex environment, featuring unmanned aerial vehicles (UAVs) and cellular-base-station-aided V2X (C-V2X) systems exploiting clustered jamming using 3-dimensional (3-D) beam-forming millimeter-wave antennas. UAVs are modeled as a 3-D Poisson point process (PPP), and macro-based tower-mounted base-stations (MBSs) are modeled as a 2-D PPP. Roads are modeled as a Poisson line process. The vehicular nodes (V-Ns) are modeled on each road as a 1-D PPP. The deviations of the UAV’s millimeter-wave band antenna beam follow a Normal distribution. In this paper, for a full-duplex setting, the probabilities of coverage and equipment-association, along with the efficiency of the spectrum associated with various UAV and tower-based connections, are explored in the presence of clustered jamming. The probability of coverage and association of multiple links is derived with respect to the jamming clusters, V-Ns, MBSs, UAVs, jammers’ power, and antenna beams. The results demonstrated that jamming degrades system’s efficiency. This efficiency is further degraded whenever higher 3-D beam-width deviations of the millimeter waveband antenna and jammers are present. Therefore, robust counter-scenarios should be designed for the cases where jamming signals and varying beams disrupt the network.

Mathematics doi: 10.3390/math14152829

Authors: Samed Bajrić

The standardised Module-Lattice-Based Key-Encapsulation Mechanism uses coefficient compression, yet an abstract residue map has no unique Boolean cube spectrum until a binary representation, domain extension, and input measure are fixed. We analyse these maps under an explicit twelve-bit reduction-based lift. Exact interval character sums yield a general high-modulus theorem for quarter-threshold indicators and show that message decoding has a uniquely dominant high-bit parity. The argument also clarifies the connection among Walsh coefficients, affine approximation, Hamming distance, nonlinearity, agreement probability, and sign correlation. A vectorial extension exhaustively certifies every nonzero scalar component of the standardised compression widths by deterministic integer Walsh–Hadamard transforms. A representation comparison then separates full-cube coefficients from centred, distribution-dependent correlations and delineates how canonical, centred, Montgomery, Barrett, shared, or compiler-generated intermediates require distinct models. Reproduction scripts regenerate the complete certificates and the spectral-gap visualisation without sampling, random choices, physical traces, floating-point decisions in the core certificates, or network access. The results identify mathematically distinguished affine predictors for later implementation-specific validation; they do not establish measured leakage, attack success, or implementation resistance.

Mathematics doi: 10.3390/math14152828

Authors: Zehra Özdemir Johan Gielis

In this work, we present a generalized geometric framework that extends quaternion-based rotation and translation operators within a generalized inner product and vector product setting defined on Gielis-type superquadrics. By incorporating the multiplicative shape factor ρ(ϕ), we formulate rotation matrices and quaternion mappings adapted to the elastic and non-Euclidean behavior of biological growth surfaces. The proposed framework extends classical Euclidean constructions to the generalized metric setting, enabling smooth, direction-dependent deformations and providing a unified description of curvature-induced growth, differential thickening, and torsional motions observed in plants. The resulting formulation offers a mathematically consistent and biologically interpretable tool with potential applications in computational botany, growth-based animation, and the design of biologically inspired structures.

Mathematics doi: 10.3390/math14152827

Authors: Xupeng Huo Yang Zhao Qizheng Zhou Weige Liang Peiyi Zhou Qingmiao Ma

The cloud model provides a unified mathematical framework for representing both randomness and fuzziness simultaneously, offering an effective mathematical tool for modeling complex systems. This paper focuses on the emergent phenomenon of spontaneous synchronous flashing in fireflies and employs Gaussian cloud model theory to develop an initialization framework. Uncertainty is quantified using three core numerical characteristics of the cloud model—expectation, entropy, and hyperentropy—while a Gaussian potential function is adopted to characterize how interaction strength varies with spatial distance. A MATLAB-based simulation platform is developed to reproduce the emergent transition of fireflies from disordered flashing to synchronous behavior across different spatial domains. Our results demonstrate that group synchronization efficiency is strongly governed by the initial spatial configuration of individuals. Introducing driving nodes (special agents) can accelerate synchronization, but increasing their quantity yields diminishing marginal returns; the optimal strategy is a single special agent placed at the domain center. Collective emergence depends more strongly on the topological structure of inter-individual interactions than on the number of driving nodes. These findings confirm that interaction topology dominates synchronization performance over driving-node quantity, providing a theoretical basis for guiding unmanned swarm coordination with minimal control nodes.

Mathematics doi: 10.3390/math14152826

Authors: Arash Pourrezaee Ali Afzali Shahryar Sorooshian

This paper aims to present efficient alternative mixed-integer non-linear programming models and a customized hybrid metaheuristic, the Genetic-Based Algorithm (GBA), for a resource-constrained project-scheduling problem with a flexible network structure (RCPSP-FNS). We also consider the cost–time trade-off in the problem with a flexible network by using activity-duration compression. We present three approaches to solve the problem, including a mixed-integer non-linear program (MINLP) using binary variables representing activity completion times (MINLP1), an alternative mixed-integer non-linear program using integer variables representing activity-completion times (MINLP2) that has not presented before in RCPSP-FNS modeling, and the GBA. A total of 35 different problems are solved to examine the computational efficiency of the solution approaches. The MINLP1 and MINLP2 models are both solved by the GEKKO solver. The results indicate that solving the MINLP2 model can reach the optimal objective value obtained by solving the MINLP1 model in significantly less time. In addition, the proposed genetic-based algorithm can solve some large problems in a more efficient way in comparison to solving MINLP1 by using GEKKO. However, solving the MINLP2 model using GEKKO is the most efficient solution approach in comparison to both MINLP1 and the proposed genetic-based algorithm. MINLP2 can be solved to proven optimality (in much less time) for problems in which the MINLP1 model can, at most, reach near-optimal solutions.

Mathematics doi: 10.3390/math14152825

Authors: Khalil Jouili Monia Charfeddine Mongi Ben Moussa

This paper addresses the voltage regulation problem for DC microgrids modeled as nonlinear dynamical systems subject to parametric uncertainties and external disturbances. A data-driven predictive hybrid control scheme is developed, combining a nonlinear sliding mode law that guarantees finite-time current convergence, an adaptive fuzzy universal approximator that compensates for unknown residual dynamics and mitigates chattering, and a recursive predictor built online via forgetting-factor recursive least squares. Real-time gain optimization is achieved through the minimization of a quadratic predictive performance index. A composite Lyapunov analysis rigorously establishes uniform ultimate boundedness of the low level Adaptive Fuzzy Sliding Mode Control (AFSMC) inner loop, assuming bounded reference currents provided by the DQL agent and characterizes the convergence residual set of the tracking error. Comparative simulations against conventional fuzzy logic and a standard (non augmented) Deep Q-Learning baseline with fixed gain SMC corroborate the theoretical guarantees, demonstrating superior voltage regulation, reduced battery deep discharges, and improved load management.

Mathematics doi: 10.3390/math14152824

Authors: Ahmed Bchatnia Saleh Fahad Aljurbua

This work studies an optimal control problem for a wave equation with a moving localized damping, modeled by the characteristic function of a ball whose center evolves according to a controlled second-order dynamical system. The objective is to minimize the H1-norm of the time derivative of the wave at a final time, while penalizing the control effort through a quadratic cost on the acceleration. We first establish the local well-posedness of the coupled system, consisting of a damped wave equation and an ordinary differential equation for the damping center. Under suitable compactness and regularity assumptions on the admissible controls, we prove the existence of at least one optimal control via the direct method of the calculus of variations. We then derive first-order necessary optimality conditions through an extended Lagrangian formalism, yielding a coupled system involving the primal state, two adjoint states, and a pointwise relation linking the optimal control to the adjoint variable. The gradient of the cost functional is computed explicitly, showing that the sensitivity is concentrated on the boundary of the moving damping zone. Finally, a numerical implementation based on a gradient descent algorithm is proposed, using a Newmark scheme for the wave equation and a Verlet scheme for the trajectory, with regularization of the nonsmooth indicator function. The resulting algorithm provides a systematic approach for computing optimal damping trajectories in applications such as vibration suppression and noise control.

Mathematics doi: 10.3390/math14152823

Authors: Yihong Wang

Multi-linear systems with Z-tensors are more general than those with M-tensors. In this paper, we consider multi-linear systems with Z-tensor coefficients. Theoretically, we provide a convergence analysis and compare the spectral radii of the preconditioned iteration tensor and the original tensor within the Z-tensor framework. Numerical examples are presented to verify the theoretical spectral radius comparisons. Additionally, numerical examples on nonsingular M-tensors demonstrate the effectiveness of the proposed preconditioner.

Mathematics doi: 10.3390/math14152822

Authors: Alaa Tharwat

This tutorial takes the reader on a historical and technical journey from the simple Perceptron (1958) to modern Convolutional Neural Networks (CNNs) that dominate spatial data processing (images and video). We start with the Perceptron’s linear classifier, then expose its inability to learn non-linear patterns (e.g., XOR), which motivates the Multi-Layer Perceptron (MLP) and the backpropagation algorithm. Next, we discuss the limitations of MLP when faced with structured data like images—parameter explosion, loss of spatial information, and lack of translation invariance—and use these limitations as a natural springboard to the core ideas of CNNs: local connectivity, weight sharing, and hierarchical feature learning. Throughout, we provide intuitive explanations, mathematical formulations, and step-by-step numerical examples (e.g., a complete forward and backward pass for a small network, and a manual 2D convolution). Clear graphical representations and examples help readers understand each concept. The tutorial concludes with a detailed walkthrough of influential CNN architectures (LeNet-5, AlexNet, VGG, GoogLeNet, ResNet, DenseNet, and EfficientNet) and also discusses more recent attention-based models (e.g., Vision Transformers and ConvNeXt), explaining why each was necessary and how it advanced the field. Aimed at students and practitioners with a basic knowledge of calculus and linear algebra, this tutorial connects foundational ideas to state-of-the-art deep learning, focusing on spatial data. It is designed for readers who want to understand why each architectural choice was made, not just what the final model looks like.

Mathematics doi: 10.3390/math14152821

Authors: Mohamed Mahmoud Mostafa Farouk Senussi Mahmoud Abdalla Mahmoud SalahEldin Kasem Hyun-Soo Kang

Weakly supervised video anomaly detection (WS-VAD) localizes anomalous events in untrimmed videos using only video-level annotations. While CLIP-based methods have advanced this task through vision–language alignment, widely adopted approaches construct text prototypes from short category-name prompts of at most five words, leaving the CLIP text encoder not fully exploited. We propose SETAS-VAD, which addresses this gap through a Category Semantic Alignment (CSA) loss function: for each anomaly category, a large language model generates multi-sentence descriptions covering complementary semantic aspects, encoded once offline into frozen prototype vectors. An InfoNCE contrastive objective pulls attention-weighted anomaly features toward ground-truth category prototypes at zero additional inference overhead (prototype generation and encoding are performed once offline as a preprocessing step, not at test time). Under fully reproducible conditions on UCF-Crime and XD-Violence, SETAS-VAD achieves state-of-the-art temporal localization (30.45% mAP on XD-Violence, 12.16% on UCF-Crime), with per-threshold gains increasing at stricter IoU values, indicating improved boundary precision rather than coarse detection sensitivity.

Mathematics doi: 10.3390/math14152820

Authors: Wanjun Han Senxiang Lu Jingwen Bai

Magnetic leakage detection is one of the most commonly used methods for pipeline inspection, which mainly uses magnetic sensors to detect the magnetic leakage field on the internal and external surfaces of the pipeline to determine whether there are defects in the pipeline. The defect quantification algorithm includes a forward model and an optimization algorithm, in which the estimation of target defects using optimization algorithms is one of the key aspects of defect inversion. Most of the existing optimization algorithms are based on particle swarm algorithms (PSOs) and genetic algorithms (GAs), which are prone to premature problems and have low convergence accuracy. To address the problems in the process of defect inversion, this paper proposes a new inversion algorithm, which obtains part of the prior knowledge from the application context of defect inversion, and adopts the decay oscillation function as the nonlinear convergence factor based on the whale optimization algorithm (WOA). In addition, referring to the concepts of “genetic” and “mutation” in the GA, a diversity variation strategy based on dynamic step size is designed. The algorithm designed has the advantages of fast operation and high search accuracy. At the end of the paper, two sets of experiments are designed to compare the improved WOA with other existing optimization algorithms. The results demonstrate that the algorithm is significantly superior to other algorithms, both in the ideal case of simulation experiments and in the practical application of defect inversion.

Mathematics doi: 10.3390/math14152819

Authors: Shengyu Cai Jianchao Bai

Federated learning enables privacy-preserving distributed training but suffers from client drift and slow convergence under statistical data heterogeneity. Most existing federated optimization methods address client drift via parameter-space constraints or aggregation-level corrections, while fewer works directly shape the gradient covariance spectral structure of the optimization landscape. This paper analyzes the convergence problem from a spectral perspective, revealing that non-IID data causes spectral diffusion in the gradient covariance matrix and degrades convergence. Guided by random matrix theory, we propose federated spectral regularization (Fed-SR), a computationally efficient method that indirectly constrains spectral spread via gradient norm regularization. Although computing the regularizer gradient requires Hessian vector products, our optimized auto-differentiation implementation avoids storing full Hessian matrices and restricts extra computational overhead to a negligible level. Experiments on CIFAR-10, CIFAR-100, and other benchmarks show that Fed-SR outperforms baselines including FedAvg, FedProx, and SCAFFOLD in non-IID scenarios, reducing communication rounds and improving accuracy and stability. Ablation studies, spectral analysis, and controlled spectral feature manipulation experiments provide consistent empirical evidence showing a strong empirical association between the “spectral concentration” effect and performance gains, offering mechanistic interpretability consistent with our proposed theoretical framework within the tested experimental settings.

Mathematics doi: 10.3390/math14152818

Authors: Wei Song Yanhong Hu Pingping Zhang

The Feigenbaum–Kadanoff–Shenker (FKS) equation describes the quasiperiod route to chaos for circle maps. To facilitate research on the FKS equation, Shi introduced the second type of this equation and provided a construction method for its flat-bottomed single-valley solutions. In the present study, we introduce a new construction method for flat-bottomed single-valley solutions of the second type of FKS equation.

Mathematics doi: 10.3390/math14152817

Authors: Yeyang Han Ke Yan

Firms are increasingly introducing AI assistants into human-led live-streaming rooms, yet it remains unclear how such assistance should be configured and when it creates economic value. We develop an analytical model that compares human-only live-streaming selling with human–AI collaborative live-streaming selling. The firm sets the selling price in both modes and, under collaboration, jointly chooses the AI capability level. The model captures two channels through which AI may create value: enhancing the effectiveness of the human host and generating demand spillover beyond the room’s baseline conversion. We derive the equilibrium price, AI capability, demand, and profit, and we identify the conditions under which collaboration outperforms human-only live-streaming selling. The results show that AI capability is more valuable when paired with a stronger host, whereas the effect of product quality on AI investment is not necessarily positive. Lower AI cost may support a higher selling price by enabling a more capable selling process. Moreover, demand improvement and profit improvement need not occur simultaneously. Extensions examine AI-only live-streaming selling and imperfect AI assistance.

Mathematics doi: 10.3390/math14152816

Authors: Hassan Alfifi

This paper investigates the effects of diffusion and two time delays on the dynamics of an HIV infection model in one-dimensional domain. The Galerkin method is employed to derived theoretical equations. A condition is established for identifying exact Hopf bifurcation points, followed by an in-depth discussion on stability analysis maps. Bifurcation maps are constructed to illustrate the system dynamics under three distinct delay scenarios, highlighting the stable and unstable areas. When the delay parameter τi>0, there are two distinct stability areas, whereas in the absence of delay, only one stable region exists. The findings indicate that the time delays and the diffusion rate significantly influence the stability regions of the system. Bifurcation maps are presented to illustrate selected examples of 3D periodic oscillations to validate all theoretical outputs shown in this paper.

Mathematics doi: 10.3390/math14152815

Authors: Jiayu Zhang Yang Liu

Uncertain delay differential equations can simultaneously describe uncertain disturbances and time-delay effects during the evolution process of a system, and parameter estimation is a crucial step in the application of uncertain delay differential equations. To address the issue of equal-weight imbalance in residual-based moment estimation, this paper investigates a weighted moment estimation method for uncertain delay differential equations based on uncertain theory. Specifically, a counterexample is presented to illustrate the equal-weight imbalance problem in the existing residual-based moment estimation, and a weighted moment estimation approach for uncertain delay differential equations is derived based on the idea of relative deviation. Subsequently, a numerical algorithm for calculating the weighted moment estimation is designed, and an empirical study modeling population birth rates is also provided to demonstrate the effectiveness of the method and numerical algorithm proposed in this paper.

Mathematics doi: 10.3390/math14152814

Authors: Mingyang Li Woraphon Yamaka Paravee Maneejuk

This study examines the asymmetric relationship between housing prices and stock market returns across China’s major economic regions. Specifically, it investigates whether positive and negative shocks exhibit different transmission dynamics and whether these dynamics vary across regions characterized by different levels of financial development and housing market maturity. Using monthly data from 2005 to 2024, the study employs region-specific asymmetric vector autoregression (VAR) models, asymmetric Granger causality tests, and generalized impulse response analysis based on asymmetric decompositions of housing prices and stock market returns. The results suggest that statistically significant housing-to-stock predictability is observed primarily following negative housing price shocks in selected regions, whereas positive shocks generally exhibit weaker or statistically insignificant predictive effects. Conversely, positive stock market shocks generally provide more consistent evidence of stock-to-housing predictability, particularly in the Eastern and Central regions, although the responses are more mixed in the Western region and vary in magnitude, statistical significance, and persistence across regional markets. Overall, the results provide evidence of heterogeneous dynamic transmission patterns across China’s major economic regions and suggest that housing-related downside risk may represent an important source of regional macro-financial vulnerability.

Mathematics doi: 10.3390/math14152809

Authors: Norah Alshehri Mohammed Guediri

This paper examines the classification of Lorentzian hypersurfaces in Minkowski space, specifically focusing on those that are characterized as good or bad and that admit Ricci solitons with a concircular vector field. First, we demonstrate that Ricci solitons of this nature do not exist in Lorentzian spaces that have non-zero sectional curvature. Additionally, we show that the shape operator can be expressed in specific canonical forms.

Mathematics doi: 10.3390/math14152811

Authors: Chunlong Fu Jiaxin Zou Guofang Liu Pingli Zheng Kaiwen Xiao Yang Deng Hongxia He Qi Jiang

In the single-depot multi-traveling salesman problem, traditional depot location methods often overlook task balance among traveling salesmen, leading to excessive load on certain units and compromising overall operational efficiency. To address this issue, this paper proposes an optimized depot location method based on clustering and multi-task balancing. The core contribution lies in the design of a multi-weight adaptive depot optimization method. This approach clusters city nodes into multiple groups through cluster analysis and dynamically synthesizes direction vectors using information such as the number of samples within each cluster and the convex perimeter. It iteratively optimizes depot locations, minimizing the total path length while enhancing workload balance across all traveling salesman routes. Additionally, a “divide-and-conquer” strategy decomposes the complex MTSP into multiple parallel TSP subproblems, which are then efficiently solved using Or-Tools. A comprehensive evaluation framework is introduced, incorporating Total-Sum distance, Min-Max distance, Workload Balance, Cluster separability, Robustness, and Running time. Experimental results on the TSPLIB standard dataset demonstrate that the proposed method exhibits significant advantages over various traditional clustering algorithms in both route optimization and route balancing, validating its effectiveness and practicality. The method’s robust performance provides a reliable solution for real-world applications such as logistics distribution, further highlighting its practical value. Experimental results show that the proposed method reduces the total travel distance and improves workload balance on multiple TSPLIB instances compared with conventional depot selection baselines.

Mathematics doi: 10.3390/math14152813

Authors: Yingbiao Hu Huinian Li Yu He Zhenfu Pan Ningxia Chen Chengcheng Yang Wei Ke

Accurate and efficient recognition of tomato leaf diseases and pests is essential for precision agriculture, yet practical deployment remains challenging due to complex backgrounds, domain shifts, and subtle inter-class visual differences. From a broader mathematical perspective, this task can be viewed as token-level representation refinement under noise, ambiguity, and distribution shift. Recent state-space-model-based vision backbones provide favorable efficiency for high-resolution imagery but lack explicit mechanisms for adaptive feature refinement under noisy conditions. To address these issues, we propose EATMamba, an evolution-inspired Vision Mamba framework for tomato leaf disease and pest classification. Rather than being limited to a task-specific classifier, EATMamba is formulated as an evolution-inspired differentiable token-refinement mechanism designed to be compatible with state-space visual recognition backbones. EATMamba introduces two lightweight and fully differentiable modules—Evolutionary Crossover–Interaction and Knowledge-Guided Mutation–Selection—which perform token-level recombination and selective refinement to emphasize discriminative disease cues while suppressing irrelevant background information. These modules are inspired by crossover, mutation, and selection concepts, but are implemented as trainable differentiable operations over visual tokens, forming a generate–recombine–select style mechanism for representation refinement. The scope of this study is low-cost RGB-based visible-symptom disease and pest classification, rather than pre-symptomatic early disease detection. Extensive experiments on two complementary tomato datasets, including a controlled high-resolution dataset and an in-the-wild farm dataset, demonstrate that EATMamba consistently outperforms representative CNN-, Transformer-, and state-space-model-based baselines. Ablation studies and visualization analyses further confirm the complementary contributions of the proposed modules. Overall, EATMamba provides an effective and efficient framework for fine-grained plant disease recognition and illustrates how evolution-inspired principles can be incorporated into modern vision backbones for robust agricultural image analysis.

Mathematics doi: 10.3390/math14152812

Authors: Minjun Kim Gyuho Choi

Advanced driver assistance systems (ADAS) require a comprehensive understanding of multiple tasks related to the physical and mental states of drivers and traffic situations. Existing ADAS studies perform driver emotion recognition (DER), driver behavior recognition (DBR), traffic context recognition (TCR), and vehicle behavior recognition (VBR) using models designed based on single-task learning, thereby failing to reflect the interactions among tasks in real driving environments. This paper proposes perception and recognition with inter-modal selective multi-task learning (PRISM-MTL), an integrated multimodal and multi-task learning framework that jointly recognizes DER, DBR, TCR, and VBR. The proposed PRISM-MTL consists of a hierarchical stage-wise attention network (HSA-Net)-based multimodal encoder that extracts spatial features from heterogeneous multimodal inputs and task-specific modality fusion (TSMF), which selectively learns effective modality information for each task. This design addresses negative transfer, a key challenge in multi-task learning. In the multimodal encoder, HSA-Net extracts visual modality tokens that emphasize global structural patterns and key spatial regions from multi-view images, while Token-SE generates joint modality tokens that reflect the spatial configuration of joint data. TSMF generates task-specific fusion features that selectively emphasize the modality cues for each task. The generated task-specific fusion features are summarized through temporal mean pooling, and final predictions of driver states and traffic situations are produced by each task head. Experimental results show that the proposed PRISM-MTL achieves state-of-the-art performance on the public AIDE database, with an mAcc of 86.25% ± 0.35 for multi-task recognition of driver states and traffic situations.

Mathematics doi: 10.3390/math14152810

Authors: Yipene Cedric Francois Bassole Yunsick Sung

End-to-end autonomous driving systems have demonstrated remarkable potential in navigating complex environments by directly mapping sensory inputs to control actions. However, many existing approaches rely on a single encoder to compress driving big data into a latent vector, which does not explicitly convert driving conditions into internal safety constraints. In addition, some approaches often depend on static, inflexible safety boundaries driven by mathematical equations, and they lack a clear mechanism to adapt control actions under varying conditions. This disconnection between conditions and control actions limits their ability to maintain safe operational controls in diverse kinds of scenarios. To address these limitations, we propose a Big Data-Driven Multi-Constraint Learning framework to generate safe driving action proxies derived from ego motion represented by steering and acceleration. Our architecture uses two encoders in cascade: a Temporal State Encoder that extracts a latent state representation from input driving states using temporal self-attention, and a Latent Constraint Generation Encoder that transforms this latent state into a compact internal constraint vector representing flexible and dynamic safety boundaries instead of static mathematical equations. This constraint vector is fused with the latent state representation to guide a Control Network for action generation, while a Condition Classifier and a Safety Discriminator enforce condition adaptability and safety. We evaluated the framework on the large-scale nuScenes big data repository, and experimental results demonstrated that our deep learning analysis successfully uncovers hidden safety patterns across massive driving logs, yielding stable convergence with low errors, notably a normalized Steering Mean Absolute Error (MAE) of 0.0189, a normalized Acceleration Mean Absolute Error of 0.0387, and strong safety discrimination with a Safety Accuracy of 0.9996, indicating that the learned internal safety constraints effectively modulate control actions under diverse driving states and uncertainties.

Mathematics doi: 10.3390/math14152808

Authors: Gyuri Kim Yeongsu Bak

This paper proposes a motor control system implementation and speed estimation method using an artificial neural network (ANN) with offline training and online inference. Recent studies have increasingly explored the application of ANN to motor drive systems. However, these studies have focused on improving control performance, and discussion on how to implement and apply ANNs to motor drive systems remains relatively limited. To address this limitation, this paper proposes the procedures required to apply an ANN to a motor drive system, including data collection under various operating conditions, ANN model design and training, and online inference. The proposed method reduces the computational burden during training and achieves fast inference in the driving environment. In addition, an ANN model suitable for the motor drive system is designed through estimation performance comparison. The validity of the proposed system is verified through simulation and experimental results.

Mathematics doi: 10.3390/math14152807

Authors: Shiqian Liu Peizheng Wang Chao Wu

Existing data valuation methods assign value to individual data points, ignoring the workflow structure through which value is generated in machine learning (ML) pipelines. In practice, value emerges from a chain of interdependent actions—collection, preprocessing, feature engineering, and model training—performed by different agents with intertwined incentives. We propose a workflow-level data valuation framework that jointly addresses ownership attribution and compensation stability. We formalize the Data Chain (DC) and derive an ownership assignment rule from incomplete contract theory: The agents with the highest marginal contributions are retained in the compensation coalition. Under supermodular characteristic functions and a stated contextual-dominance condition, this choice does not increase the Least-core deficit relative to any competing retained set reachable by single-action swaps. Recognizing that workflow structures demand coalition stability over distributional fairness, we adopt the Least-core with contribution-aware coefficients to allocate compensation. We prove that the surplus game transformation preserves the core structure (core translation invariance), that the Data Chain Shapley value satisfies the efficiency axiom, and that our LP formulation with Wi=|ψi| guarantees feasibility. Experiments on multiple datasets show that, with a linear base classifier, competitive integration achieves full or near-full coalition-constraint satisfaction while preserving per-agent feasibility. We also assess supermodularity empirically and identify a narrower stability regime when high-capacity models are substitutable. These results define the framework’s present scope and scalability limits.

Mathematics doi: 10.3390/math14152806

Authors: Axel Schulze-Halberg

We construct classes of indefinite integrals that involve exceptional orthogonal polynomials of Laguerre, Jacobi, or Hermite types. By means of a recently devised method, these integrals can be represented in closed form. As a byproduct, we obtain integrals that involve confluent Heun or Heun polynomials.

Mathematics doi: 10.3390/math14152805

Authors: Tristan Guillaume

We study the small-radius topology of the Euclidean tube generated by a smooth planar Gaussian path. Let X=(Xt)0≤t≤T be a sufficiently regular Gaussian process with values in R2, and define its radius-ε tube by Tε={x∈R2:dist(x,X([0,T]))≤ε}. When the sample path is a smooth immersed curve with finitely many transversal double points and no higher-order intersections, the tube is, for all sufficiently small radii, a regular neighbourhood of the finite planar graph traced by the curve. Consequently, its Euler characteristic is determined by the number N of self-intersections: χ(Tε)=1−N for all sufficiently small ε, almost surely. We combine this deterministic topological observation with the Kac–Rice formula applied to the two-parameter Gaussian difference field F(s,t)=Xt−Xs. This yields an exact integral expression for the expected limiting Euler characteristic E[χ0], where χ0=limε↓0χ(Tε). For stationary Gaussian coordinates with covariance r, the formula reduces to a one-dimensional integral depending only on r, r′, and r″. We then specialize to the squared-exponential covariance r(u)=exp(−u2/2ℓ2), obtaining a fully explicit dimensionless quadrature depending only on T/ℓ. For the squared-exponential kernel, the expected number of limiting small-radius holes satisfies E[N]∼18π(T/ℓ)2 in the short-correlation regime. Finally, we add a constant linear drift and show that the expected number of self-intersections is modified by two competing mechanisms: a Gaussian density-damping term and a noncentral velocity-amplification term.

Mathematics doi: 10.3390/math14152804

Authors: Yi-Chung Hu Geng Wu Yu-Chao Cheng

Previous studies in passenger flow forecasting commonly employ decomposition ensemble models with linear addition, treating individual single-component forecasts with equal weights, to produce ensemble forecasts. It is known that fuzzy cognitive maps (FCMs) in multiple criteria decision-making are capable of modeling system dynamics by catching the causal relationships between the concepts describing a system. To enhance the forecasting ability of decomposition ensemble models with linear addition, this study aims to develop weighting schemes based on FCMs and grey relational analysis (GRA). Time series are decomposed into several components, and neural networks are applied to forecast individual components. Then, GRA is applied to assess the weights for individual single-component forecasts. To obtain ensemble forecasts, an optimal FCM determined by a genetic algorithm is used to determine the final combination weights for individual single-component forecasts. In comparison with benchmark models, the results show that the proposed FCM-based decomposition ensemble models significantly effectively improve the forecasting accuracy of air passenger flow in Taiwan across different forecasting horizons.

Mathematics doi: 10.3390/math14152803

Authors: Shin-ichi Inage

A persistent concentration of scale-critical quantities is widely regarded as a necessary precursor to finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equations. This paper develops a conditional structural closure framework for excluding a dynamically sustained persistent concentration within an admissible low-Mach realizable class. A localized amplitude-modulation decomposition of High–High transfer activity is introduced together with a localized Littlewood–Paley bridge linking transfer-amplitude concentration to mechanical concentration. A concentration-representation principle and a low-Mach inheritance mechanism then show that a persistent mechanical concentration necessarily generates a nontrivial thermo-acoustic trace. A central result is a localized structural necessity principle establishing that a persistent concentration necessarily generates a localized transfer-amplitude concentration. Combining the localized Littlewood–Paley bridge, the concentration-representation mechanism, the low-Mach inheritance principle, and a thermo-acoustic ε-regularity framework, we show that the resulting thermo-acoustic concentration scenario is excluded within the admissible low-Mach realizable class. Consequently, a dynamically sustained persistent concentration cannot occur in this class. The result is a conditional structural closure theorem: it excludes a persistent concentration only within the admissible low-Mach realizable class and does not constitute an unconditional regularity theorem for arbitrary solutions of the three-dimensional incompressible Navier–Stokes equations. The sole non-universal assumption is admissible low-Mach realizability, which is shown to follow whenever an admissible entropy-coercive, weakly compressible approximation exists. The principal remaining open problem is the universality of admissible low-Mach realizability.

Mathematics doi: 10.3390/math14152802

Authors: Donghyuk Choo Haill An Younhyun Jung

Sparse-view computed tomography (CT) reconstruction aims to recover high-quality CT volumes from a limited number of X-ray projection images, thereby reducing radiation exposure during image acquisition. However, this problem is inherently ill-posed because each projection provides only indirect line-integral supervision, and different attenuation distributions can explain similar sparse measurements. Existing analytic and iterative methods often suffer from streak artifacts and unstable solutions, while supervised learning-based methods require paired training data and may generalize poorly across anatomical regions or acquisition settings. Neural Radiance Field (NeRF)-based methods have recently shown promise by representing the attenuation field as a continuous coordinate-based function optimized directly from projection images. Nevertheless, these methods mainly enforce projection consistency and do not explicitly use volume-domain uncertainty to guide subsequent reconstruction. In this work, we propose EpiC-NeRF, a CT-specific closed-loop framework that actively feeds estimated epistemic uncertainty back into sparse-view reconstruction. EpiC-NeRF adapts evidential uncertainty estimation and aggregation to the X-ray CT line-integral formulation and maintains the resulting spatial uncertainty in a persistent three-dimensional Epistemic Grid Map. The accumulated uncertainty is used by Epistemic-Adaptive Layer Normalization to modulate intermediate features and by dual active sampling to guide ray- and point-level sample allocation. The newly estimated uncertainty then updates the grid map and guides subsequent optimization iterations, forming a unified feedback loop between uncertainty estimation and CT reconstruction. Experiments on four CT volume datasets demonstrate that EpiC-NeRF achieves improved reconstruction fidelity over existing analytic, iterative, and neural implicit reconstruction methods.

Mathematics doi: 10.3390/math14152801

Authors: Numan Yalcin

Classical Laplace-, Sumudu-, and Elzaki-type transforms are formulated within additive analytical frameworks and do not naturally accommodate multiplicative scaling structures arising in non-Newtonian calculus. Motivated by this limitation, this study introduces a non-Newtonian Laplace–Sumudu–Elzaki transform (NNLSET) based on logarithmic scaling mechanisms, multiplicative measures, and power-type kernels. The proposed framework is constructed by replacing the classical measure dt with the multiplicative measure dt/t and the linear scaling structure fut with the nonlinear scaling structure ftu. Using the logarithmic transformation t=ex, a canonical kernel representation of the form t−αu is derived, establishing a correspondence between multiplicative power-type kernels and weighted exponential structures in the logarithmic domain. Within an admissible weighted function framework, several analytical properties of the transform are established, including existence, boundedness, stability, uniqueness, restricted recoverability, and a logarithmic derivative representation associated with expressions of the form tf′t. A comparative analysis with the classical Laplace–Sumudu–Elzaki framework, together with illustrative differential-equation examples, a representative nonlinear MEMS oscillator, and a numerical computation, is presented. The obtained results demonstrate that the NNLSET provides a mathematically consistent framework for the analysis of multiplicative structures, logarithmic scaling phenomena, and logarithmically structured differential equations. Its applicability is further illustrated through the analysis of a representative nonlinear MEMS oscillator.

Mathematics doi: 10.3390/math14152800

Authors: Alanoud Subahi

Smart-manufacturing Industrial Internet of Things (IIoT) deployments increasingly depend on low-latency device-to-device (D2D) communication among resource-constrained, physically exposed field devices. This setting makes mutual authentication and session-key establishment difficult: public-key-intensive or cloud-dependent schemes add overhead, availability dependence, and single points of failure, while weak PUF-based designs may expose challenge-response pairs (CRPs) to replay, disclosure, and modeling attacks. This paper proposes PASMAP, a lightweight PUF-ASCON mutual authentication protocol for gateway-assisted D2D communication in smart-manufacturing IIoT. PASMAP combines SRAM-PUF key reconstruction, fuzzy-extractor helper data, hash- and XOR-based obfuscation, and ASCON authenticated encryption with associated data (AEAD) to protect hardware-rooted identities, hide raw PUF responses, and establish fresh session keys for post-authentication data exchange under an explicitly trusted local-gateway model. The protocol is evaluated against physical, protocol-level, and insider threats, including cloning, tampering, replay, man-in-the-middle, CRP disclosure, PUF modeling, stolen-verifier, and known-key attacks. A real-or-random (ROR) analysis bounds the adversary’s session-key advantage using hash collisions, PUF-response prediction, online guessing, and ASCON AEAD security. A mixed-platform evaluation based on ESP32 primitive timings for the edge devices and desktop timings for the resource-rich gateway yields an estimated total computation cost of 4.762 ms. The initiator and responder require 2.006 ms/264.79 μJ and 2.679 ms/353.63 μJ of computational energy, respectively, while the five-message exchange carries 4704 bits. These results indicate low computational overhead under the stated benchmark and power-model assumptions. However, the protocol totals are operation-count-based estimates, the PUF and fuzzy-extractor operations are simulated, and the energy model excludes several platform- and communication-dependent costs. A complete embedded implementation is therefore required to validate end-to-end latency, memory use, energy consumption, communication-stack overhead, SRAM-PUF reliability, and fuzzy-extractor performance.

Mathematics doi: 10.3390/math14152798

Authors: Weitao Hu Dashan Fan

Commutators of fractional integral operators play an important role in harmonic analysis due to their close connections with function regularity and partial differential equations. In this paper, we study higher-order commutators of fractional integral operators with rough variable kernels. Compared with first-order commutators, the higher-order setting involves more complicated interactions between the oscillation of the underlying BMO function and the fractional integral structure, which requires new ideas and techniques. We establish boundedness properties for these higher-order commutators under sharp conditions on the angular integrability of the variable kernels. Our results extend the existing boundedness theory of first-order commutators to higher-order cases and demonstrate the applicability of the developed techniques to a broader class of fractional integral operators with rough variable kernels.

Mathematics doi: 10.3390/math14152799

Authors: Antonio Joaquín Segura García Ziwei Shu Ramón Alberto Carrasco

Probabilities in sports forecasting are shaped by betting-market mechanisms, where expectations are exchanged through prices and transformed into implied probabilities. Although probabilistic predictions are usually treated as quantitative variables, rare events may be represented with lower resolution, making some probability ranges behave closer to ordinal structures. This paper investigates this hypothesis using NBA betting odds. We directly assess ordinality by examining the loss of linearity between implied probabilities and empirical outcome frequencies while evaluating whether monotonicity is preserved across probability ranges. The results show that low-probability odds deviate from the linear behavior expected under quantitative assumptions, while largely maintaining their ordinal ordering. The ordinal hypothesis is independently evaluated through a predictive experiment based on a Gated Recurrent Unit model. The model uses team-level temporal sequences as input features to predict the probability of a target game. We compare the behavior of the ordinal-aware CORAL loss across the analyzed subsets. The results suggest that the subset exhibiting ordinal characteristics attains a lower statistical risk under this ordinal inductive bias than the subset characterized by quantitative behavior. Finally, we discuss the broader applicability of comparative training with loss-function-induced inductive biases as a practical method for identifying the measurement level of target variables.

Mathematics doi: 10.3390/math14152797

Authors: Reem Alrebdi

In this article, a nonlinear coupled system involving Caputo difference operators of distinct orders subject to novel coupled boundary conditions is studied. Existence and uniqueness solutions are established by applying the standard fixed-point theorem under suitable assumptions. The stability behavior of the considered system is analyzed using the Hyers–Ulam stability approach, and sufficient limitations ensuring the stability of the results are obtained. To demonstrate the validity of the theoretical solutions, two illustrative examples are presented. The first example introduces the assumptions required for the existence, uniqueness, and stability results. The second example verifies a financial discrete coupled system solved numerically via a discrete iterative method. Numerical simulations are performed for different fractional orders to examine the influence of memory effects on solution dynamics.

Mathematics doi: 10.3390/math14152796

Authors: Feiyan Li Jiahao Li Hongfei Jia Xinxin Zhang Tianci Gao Zetong Qin Hangtian Du

Accurate vehicle trajectory prediction is essential for autonomous driving safety. However, existing data-driven models often ignore kinematic constraints, causing lateral jitter and trajectory distortion, while purely kinematics-based models lack flexibility in complex interactions. To address this, this paper presents a multimodal trajectory prediction method combining kinematics with dynamic interaction features. Operating in the Frenet coordinate system, the proposed model extracts historical features via a Bidirectional Gated Recurrent Unit (Bi-GRU) and utilizes an Adaptive Social Gating Network (ASGN) with multi-head attention to filter irrelevant interaction noise. This paper introduces a Stochastic Gating Decoder for multimodal latent variable sampling, adaptively fusing kinematics and data-driven paths to capture driver intention uncertainty while maintaining kinematic consistency. The model is trained using a composite loss function (Focal Loss and Best-of-K) to mitigate dataset long-tail distribution and trajectory divergence. Experiments on the HighD dataset show the proposed model achieves a minADE of 0.425 m and a minFDE of 0.955 m, outperforming baselines and reducing Lat-ADE by 53.9% compared to Social-GAN. These results confirm the model generates smoother, kinematically interpretable trajectories with higher accuracy in long-tail lane-changing scenarios.

Mathematics doi: 10.3390/math14152795

Authors: Wu Wen Jinghui Luo Kailun Qiu Zhong Xiao Gen Lai

Self-Supervised Learning (SSL) has attracted increasing attention in remote sensing image understanding because it can learn transferable representations from unlabeled images. However, two issues remain insufficiently examined in collaborative SSL for remote sensing. First, when high-ratio masking removes entire small objects or structurally informative regions, the remaining visible patches may provide insufficient evidence for semantically coherent reconstruction. Second, heterogeneous self-supervised objectives may exhibit different loss scales, gradient magnitudes, and convergence behaviors such that fixed coefficients can produce uneven branch contributions during training. To address these issues, this paper proposes Context-Consistency Enhanced Collaborative Self-Supervised Learning (C2DSSL) for remote sensing image understanding. C2DSSL introduces a teacher–student context-consistency constraint, in which multi-scale reconstruction features from the complete teacher observation serve as contextual targets for the student network when reconstructing the corresponding masked observation. In addition, gradient-sensitive dynamic weighting uses temporally smoothed loss-gradient magnitudes as empirical signals to adjust the relative contributions of heterogeneous self-supervised objectives. Under the evaluated settings, adding the context-consistency constraint improves KNN representation evaluation, UCMerced classification, Potsdam semantic segmentation, and DOTA oriented object detection, while maintaining the Baseline performance on the Million-AID subset. Dynamic weighting shows task-dependent effects, including a performance gain on the Million-AID subset classification task when combined with CCL.

Mathematics doi: 10.3390/math14152794

Authors: Kai-Cheng Wang

We establish explicit direct and inverse approximation estimates for biorthogonal multiresolution projections on weighted Besov spaces over Muckenhoupt Ap weights. A Jackson-type direct estimate bounds the weighted Lp projection error by 2−Js times the weighted Besov norm, and a matched Bernstein-type inverse estimate bounds the weighted Besov seminorm of a resolution-space element by 2Js times its weighted Lp norm. Every constant is displayed in factorized form: each factor is either given in closed form or is the operator norm of the Hardy–Littlewood maximal operator on the weighted Lebesgue space, through which the entire dependence on the Muckenhoupt characteristic is routed. For a piecewise regular class combining a Besov-smooth component with finitely many net-zero jumps, the projection error separates into a smooth part decaying at 2−Js and a localized jump part carrying the weighted measure of a shrinking interval about each jump; when the weight is locally comparable to Lebesgue measure near the jumps, this yields the effective rate min(s,1/p). A deterministic numerical experiment confirms this rate within one fixed biorthogonal analysis, and the same projection is illustrated on an empirical higher-education finance series.

Mathematics doi: 10.3390/math14152793

Authors: Futi Liu Zian Zhao

Portfolio optimization is a fundamental problem in financial decision-making, and it is concerned with balancing expected return and investment risk. With the growing emphasis on sustainable investing, environmental, social, and governance (ESG) criteria have been incorporated into portfolio optimization. In practice, ESG-aware portfolio optimization faces parameter ambiguity from market fluctuations, delayed ESG disclosure, and rating disagreement, and the exposure to return uncertainty may depend on portfolio decisions rather than being fully exogenous. Existing studies, however, generally specify uncertainty sets independently of portfolio decisions. To address this limitation, an adjustable robust approach is proposed for ESG-aware portfolio optimization under decision-dependent return uncertainty. A joint polyhedral uncertainty set is constructed to capture the ambiguity in asset returns and ESG scores, where the return bounds depend on first-stage portfolio weights through ESG-related holdings, whereas ESG score uncertainty remains decision-independent. A two-stage robust framework with recourse rebalancing and proportional transaction costs is formulated, with financial loss and ESG performance balanced in the objective and tail risk controlled by a CVaR constraint embedded in a column-and-constraint generation scheme. The resulting minimax problem is solved by a column-and-constraint generation algorithm with a Rockafellar–Uryasev linearization of CVaR over iteratively generated scenarios. Numerical experiments using real stock data are designed to evaluate downside-risk control and portfolio ESG performance relative to deterministic and classical robust benchmarks.

Mathematics doi: 10.3390/math14152792

Authors: Almaz Mustafin

Traditional frameworks of innovation diffusion, such as epidemic and Lotka–Volterra–Gause models, treat technological substitution primarily as a population-driven or social communication process, frequently overlooking the critical constraints imposed by external factor scarcities. To address this fundamental economic gap, this study performs a qualitative analysis of exploitative competition between two distinct technologies sharing two complementary resources, modeled via a non-linear system of chemostat-type consumer–resource ordinary differential equations. Technologies are represented as homogeneous populations of elemental firms, where individual output is governed by a ratio-dependent, fixed-proportions Leontief production function integrated with a hyperbolic clearing response. Operating within an open industrial system, the model accounts for resource supply rates and firm exit dynamics. We analytically derive the coordinates of both boundary and interior fixed points within the non-negative orthant of the phase space. By investigating the eigenvalues of the associated Jacobian matrix, we establish necessary and sufficient conditions for local asymptotic stability, competitive exclusion, and technological coexistence, demonstrating that efficiency is determined by a break-even resource availability threshold. Our results reveal that structural reconfigurations of the industry supply plane trigger bifurcations between local dominance and multistability. The latter manifests as a path-dependent, Quastlerian selection of initial conditions rather than inherent technological superiority. Finally, we establish the geometric boundaries of the stable assemblage niche, proving that technological diversity is regulated by resource supply rates. By explicitly incorporating resource scarcity into a dynamical predator–prey framework, the proposed model offers a more robust economic and mathematical foundation for innovation diffusion, providing policymakers with structural insights into the resource allocation strategy and the long-term management of industrial diversity.

Mathematics doi: 10.3390/math14152791

Authors: Ivan Mezník

The class of ∞—languages (IG-languages) generated by incomplete generating machines (IG-machines) is investigated. The cardinality characterizations of words, ω-words and IG-languages generated by IG-machines are given, and some closure properties of IG-languages are studied.

Mathematics doi: 10.3390/math14152790

Authors: Yuri V. Tyutyunov Vasily N. Govorukhin Vyacheslav G. Tsybulin

We formulate and study a predator–prey model with pursuit-evasion spatial behaviour, incorporating the fear effect in a sexually-reproducing prey population. The pursuit-evasion movements are described as indirect taxis: populations respond to diffusively dispersed and decaying kairomonal cues of their antagonists. The prey-emitted kairomone attracts predators, while the predator-emitted kairomone repels prey and locally reduces prey reproduction rate, mimicking a physiological fear response. To isolate the net effect of predator’s prey-taxis, we assume predator birth/death rates are negligible, implying a constant predator abundance. Linear stability analysis yields a condition for taxis-driven oscillatory instability of the homogeneous steady state. Numerical simulations reveal spatially heterogeneous dynamics, coexistence of periodic travelling waves, and transitions to spatiotemporal chaos. The results highlight interrelations between fear responses, spatial movements, spatiotemporal heterogeneity, and viability of the trophic system.

Mathematics doi: 10.3390/math14152789

Authors: Gianina-Maria Petrașcu Ioana Bîrlan Cristina-Rodica Boboc Adriana AnaMaria Davidescu

Rapid AI adoption has intensified public and media attention toward AI-related developments, making news sentiment an increasingly important indicator of expectations and perceptions. This study constructs a global daily AI News Sentiment Index using data from the GDELT Global Knowledge Graph and examines the forecasting properties of the index using statistical, sequential deep learning and image-based forecasting methods. The period of observation of the dataset spans from January 2016 to December 2025, and the sample consists of 3635 daily observations. A sentiment index is constructed from the positive and negative sentiments and is analyzed together with its constituent series. For the empirical framework, ARIMA, ARIMA–GARCH, ETS, LightGBM, LSTM, CNN, TCN, GAN, and convolutional models based on Gramian Angular Summation Field (GASF) and Gramian Angular Difference Field (GADF) transformations are compared using accuracy evaluation on the out-of-sample test set. In this study, we find no evidence that increasing model complexity leads to better forecasting performance for the AI news sentiment series. In most cases, sequential forecasting models perform better than their image-based counterparts, while GASF and GADF transformations do not deliver consistently better forecasting performance for sentiment series of different types and different forecasting horizons. This result indicates that transforming noisy sentiment time series into an image may hide rather than preserve useful information for forecasting purposes. The study contributes to the growing literature on AI news sentiment forecasting by providing a comprehensive comparison of statistical, sequential, and image-based forecasting paradigms and offers practical insights for researchers, policymakers, and practitioners interested in monitoring AI-related expectations and sentiment dynamics.

Mathematics doi: 10.3390/math14152782

Authors: Ning Jin Zhiying Wang Ruohan Ma

Short-term prediction of hazardous gas concentrations is crucial for industrial air monitoring, but conventional approaches often fail to capture abrupt local fluctuations and nonlinear temporal dependencies, limiting prediction accuracy. To address these limitations, this study develops a multi-task residual Transformer-based framework for short-term concentration forecasting. First, historical high-frequency H2S measurements are processed using a sliding-window approach to form input sequences for the model. Next, a shared Transformer encoder extracts temporal features, while task-specific branches perform residual concentration prediction and concentration-based emission-state classification. Within this multi-task framework, an adaptive weighting mechanism emphasizes high-variation samples during training to improve sensitivity to rapid concentration changes. Experiments conducted on data from the South Coast Air Quality Management District demonstrate that, averaged over three random seeds, the model achieves an MAE of 0.133±0.001, an RMSE of 0.237±0.000, and an R2 of 0.810±0.001 for one-observation-step forecasting. These results show that the proposed framework effectively captures abrupt rises and peak concentrations, providing a reliable tool for industrial emission monitoring and early warning applications.

Mathematics doi: 10.3390/math14152788

Authors: Ymnah Alruwaily Slim Dhahri Foued Mtiri

In this paper, the convergence-to-zero and finite-horizon guaranteed-cost synchronization criteria are developed for linear Caputo–Hadamard fractional-order systems with an admissible time-varying delay. The delay assumption is expressed in such a way that is consistent with the Caputo–Hadamard Halanay inequality and the memory structure of the model, which is logarithmic in time. This analysis includes a quadratic Caputo–Hadamard Lyapunov method, Schur-complement bounds for the delayed channel and a supremum argument in logarithmic time. An important novelty in the proposed approach is that the current state, the delayed state and the Caputo–Hadamard derivative are not considered as independent augmented variables; this prevents the structural feasibility obstacle from occurring when using full-space residual LMI formulations. The convergence-to-zero condition is first established for the drive system. Next, a fixed-gain guaranteed-cost synchronization theorem is established and, by using a standard change of variables, a convex controller-synthesis condition is arrived at. An explicit logarithmic-time form of the finite-horizon cost estimate is derived. The criteria are further extended to systems with several admissible delays and to systems with norm-bounded parametric uncertainty. Four numerical examples are reported, in which feasible matrices, the controller gain, strict eigenvalue margins and a comparison of the simulated cost and the theoretical upper bound are given, together with a quantitative comparison against augmented-state linear matrix inequality formulations, a scalability study up to a dimension of 30 and a sensitivity study. The simulations show the dynamics that the theory predicts; the convergence to the asymptotics is valid for the LMI certificates checked in the simulations and for the Caputo–Hadamard Halanay inequality. In conclusion, the paper delivers a complete and numerically verifiable design chain for Caputo–Hadamard synchronization: admissibility of a possibly unbounded time-varying delay is checked directly, a stabilizing gain is obtained from a convex program whose largest block has size 2n instead of 3n, and an a priori cost certificate JT* is produced from the same feasible variables; on the reported benchmark, the method retains 95.7% of the admissible delay-channel gain of an augmented-state formulation while solving up to 21 times faster at dimension 30.

Mathematics doi: 10.3390/math14152784

Authors: Gürsel Yeşilot Elif Özel Ay

We introduce and study square-difference factor absorbing primary hyperideals (sdf-absorbing primary hyperideals) in commutative multiplicative hyperrings. A proper hyperideal I is called sdf-absorbing primary if x2−y2⊆I implies x+y∈I or x−y∈I. This class provides a proper common generalization of both primary hyperideals and sdf-absorbing hyperideals. We establish a comprehensive characterization; investigate the behavior under homomorphisms, localization, intersections, ascending chains, and Cartesian products; and identify the precise conditions under which the three notions coincide.

Mathematics doi: 10.3390/math14152786

Authors: Mohammad Aldossary Jaber Almutairi Ibrahim Alzamil

Predictive maintenance of electric vehicle (EV) fleets requires accurate estimation of Remaining Useful Life (RUL), Time-to-Failure (TTF), and State-of-Health (SOH) from heterogeneous Internet of Things (IoT) telemetry. However, real-world degradation patterns are nonlinear, nonstationary, and highly imbalanced near failure. This study proposes Q-TRACNet, a temporal attention framework that combines causal maintenance-aware preprocessing, adaptive temporal condensation, residual refinement, learnable phase modulation, and hybrid Particle Swarm Optimization–Quantum-Guided Descent parameter tuning. The framework is evaluated on the EV-HLM-RUL dataset and three established prognostics benchmarks: NASA CMAPSS, PHM 2012, and XJTU-SY. Chronological training, validation, and testing partitions are used to preserve temporal causality. On EV-HLM-RUL, Q-TRACNet achieves an MAE of 9.8, an RMSE of 14.7, an R2 of 0.979, and a Critical Degradation Awareness Index (CDAI) of 0.91. It reduces RMSE by 20.11% relative to the strongest competing baseline and achieves an NRMSE of 0.102 and a Kendall correlation of 0.89 (p<10−4). Cross-dataset experiments demonstrate stable performance for RUL, TTF, and short- and long-horizon SOH prediction. Ablation and sensitivity analyses further confirm the contributions of the temporal and attention components and the stability of degradation-aware evaluation. Q-TRACNet also provides lower training cost and inference latency than competing architectures, supporting practical maintenance planning, inspection prioritization, and resource allocation in connected EV fleets.

Mathematics doi: 10.3390/math14152787

Authors: Zhazira Yerkisheva Kulzina Nazarova Kairat Usmanov

We study a nonlocal boundary value problem for a system of functional-differential equations with involution and an additional parameter. The proposed approach is based on a parameterization method developed by D. Dzhumabaev. The original boundary value problem is transformed into an equivalent Cauchy problem posed at the midpoint of the interval, together with a system of algebraic equations for the unknown parameters. By exploiting the symmetry and antisymmetry properties of the solution, we reduce the resulting Cauchy problem to a system of coupled Volterra integral equations of the second kind. This reduction makes it possible to derive explicit solution representations and to establish necessary and sufficient conditions for unique solvability in terms of the invertibility of the matrix generated by the boundary conditions. Finally, a first-order functional-differential equation is analyzed to demonstrate the applicability of the proposed method and to illustrate the obtained solvability conditions.

Mathematics doi: 10.3390/math14152785

Authors: Yu-Min Lian Jun-Home Chen

This study extends analytical barrier-option pricing models by jointly incorporating Markov-switching liquidity risk, asymmetric double-exponential jump risk, and a state-dependent Heath–Jarrow–Morton interest-rate structure. The underlying stock price dynamics under imperfect liquidity are driven by a Markovian regime-switching liquidity-adjusted double-exponential jump-diffusion model, and the risk-neutral valuation is obtained through an Esscher transform. Compared with existing DEJD barrier-option, liquidity-adjusted option-pricing, and Markov-modulated stochastic-interest-rate models, the proposed models highlight the joint effects of liquidity conditions, jump risk, regime switching, and state-dependent forward rates on European-style barrier option prices. Numerical illustrations based on Monte Carlo simulation are provided to examine model implications and benchmark special cases.

Mathematics doi: 10.3390/math14152783

Authors: Ainur Zhumali Dauren Zhakebayev Kairzhan Karzhaubayev

The present work introduces a fully resolved three-dimensional thermal Lattice Boltzmann framework developed to investigate the impact of freely moving, finite-size spherical particles on natural convection within a cubic enclosure. The fluid-phase momentum and energy fields are resolved using coupled double-distribution function kinetic approach, while the solid phase is governed by explicitly coupled linear, angular, and thermal conservation equations. To accurately map the moving spherical surfaces onto the Eulerian lattice grid, a second-order linear interpolated bounce-back scheme is implemented. The conjugate heat transfer between the phases is simplified via a lumped capacitance model, assuming negligible internal thermal resistance within the solid spheres. Short-range particle–particle and particle–wall interactions are handled using Glowinski’s repulsive force model. The spatial accuracy of the framework is validated using a circular Taylor–Couette flow benchmark—demonstrating second-order spatial convergence and a differentially heated natural convection in a cubic cavity benchmark, yielding bulk Nusselt numbers within 1% of established literature data. This validated tool is subsequently used to analyze the complex interplay between particulate motion and bulk thermal transport efficiency. Analysis of the temperature fields reveals that the overall thermal structure is governed primarily by the Rayleigh number, while the low particle concentration produces only minor modifications to the convective heat transfer. In contrast, the particle distribution exhibits a strong dependence on the flow intensity.

Mathematics doi: 10.3390/math14152781

Authors: Meng Wang

This paper studies the Vehicle Routing Problem with Simultaneous Pickup and Delivery and Time Windows (VRPSPDTW). The problem arises from reverse logistics, last-mile distribution, and circular logistics, where vehicles must serve delivery and pickup demands while respecting vehicle capacity and customer time-window constraints. This work develops DBLS-SP. The method combines an improved insertion-based initialization procedure, a repair-oriented dynamic scoring strategy, ejection-based reinsertion, multi-armed-bandit control of the route-inheritance ratio, vehicle-layer archive-guided route descent, and quality-first nearest-neighbor pool replacement. Computational experiments on the Wang–Chen (WC) and JD Logistics (JD) benchmark instances show that DBLS-SP obtains the best algorithmic solution on 67 of 68 WC instances and the best or tied-best objective value on 16 of 20 JD instances. The ablation study confirms the contribution of the main components, and the proposed reduction strategy removes more infeasible paths than the original reduction rule while preserving feasibility.