Mathematics

17 pages, 6134 KB

Open AccessArticle

Distributed Cooperative Multi-Target Search for an Autonomous Underwater Vehicle Swarm in Unknown 3D Underwater Environments

by You Zhou, Mao Wang and Shaowu Zhou

Mathematics 2026, 14(12), 2236; https://doi.org/10.3390/math14122236 (registering DOI) - 22 Jun 2026

Viewed by 96

Abstract

This paper investigates the problem of multi-target search by an Autonomous Underwater Vehicle (AUV) swarm in unknown three-dimensional (3D) underwater environments with obstacles under limited communication conditions. To address this problem, a distributed cooperative search framework is proposed. Within this framework, an adaptive [...] Read more.

This paper investigates the problem of multi-target search by an Autonomous Underwater Vehicle (AUV) swarm in unknown three-dimensional (3D) underwater environments with obstacles under limited communication conditions. To address this problem, a distributed cooperative search framework is proposed. Within this framework, an adaptive dual-state search mechanism driven by a target response function is designed. This mechanism enables the swarm to transition between independent large-scale roaming search and precise cooperative search. On this basis, a multi-target search method is developed by integrating a virtual force model, motion-constrained 3D Particle Swarm Optimization (PSO), and a sectional 3D tangent-plane obstacle-avoidance method. Simulation results demonstrate the effectiveness and engineering feasibility of the proposed framework. Under the conditions of unknown terrains and communication limits, the AUV swarm can adaptively execute state transitions, safely avoid 3D obstacles, and complete multi-target search tasks. Specifically, as the swarm size increases from 30 to 60 AUVs, the mean number of iterations drops from 432.97 to 269.73, while the total energy consumption expectedly rises from 11.79 × 10⁴ to 15.51 × 10⁴, reflecting a well-balanced trade-off between efficiency and cost. This study provides a practical distributed control reference for AUV swarms in complex communication-constrained underwater scenarios. Full article

(This article belongs to the Special Issue Recent Advances in Nonlinear Control Theory and System Dynamics)

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17 pages, 3611 KB

Open AccessArticle

Advanced Negative-Derivative Feedback Control for Nonlinear Resonance Suppression in 2-DOF AFM Systems

by Khalid Alluhydan and M. N. Abd EL-Salam

Mathematics 2026, 14(12), 2235; https://doi.org/10.3390/math14122235 (registering DOI) - 22 Jun 2026

Viewed by 63

Abstract

A Negative Derivative Feedback (NDF) controller is designed for vibrations suppression of an atomic force microscope (AFM) model. The controlled system is modeled as a two-degree-of-freedom (2-DOF) closed-loop dynamic system. The average method was used to derive approximate analytical solutions. All possible resonance [...] Read more.

A Negative Derivative Feedback (NDF) controller is designed for vibrations suppression of an atomic force microscope (AFM) model. The controlled system is modeled as a two-degree-of-freedom (2-DOF) closed-loop dynamic system. The average method was used to derive approximate analytical solutions. All possible resonance conditions were identified, with particular attention given to the simultaneous resonance case

Ω = ω_{1}, Ω_{1} = {2 ω}_{1}, ω_{2} = ω_{1}

, identified as the most critical. For validation and proper insights, the system was also solved numerically using the fourth-order Rung–Kutta method. The time response of the AFM system in contact mode was analyzed before and after applying the NDF controller under the worst-case resonance conditions. A comprehensive parametric study was conducted to evaluate the controller’s robustness and effectiveness. The results demonstrate a high degree of agreement between the numerical simulations and the analytical approximations, confirming the reliability of the approach. Full article

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33 pages, 630 KB

Open AccessArticle

A New Class of Conway–Maxwell–Poisson Liu-Type Regression Estimators for Effectively Modeling Multicollinear Count Data

by Fatimah A. Almulhim, A.T. A. Hammad, Fathy H. Riad and M. A. El-Qurashi

Mathematics 2026, 14(12), 2234; https://doi.org/10.3390/math14122234 (registering DOI) - 22 Jun 2026

Viewed by 56

Abstract

One of the most widely used regression models for count data is the Conway–Maxwell–Poisson regression model (CMPRM), which often provides a better fit for over- and underdispersed count data than traditional models, such as Poisson regression and negative binomial regression. Parameter estimation in [...] Read more.

One of the most widely used regression models for count data is the Conway–Maxwell–Poisson regression model (CMPRM), which often provides a better fit for over- and underdispersed count data than traditional models, such as Poisson regression and negative binomial regression. Parameter estimation in the CMPRM is typically performed using the maximum likelihood estimation (MLE) method. However, when explanatory variables are highly correlated, a phenomenon known as multicollinearity arises, posing a significant challenge to the analysis. Multicollinearity makes it difficult to identify the individual effects of explanatory variables, leading to inflated variances and larger standard errors of the MLEs. To address the issue of multicollinearity, this paper introduces a new class of Liu-type estimators within the CMPRM. The proposed estimators aim to improve the estimation accuracy and reliability of the CMPRM compared with existing biased estimation methods. The efficiency of the proposed estimator is evaluated through theoretical comparisons and Monte Carlo simulation experiments conducted under various conditions. Furthermore, two real-data applications are presented to demonstrate the practical usefulness of the proposed estimation method. The results from the theoretical analysis, simulation study, and empirical applications indicate that the proposed estimators outperform existing methods in terms of achieving more accurate and reliable estimates. Full article

(This article belongs to the Special Issue Statistical Theory and Application, 2nd Edition)

44 pages, 535 KB

Open AccessArticle

Auto Ball Covariance and Correlation for Fixed-Lag Nonlinear Dependence in Time Series

by Qiang Zhang and Chaobang Gao

Mathematics 2026, 14(12), 2233; https://doi.org/10.3390/math14122233 (registering DOI) - 22 Jun 2026

Viewed by 59

Abstract

Classical tools for time series dependence analysis are primarily designed for linear dependence and may fail to detect serial structure when a series is uncorrelated but not independent. To address this problem, we propose the auto ball covariance function and the corresponding auto [...] Read more.

Classical tools for time series dependence analysis are primarily designed for linear dependence and may fail to detect serial structure when a series is uncorrelated but not independent. To address this problem, we propose the auto ball covariance function and the corresponding auto ball correlation function for measuring lag-specific nonlinear dependence in strictly stationary time series taking values in a separable Banach space. The proposed diagnostic uses metric-ball probabilities to measure fixed-lag distributional dependence without moment requirements, making it suitable for vector-, function-, and norm-induced object-valued time series. Under suitable conditions, we show that the proposed measure is zero if and only if the lagged components are independent. We further develop sample versions of the proposed statistics and establish their large-sample properties, including strong consistency under absolute regularity and a fixed-lag null asymptotic law under a finite-range dependence condition on the lagged-pair process. Simulation studies demonstrate that the proposed method performs well in a variety of settings, especially for nonlinear, heavy-tailed time series. A real-data analysis of annual sunspot numbers further illustrates how the proposed diagnostic can reveal nonlinear residual dependence that is not visible from ordinary autocorrelation diagnostics. Full article

(This article belongs to the Section D1: Probability and Statistics)

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11 pages, 4447 KB

Open AccessArticle

Design of a Tunable Multi-Band Transmitting and Band-Stop Photonic Crystal IR Filter Utilizing Lucas Numbers

by Çiğdem Seçkin Gürel and Berker Yalçın

Mathematics 2026, 14(12), 2232; https://doi.org/10.3390/math14122232 (registering DOI) - 22 Jun 2026

Viewed by 69

Abstract

In this study, a novel aperiodic photonic crystal (PC) structure is designed using the golden ratio-based Lucas sequence, and its infrared (IR) transmission characteristics are investigated. Transmission behavior demonstrates a strong dependence on the number of unit cells and parity (even or odd) [...] Read more.

In this study, a novel aperiodic photonic crystal (PC) structure is designed using the golden ratio-based Lucas sequence, and its infrared (IR) transmission characteristics are investigated. Transmission behavior demonstrates a strong dependence on the number of unit cells and parity (even or odd) of the defect layer repetitions, enabling the formation of a predetermined number of resonant modes around the operating wavelength and broad photonic stopbands at longer wavelengths with sharp defect modes. With its high spectral tunability, the proposed new Lucas sequence-based structure represents a viable candidate for the design of high-performance optical filters and components. These findings indicate that novel Lucas sequence-based PC designs will provide new opportunities for manipulating light–matter interactions in future studies. Full article

(This article belongs to the Special Issue Mathematics and its Applications in Science and Engineering, 4th Edition)

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46 pages, 1464 KB

Open AccessArticle

Mathematical Modeling and Dynamical Analysis of a Nonlinear Coupled Stress-Mitigation System with Signed Threshold-Relative Policy Feedback and Physics-Informed Neural Network Simulation

by Khaled Aldwoah, Faez A. Alqarni, Osman Osman, L. M. Abdalgadir, Amel Touati and Waleed Adel

Mathematics 2026, 14(12), 2231; https://doi.org/10.3390/math14122231 (registering DOI) - 22 Jun 2026

Viewed by 62

Abstract

This study develops and analyzes a four-state nonlinear policy–feedback dynamical system that couples a system stressor, an accumulated burden, a signed mitigation–response variable, and a signed policy-pressure variable. The proposed model represents governance response through a smooth threshold-centered feedback mechanism, in which the [...] Read more.

This study develops and analyzes a four-state nonlinear policy–feedback dynamical system that couples a system stressor, an accumulated burden, a signed mitigation–response variable, and a signed policy-pressure variable. The proposed model represents governance response through a smooth threshold-centered feedback mechanism, in which the policy-pressure dynamics depend continuously on the deviation of the stressor from a prescribed reference threshold. Unlike reduced-order formulations with purely exogenous interventions, the present framework generates endogenous interactions among stress accumulation, burden evolution, mitigation response, and policy adjustment. The qualitative analysis establishes local well-posedness in the admissible phase domain, conditional nonnegativity of the accumulated burden, and boundedness of trajectories on admissible intervals. An autonomous effective system is then derived to characterize quasi-stationary mean behavior of the periodically forced dynamics. For this effective system, local stability is investigated using Gershgorin estimates and Routh–Hurwitz criteria, leading to explicit analytical conditions for local asymptotic stability and a critical policy-responsiveness threshold associated with possible Hopf-type oscillatory transitions. The analysis highlights the stabilizing role of mitigation damping and cubic saturation in regulating the feedback loop. To approximate the nonlinear system, a Physics-Informed Neural Network (PINN) surrogate is constructed by embedding the governing equations into a differentiable residual loss while enforcing the initial conditions analytically. The accumulated burden is represented through an admissible neural-network ansatz to preserve the well-definedness of the logarithmic coupling term, while the mitigation–response and policy-pressure variables remain signed in accordance with the model formulation. Numerical validation against reference ode45 solutions across two governance regimes shows maximum absolute errors of order

10^{- 3}

, indicating that the PINN provides a reliable differentiable surrogate for the coupled policy–feedback dynamics. The resulting framework offers a foundation for future inverse modeling, parameter estimation, and data-assimilation studies involving policy responsiveness, intervention thresholds, and burden- suppression effects. Full article

(This article belongs to the Section C2: Dynamical Systems)

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38 pages, 464 KB

Open AccessFeature PaperArticle

Subsonic Thermo-Acoustic Continuation Framework for the Compressible Navier–Stokes–Fourier System: Fourier–Triadic Concentration Exclusion and Thermodynamic Regularization

by Shin-ichi Inage

Mathematics 2026, 14(12), 2230; https://doi.org/10.3390/math14122230 (registering DOI) - 22 Jun 2026

Viewed by 150

Abstract

This paper studies the continuation problem for the three-dimensional compressible Navier–Stokes–Fourier system inside an admissible thermo-acoustic regime under periodic boundary conditions. The analysis considers strong solutions in the Sobolev class

H^{s} (Ω)

,

s >

5/2, with positive density and temperature and [...] Read more.

This paper studies the continuation problem for the three-dimensional compressible Navier–Stokes–Fourier system inside an admissible thermo-acoustic regime under periodic boundary conditions. The analysis considers strong solutions in the Sobolev class

H^{s} (Ω)

,

s >

5/2, with positive density and temperature and strict subsonic evolution. Using dyadic Fourier–triadic decomposition together with localized Littlewood–Paley analysis, the nonlinear transfer structure is decomposed into perturbative interaction classes and coherent same-scale High–High interactions. Within the present framework, coherent same-scale High–High persistence is identified as the only currently identified potentially nonperturbative concentration mechanism. A transport–acoustic alternative structure is then derived connecting persistent transport concentration with nonvanishing pressure response. The resulting pressure response is decomposed into thermodynamic and acoustic branches. The transonic acoustic branch is shown to be incompatible with the strict subsonic admissible class. The remaining interaction structure is controlled through entropy-driven thermodynamic dissipation and localized thermo-acoustic regularization. The exclusion of dynamically sustained critical thermo-acoustic concentration yields a localized ε-regularity framework combining thermodynamic dissipation, Campanato decay, and interior parabolic regularization. The resulting estimates provide localized Lipschitz control sufficient for the continuation of admissible strong solutions within the same thermo-acoustic class. The framework further remains compatible with weak–strong stability and irreversible long-time thermodynamic relaxation through the relative entropy structure and free-energy dissipation. Full article

27 pages, 4131 KB

Open AccessArticle

An Efficient Selection and Evaluation Hyper-Heuristic for Stochastic Underground Mine Production Scheduling

by Jianli Cao, Bingchen Han, Zirui Xiang, Yongyi Fang, Kejie Zou, Hangxing Ding and Xinyu Liu

Mathematics 2026, 14(12), 2229; https://doi.org/10.3390/math14122229 (registering DOI) - 22 Jun 2026

Viewed by 130

Abstract

Underground mine production scheduling under uncertainty is a complex and multi-field coupling system project. In this study, underground mine production scheduling seeks to determine the optimal start time of extraction-related projects, with the objectives of maximizing net present value, minimizing makespan, and maximizing [...] Read more.

Underground mine production scheduling under uncertainty is a complex and multi-field coupling system project. In this study, underground mine production scheduling seeks to determine the optimal start time of extraction-related projects, with the objectives of maximizing net present value, minimizing makespan, and maximizing resource utilization rate. The Copula function is adopted to formulate the correlation between uncertain project duration and cost and generate a set of stochastic scenarios. Then, the K-means algorithm classifies the scenarios into multiple scenario families, and the SBR algorithm is adopted to perform scenario reduction. Moreover, a rank choice function-based hyper-heuristic algorithm is extended to solve the multi-objective optimization model, which makes an excellent balance among the three objective functions. For determining the optimal scheduling plan, the cross-efficiency DEA algorithm is used to evaluate the archive set, sort the optimal solution, and guide the next iteration. The computational case verifies the effectiveness and efficiency of the multi-objective underground mine scheduling model, stochastic scenario and technical and hyper-heuristic algorithm. Full article

(This article belongs to the Special Issue Advanced Optimization Modeling and Algorithms for Planning and Scheduling)

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18 pages, 898 KB

Open AccessArticle

Methodical Aspects of Calculation of Technical Energy Losses in a Direct Current Electric Network

by Alexey Kirpikov, Vladislav Oboskalov, Murodbek Safaraliev, Ismoil Odinaev, Mihail Senyuk and Svetlana Beryozkina

Mathematics 2026, 14(12), 2228; https://doi.org/10.3390/math14122228 (registering DOI) - 22 Jun 2026

Viewed by 130

Abstract

This paper addresses probabilistic and statistical methods for calculating technical energy losses in direct current (DC) networks. A DC network model is adopted as the basis for the analysis, and several approaches are compared in terms of qualitative features and computational efficiency. The [...] Read more.

This paper addresses probabilistic and statistical methods for calculating technical energy losses in direct current (DC) networks. A DC network model is adopted as the basis for the analysis, and several approaches are compared in terms of qualitative features and computational efficiency. The load profile is described using probabilistic indicators, emphasizing the importance of accounting for correlation moments (CMs) between node powers and CMs between voltages to reduce calculation errors. A correction procedure for the mathematical expectation of node voltages is proposed, which significantly improves the accuracy of loss estimation. Simulation studies on representative four-node DC test networks show that the proposed method reduces the root mean square error in loss estimation by up to 15–20% compared with traditional approaches based solely on mean load values. The results confirm that the correction of node voltage expectations provides a good balance between accuracy and computational cost and can be recommended as an independent procedure within existing probabilistic frameworks for loss assessment. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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23 pages, 896 KB

Open AccessArticle

From Wikidata to Smart Tourism: A Reproducible Pipeline Based on AI and Fuzzy Logic for Interpretable Multi-Category Classification of Points of Interest

by Aristea Kontogianni, Konstantina Chrysafiadi, Maria Virvou and Efthimios Alepis

Mathematics 2026, 14(12), 2227; https://doi.org/10.3390/math14122227 (registering DOI) - 22 Jun 2026

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Abstract

Wikidata provides extensive coverage of tourism-related Points of Interest (POIs), yet its heterogeneous type system and uneven metadata limit its direct use in smart tourism applications. This paper presents an end-to-end pipeline that transforms Wikidata POIs into a compact and interpretable tourism-oriented representation [...] Read more.

Wikidata provides extensive coverage of tourism-related Points of Interest (POIs), yet its heterogeneous type system and uneven metadata limit its direct use in smart tourism applications. This paper presents an end-to-end pipeline that transforms Wikidata POIs into a compact and interpretable tourism-oriented representation supporting multi-category assignments. We collect POIs from six countries—Greece, Italy, Spain, Norway, Sweden, and Denmark—and construct a dataset that integrates core identifiers with textual descriptions, type information, heritage indicators, geographic coordinates, and Wikipedia sitelinks. We introduce an eight-category tourism taxonomy capturing key themes, including cultural venues, archaeological and historic sites, monuments, fortifications, religious sites, protected areas, natural features, and coastal or water locations. As a reproducible baseline, category likelihoods are estimated using sentence embeddings and similarity to category anchor descriptions, producing a probability vector for each POI. Building on this baseline, we propose a fuzzy inference layer that integrates embedding-based probabilities with structured Wikidata signals to generate interpretable membership degrees across categories and enable principled multi-category classification. This fusion is particularly valuable for smart tourism applications, as it supports robust faceted exploration and personalized recommendations (e.g., “historic + coastal”), while providing evidence-based explanations that enhance user trust and facilitate curator oversight when POI metadata is sparse or ambiguous. The resulting pipeline produces ranked POI catalogs by country and category, country-level tourism profiles, and diagnostic views for examining uncertain cases. The approach is fully reproducible and readily adaptable to other geographic regions or domain taxonomies. Full article

(This article belongs to the Special Issue Advanced Fuzzy Logic in Artificial Intelligence)

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26 pages, 420 KB

Open AccessArticle

Asymmetric Quantum Codes from τ-Paired Matrix-Product Codes

by Sami H. Saif and Shayea Aldossari

Mathematics 2026, 14(12), 2226; https://doi.org/10.3390/math14122226 (registering DOI) - 21 Jun 2026

Viewed by 105

Abstract

Asymmetric quantum codes are useful for quantum channels in which phase and bit errors occur with different probabilities, since the two distances,

d_{z}

and

d_{x}

, can be controlled separately. We develop a permutation-paired matrix-product construction for such codes over [...] Read more.

Asymmetric quantum codes are useful for quantum channels in which phase and bit errors occur with different probabilities, since the two distances,

d_{z}

and

d_{x}

, can be controlled separately. We develop a permutation-paired matrix-product construction for such codes over

F_{q}

. The main task is to build classical code pairs

C, D \subseteq F_{q^{2}}^{k n}

satisfying the Hermitian inclusion

D^{⊥_{H}} \subseteq C

, while keeping explicit dimension and distance bounds. Let

A \in F_{q^{2}}^{k \times k}

be a non-singular-by-columns (NSC) matrix with

A A^{†} = D P_{τ}

, where D is an invertible diagonal and

P_{τ}

corresponds to an involution

τ

. For

C = [C_{1}, \dots, C_{k}] A

and

D = [D_{1}, \dots, D_{k}] A

, we prove

D^{⊥_{H}} = [D_{τ (1)}^{⊥_{H}}, \dots, D_{τ (k)}^{⊥_{H}}] A

. Thus, the global inclusion

D^{⊥_{H}} \subseteq C

is equivalent to the shorter paired inclusions

D_{τ (i)}^{⊥_{H}} \subseteq C_{i}

. This yields asymmetric quantum codes with parameters

{[[k n, \sum_{i = 1}^{k} (r_{i} + s_{i}) - k n, d_{z} / d_{x}]]}_{q}

, where the bounds for

d_{z}

and

d_{x}

follow from NSC matrix-product distance estimates. For nested maximum distance separable (MDS) constituents, the paired conditions reduce to

r_{i} + s_{τ (i)} \geq n

, giving explicit infinite families. Concrete

τ

-OD matrices and numerical examples show that nontrivial permutations can increase the quantum dimension while preserving prescribed lower bounds for

d_{z}

and

d_{x}

. Full article

29 pages, 11459 KB

Open AccessArticle

Spatiotemporally Coordinated Operation in Multiple Data Centers Based on Adaptive Large Neighborhood Search Algorithm with Hierarchical Collaboration

by Yanghui Liu, Bowen Zhou, Liaoyi Ning and Juan Yan

Mathematics 2026, 14(12), 2225; https://doi.org/10.3390/math14122225 (registering DOI) - 21 Jun 2026

Viewed by 83

Abstract

Data centers have become essential infrastructure for digital services, while their rapidly growing electricity demand makes coordinated workload and power management an important optimization problem. This paper studies the multi-data-center operation problem under time-of-use electricity pricing and formulates it as a multi-data-center mixed-integer [...] Read more.

Data centers have become essential infrastructure for digital services, while their rapidly growing electricity demand makes coordinated workload and power management an important optimization problem. This paper studies the multi-data-center operation problem under time-of-use electricity pricing and formulates it as a multi-data-center mixed-integer nonlinear programming model (MDC-MINLP). The model jointly represents binary task scheduling decisions, including temporal workload shifting and spatial task migration, and continuous power-side variables, including device-level utilization, IT and auxiliary power consumption, energy storage dynamics, grid power procurement, and quality-of-service constraints. The objective is to minimize the total operating cost by integrating electricity purchasing cost, IT operation loss, storage degradation cost, and migration cost. To solve the resulting large-scale discrete–continuous coupled problem, an Adaptive Large Neighborhood Search algorithm with Hierarchical Collaboration (HC-ALNS) is proposed. HC-ALNS reconstructs feasible task action sets, employs a surrogate objective for fast candidate screening, performs accurate power-layer evaluation for selected solutions, and adaptively adjusts search intensity according to convergence behavior. Numerical results show that HC-ALNS reduces the total operating cost by 3.67% and achieves better convergence and solution quality than NSGA-II and PSO. These findings demonstrate that the proposed MDC-MINLP and HC-ALNS provide an effective mathematical optimization framework for coordinated computation–power scheduling. Full article

(This article belongs to the Section E: Applied Mathematics)

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14 pages, 4182 KB

Open AccessArticle

Automatic Bevacizumab Response Prediction in Ovarian Cancer from Digital Pathology Images via Novel AI-Based Computational Pipeline

by Abdullah Alsaiari, Turki Turki and Y-h. Taguchi

Mathematics 2026, 14(12), 2224; https://doi.org/10.3390/math14122224 (registering DOI) - 21 Jun 2026

Viewed by 159

Abstract

Ovarian cancer is a gynecological cancer, which, if metastasized and not detected early, can cause death among women. Therefore, accurate prediction of drug responses to ovarian cancer is needed. A gynecological pathologist inspects abnormality in tissues and provides a report for patients; however, [...] Read more.

Ovarian cancer is a gynecological cancer, which, if metastasized and not detected early, can cause death among women. Therefore, accurate prediction of drug responses to ovarian cancer is needed. A gynecological pathologist inspects abnormality in tissues and provides a report for patients; however, this diagnostic process (1) is difficult to undertake; (2) requires experience; and (3) is time-consuming. Moreover, existing tools are imperfect. Hence, we present a computational pipeline to improve predictions of drug response pertaining to ovarian cancer. First, we downloaded digital pathology images pertaining to ovarian responses to bevacizumab from the Cancer Imaging Archive Repository. We employed a histogram of oriented gradients for images, constructed feature vectors, and used Fisher’s linear discriminant analysis to alter data representations through dimensionality reduction. This reduced-dimensionality data was used for regression analysis, employing support vector regression coupled with various kernels and calculating the area under the ROC curve (AUC). Experimental results were validated using transformer-based models (ViT and Swin) and other deep learning (DL) models (VGG16, ResNet50, InceptionV3, MobileNetV2, and EfficientNetB6). Our approach using a radial kernel (named SVRD + R) improved AUC performance by 17% compared to the best-performing transformer-based model (ViT). Likewise, AUC performance improved by 14.9% when compared against the best DL-based model (MobileNetV2). These results demonstrate feasibility, showing that induced models via the presented AI-based pipeline can lead to superior performance when investigating prediction problems pertaining to gynecologic cancer studies. Full article

(This article belongs to the Special Issue Machine Learning and Computational Methods in Bioinformatics and Biology)

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11 pages, 276 KB

Open AccessArticle

On the Supremum of Singleton Ratios in Submodular Functions

by Laszlo Csirmaz

Mathematics 2026, 14(12), 2223; https://doi.org/10.3390/math14122223 (registering DOI) - 21 Jun 2026

Viewed by 76

Abstract

Let N be a finite set of cardinality n, and let

a \in N

. A submodular function f on N with

f (a) = 1

is defined to be a-reduced if, for any decomposition [...] Read more.

Let N be a finite set of cardinality n, and let

a \in N

. A submodular function f on N with

f (a) = 1

is defined to be a-reduced if, for any decomposition

f = g + h

into submodular functions, where h does not depend on a, it follows that h is identically zero. The maximal possible value of f on the remaining singletons defines a quantity

λ

that characterizes the degree to which one variable can constrain the value of another; geometrically, it also limits the possible elongation of the associated submodular base polytope. The parameter has concrete relevance: it caps the share-size lower bounds provable for secret-sharing schemes via the basic Shannon inequalities, and it controls the geometry of the base polytopes on which greedy submodular-optimization algorithms operate. We construct an example demonstrating that

λ

can be as large as

Ω (n / log n)

. Furthermore, we establish a doubly exponential upper bound on

λ

. The problem of narrowing the gap between these bounds remains open. Full article

(This article belongs to the Section E: Applied Mathematics)

26 pages, 6705 KB

Open AccessArticle

Intelligent Analysis of the Geomechanical State of Rock Masses During Underground Mining

by Dmytro Babets, Amirbek Yerkinbekov, Serik Moldabayev, Samal Assylkhanova, Volodymyr Hnatushenko and Olena Sdvyzhkova

Mathematics 2026, 14(12), 2222; https://doi.org/10.3390/math14122222 (registering DOI) - 20 Jun 2026

Viewed by 169

Abstract

This study presents an intelligent framework for the analysis of multidimensional geomechanical states in underground mining systems based on numerical simulation and machine learning methods. A three-dimensional geomechanical model of the Zholymbet deposit was developed in the RS3 environment using the generalized Hoek–Brown [...] Read more.

This study presents an intelligent framework for the analysis of multidimensional geomechanical states in underground mining systems based on numerical simulation and machine learning methods. A three-dimensional geomechanical model of the Zholymbet deposit was developed in the RS3 environment using the generalized Hoek–Brown failure criterion. Numerical simulations were performed for representative mining scenarios characterized by complex excavation interaction and stress redistribution. The modelling results were transformed into a multidimensional geomechanical dataset containing stress, deformation, displacement, and yielding parameters. Principal component analysis (PCA) was applied to investigate the internal structure of the geomechanical state space and identify dominant patterns controlling the rock mass behavior. Clustering analysis revealed several geomechanical regimes corresponding to stable, transitional, and instability-prone conditions. Isolation Forest anomaly detection demonstrated that atypical geomechanical states are not randomly distributed but spatially localized near excavation systems and mining horizons. The obtained results indicate that hazardous geomechanical conditions are governed by complex interactions between stress concentration, deformation intensity, yielding processes, and excavation geometry. The proposed approach provides a basis for intelligent interpretation of large-scale numerical modelling results and may support geomechanical risk assessment in underground mining operations. Full article

(This article belongs to the Special Issue Advances in Numerical Computation and Mathematical Modelling for Mechanics and Dynamics in Geotechnical Engineering)

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22 pages, 2034 KB

Open AccessFeature PaperArticle

Fixed-Point Analysis of Supra-Contractions with Applications to Nonlinear Economic Systems

by G. Sudhaamsh Mohan Reddy, Lateef Ahmad Wani, Mudasir Younis and Saiful R. Mondal

Mathematics 2026, 14(12), 2221; https://doi.org/10.3390/math14122221 (registering DOI) - 20 Jun 2026

Viewed by 116

Abstract

In this article, we construct a framework for analyzing the equilibrium and stability of networked multi-sector economic systems via fixed-point analysis. We represent directional intersectoral dependencies, nonlinear feedback effects, and heterogeneous adjustment dynamics in the model by the coupled and tripled fixed-point theory [...] Read more.

In this article, we construct a framework for analyzing the equilibrium and stability of networked multi-sector economic systems via fixed-point analysis. We represent directional intersectoral dependencies, nonlinear feedback effects, and heterogeneous adjustment dynamics in the model by the coupled and tripled fixed-point theory in the graphically extended suprametric spaces. The graphical structure encodes supply-chain and influence networks, whereas asymmetric and nonuniform interaction strengths are encoded in the suprametric setting. Furthermore, we prove the existence, uniqueness, and convergence of equilibrium solutions under new generalized contraction conditions. We apply the theoretical findings in nonlinear state systems in which prices in interdependent markets are adjusted using integral equations. The results of numerical simulations show consistent convergence, and the sensitivity parameter of the network structure significantly influences the determination of economic stability and speed of adjustment. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Applications)

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20 pages, 803 KB

Open AccessArticle

Multi-Objective Just-in-Time Permutation Flow Shop: Tools for Analysis of Different Conflict Scenarios

by Nícolas Samuel Assis, Socorro Rangel and Helio Yochihiro Fuchigami

Mathematics 2026, 14(12), 2220; https://doi.org/10.3390/math14122220 (registering DOI) - 20 Jun 2026

Viewed by 93

Abstract

Permutation flow shop scheduling is an important production planning problem handled in different contexts. Just-in-time measures have been significant in the optimization of real problems and one is specifically addressed here: the total earliness and tardiness of jobs. The most used approach in [...] Read more.

Permutation flow shop scheduling is an important production planning problem handled in different contexts. Just-in-time measures have been significant in the optimization of real problems and one is specifically addressed here: the total earliness and tardiness of jobs. The most used approach in the literature to mathematically express this measure is to sum them up using unit weights thus obtainning a mono-objective function. In this paper it is shown that this is a simplification of a problem that is inherently multi-objective, highlighting how a more comprehensive approach can better support decision-making. A bi-objective mathematical optimization model and tools capable of analyzing the mono-objective solution within the multi-objective perspective are proposed. A computational study to analyze the benefits and difficulties of the solution using the bi-objective approach is presented. The results show that for large-scale instances in which the tardiness factor is small, the conflict between the objectives of minimizing the total earliness and minimizing the total tardiness of jobs increases significantly. Specifically, the mono-objective solution is unbalanced in 50.00% of the analyzed instance structures. However, in 48.12% of the instances, alternative Pareto-optimal trade-offs can be achieved with zero increase to the mono-objective optimal value. Therefore, the multi-objective approach has a greater potential to support decision-makers. Furthermore, we show that the choice of the solution method must be carefully considered, since the Pareto frontier associated with most instances has many non-supported points, representing up to 66.71% of the non-dominated set. Full article

(This article belongs to the Special Issue Innovations in Optimization and Operations Research, 2nd Edition)

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14 pages, 1102 KB

Open AccessArticle

Statistical Properties of Rosenthal’s Fail-Safe Number in Meta-Analysis

by Vanusa Rocha, Miguel Felgueiras and Vera Afreixo

Mathematics 2026, 14(12), 2219; https://doi.org/10.3390/math14122219 (registering DOI) - 20 Jun 2026

Viewed by 110

Abstract

Rosenthal’s fail-safe number is widely used to assess the robustness of meta-analysis results against publication bias; however, its statistical properties remain insufficiently understood. This paper re-evaluates the coverage performance of confidence intervals for the Rosenthal’s fail-safe number using an updated simulation framework that [...] Read more.

Rosenthal’s fail-safe number is widely used to assess the robustness of meta-analysis results against publication bias; however, its statistical properties remain insufficiently understood. This paper re-evaluates the coverage performance of confidence intervals for the Rosenthal’s fail-safe number using an updated simulation framework that incorporates zero truncation, an epsilon correction to the expected value, and a restriction to statistically significant meta-analyses. In addition to the standard normal bootstrap approximation, bias-corrected and accelerated bootstrap confidence intervals are considered. Simulation results show that standard bootstrap intervals tend to be conservative under symmetric settings and exhibit substantial deviations under asymmetric distributions. The bias-corrected and accelerated bootstrap method improves coverage accuracy, particularly under asymmetry and moderate sample sizes, although both methods exhibit conservative behavior in several scenarios. Overall, reliable inference for the fail-safe number depends on both appropriate parameter specification and bootstrap procedures that account for bias and asymmetry. Full article

(This article belongs to the Special Issue Advances in Statistics, Biostatistics and Medical Statistics)

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43 pages, 6832 KB

Open AccessFeature PaperArticle

The Geometry of Quantum Walks on Graphs—Theory and Applications

by Ernesto Estrada

Mathematics 2026, 14(12), 2218; https://doi.org/10.3390/math14122218 (registering DOI) - 20 Jun 2026

Viewed by 103

Abstract

We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex [...] Read more.

We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex volumes, allowing interference, localization, and spreading to be analyzed within a unified metric-angular formalism. We prove that, in contrast to classical diffusion, which collapses to a spherical geometry, quantum dynamics generate a generically non-spherical affine geometry with persistent anisotropy. Applying this theory to real-world networks—including transportation systems, semantic graphs, and neuronal connectomes—we show that quantum geometry reveals dynamically meaningful backbones, interference-based “communities”, and vulnerability structures that are invisible to classical random-walk and spectral methods. In particular, angular and radial quantum descriptors isolate functional hubs, control cores, and coherence classes without any topological or dimensionality assumptions. Together, these results demonstrate that quantum-walk-induced geometry provides a powerful new lens for understanding structure and function in complex networks. Full article

(This article belongs to the Special Issue Information-Induced Geometries in Statistics, Data Analysis, and Applied Research)

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17 pages, 338 KB

Open AccessArticle

Multi-Criteria Financial Screening Under Data Uncertainty: An LLM-Extraction and Min–Max TOPSIS Approach for SMEs

by Vinicius Minatogawa, Mitsuyoshi Fukushi, Jose Garcia, Jorge Rojas, Jose Gornall, Alfredo Angulo and Jefferson Pinto

Mathematics 2026, 14(12), 2217; https://doi.org/10.3390/math14122217 (registering DOI) - 20 Jun 2026

Viewed by 163

Abstract

Small and medium enterprises routinely face a paradox in financial monitoring: their accounting documents exist, but the cost of converting heterogeneous PDFs into timely financial signals is prohibitive without dedicated analytical staff or specialized software. This paper presents a two-layer artifact, designed under [...] Read more.

Small and medium enterprises routinely face a paradox in financial monitoring: their accounting documents exist, but the cost of converting heterogeneous PDFs into timely financial signals is prohibitive without dedicated analytical staff or specialized software. This paper presents a two-layer artifact, designed under Design Science Research, that bridges this gap using only public-web large language models (LLMs) and a parsimonious multi-criteria decision routine. Layer 1 implements a structured LLM-driven workflow that extracts account–value pairs from annual tax balance sheets without code, APIs, or fine-tuning. Layer 2 reconstructs auditable accounting aggregates and ranks yearly financial condition through TOPSIS with min–max normalization—a deliberate replacement for classical vector normalization, which fails when profitability indicators are negative, as routinely occurs in distress years. To avoid size effects and algebraic redundancy, the decision matrix uses only three criteria spanning liquidity, profitability, and solvency. The artifact is demonstrated in a four-year case study of an anonymized construction SME (2021–2024), with accountant-verified document-level match rates of 0.810, 0.998, 0.950, and 0.909. Equal weighting is the only weighting configuration used; a supplementary entropy-based dispersion diagnostic yields the same ordinal ranking—2024 > 2023 > 2021 > 2022—and 10,000 Monte Carlo replications, with uncertainty injected at the reconstructed-aggregate level, confirm that the extreme ranks are invariant across all runs. The contribution is methodological and practical: a transparent, low-infrastructure pipeline that brings first-pass financial screening within reach of SMEs operating under severe data and budget constraints. Full article

(This article belongs to the Special Issue Applications of Mathematics Analysis in Financial Marketing)

29 pages, 35248 KB

Open AccessArticle

Optimal Sensor Placement Based on Fisher Information Matrix and Improved Particle Swarm Optimization Algorithm for Typical Tensile Membrane Structures

by Qiu Yu, Xin Zhang, Zhiyang Jia and Chen Peng

Mathematics 2026, 14(12), 2216; https://doi.org/10.3390/math14122216 (registering DOI) - 20 Jun 2026

Viewed by 90

Abstract

Large-amplitude and long-term vibration deformation under external environmental loads often occurs on tensile membrane structures. Proper sensor placement plays a vital role in effectively achieving the objectives of a structural health monitoring system. In order to optimize the sensor placement to identify the [...] Read more.

Large-amplitude and long-term vibration deformation under external environmental loads often occurs on tensile membrane structures. Proper sensor placement plays a vital role in effectively achieving the objectives of a structural health monitoring system. In order to optimize the sensor placement to identify the modal vibration parameters for tensile membrane structures, this paper proposes an optimal sensor placement method based on the Fisher information matrix (FIM) and improved random strategy discrete particle swarm optimization algorithm (IRSDPSO). Firstly, the structural modal order is selected by using the two-norm difference and trace change rate of FIM, and the number of sensors is determined based on the QR decomposition and MAC criterion. Secondly, an improved particle swarm optimization algorithm named IRSDPSO, which has the discrete characteristic, is proposed to arrange the placement of sensors. Finally, the convergence, stability and sensitivity are used to evaluate the effectiveness of optimal sensor placement results using a numerical modal test example of the plane bidirectional tensile membrane structure. The results show that the first nineteen modal frequencies can be accurately identified. This indicates that the proposed optimal sensor placement method can determine the number of sensors and arrange the placement of the sensors. The work is reasonable and feasible in the optimal sensor placement for the tensile membrane structure. Full article

(This article belongs to the Special Issue Advances in Optimization and Soft Computing Methodologies with Applications)

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21 pages, 333 KB

Open AccessArticle

Quillen–Suslin Theorem for Connected Cochain DG Algebras

by Xuefeng Mao and Biyan Zhu

Mathematics 2026, 14(12), 2215; https://doi.org/10.3390/math14122215 (registering DOI) - 20 Jun 2026

Viewed by 80

Abstract

Let

A

be a connected cochain differential graded algebra and P a finitely generated differential graded

A

-module. We show that P is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considered as [...] Read more.

Let

A

be a connected cochain differential graded algebra and P a finitely generated differential graded

A

-module. We show that P is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considered as a generalization of the well-known Quillen–Suslin Theorem in differential graded context. As an application, we show that the ghost length and the cone length of a compact differential graded module coincide. Full article

19 pages, 6096 KB

Open AccessArticle

A Novel Hybrid Modeling Framework Integrating Feature Engineering for Battery Remaining Useful Life Prediction

by Ru Xiao, Jiyang Xu and Jiabo Li

Mathematics 2026, 14(12), 2214; https://doi.org/10.3390/math14122214 (registering DOI) - 20 Jun 2026

Viewed by 154

Abstract

Accurate remaining useful life (RUL) prediction is critical for the reliable operation of lithium-ion batteries. Traditional data-driven methods often suffer from parameter redundancy and error accumulation in state prediction. This paper proposes a hybrid data-driven RUL prediction framework based on Gaussian process regression [...] Read more.

Accurate remaining useful life (RUL) prediction is critical for the reliable operation of lithium-ion batteries. Traditional data-driven methods often suffer from parameter redundancy and error accumulation in state prediction. This paper proposes a hybrid data-driven RUL prediction framework based on Gaussian process regression (GPR) optimized by the lightning search algorithm (LSA). First, both local and global indirect health features (HFs) are extracted from the external characteristic parameter curves and the incremental capacity curves during battery charging/discharging. Second, the Pearson correlation coefficient is applied to select highly relevant features, forming a compact feature set. Third, a GPR model is developed, and the LSA is introduced to optimize its hyperparameters, overcoming the tendency of the conjugate gradient method to fall into local optima or fail to converge. Experimental results under identical conditions show that the proposed LSA–GPR model achieves a prediction error of 3% or less. Full article

(This article belongs to the Special Issue AI-Enabled Prognostics and Health Management: Mathematical Aspect and Engineering Applications)

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23 pages, 1466 KB

Open AccessArticle

A Spreadsheet Environment for Force, Torque and Strength of Materials Modeling: Bridging Analytical Mathematics and Engineering Practice

by Elisa Munich, Jérémie Schutz, Christophe Sauvey and Yves Gillet

Mathematics 2026, 14(12), 2213; https://doi.org/10.3390/math14122213 (registering DOI) - 19 Jun 2026

Viewed by 221

Abstract

This paper presents and validates a unified spreadsheet-based framework for engineering mechanics education and preliminary design. Three modules are integrated within a single openly available workbook: multi-point resultant force and moment computation; axial normal stress with stress concentration effects for three geometric configurations [...] Read more.

This paper presents and validates a unified spreadsheet-based framework for engineering mechanics education and preliminary design. Three modules are integrated within a single openly available workbook: multi-point resultant force and moment computation; axial normal stress with stress concentration effects for three geometric configurations (plate with hole, shoulder plate, stepped shaft); and beam deflection for simply supported and cantilever configurations under point loads. All governing equations are implemented as explicit closed-form expressions validated against analytical reference solutions for six independent cases; relative errors fall below

10^{- 10}

in all cases. Three worked exercises demonstrate the practical scope of the framework. A biomechanical multi-point force system yields joint moments of

- 6880

, −33,421, and −58,241 N·mm at the wrist, elbow, and shoulder, respectively. A tensile shoulder plate with

K_{t} \approx 1.85

produces

σ_{max} = 232

MPa against

σ_{y} = 200

MPa, identifying a design failure; a parametric redesign with fillet radius

r = 10

mm reduces

K_{t}

to approximately 1.59 and

σ_{max}

to approximately 198.7 MPa, restoring structural safety. A cantilever beam subjected to a 20,000 N tip load yields a maximum deflection of 13,133 μm. The framework constitutes a validated intermediate layer between manual analytical derivations and high-fidelity numerical simulations, applicable to preliminary design, parametric sensitivity studies, and engineering education at the linear elastic level. Full article

(This article belongs to the Special Issue Modeling and Simulation in Engineering, 4th Edition)

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12 pages, 287 KB

Open AccessArticle

Geometric Structures and Inclusion Properties of Multivalent Mittag-Leffler-Type Poisson Subfamilies

by Feras Yousef, Tariq Al-Hawary and Ibtisam Aldawish

Mathematics 2026, 14(12), 2212; https://doi.org/10.3390/math14122212 (registering DOI) - 19 Jun 2026

Viewed by 111

Abstract

The interplay between special functions and geometric function theory continues to inspire significant advances in the study of analytic and multivalent functions. In this work, we introduce and investigate several new subfamilies of multivalent functions associated with the generalized Mittag-Leffler-type Poisson distribution in [...] Read more.

The interplay between special functions and geometric function theory continues to inspire significant advances in the study of analytic and multivalent functions. In this work, we introduce and investigate several new subfamilies of multivalent functions associated with the generalized Mittag-Leffler-type Poisson distribution in the open unit disk. We establish necessary and sufficient conditions characterizing membership in these classes and derive meaningful inclusion relationships among them. Furthermore, we define a novel integral operator linked to the generalized Mittag-Leffler-type Poisson distribution and examine its mapping properties and structural connections with the proposed function classes. The results presented herein not only unify and extend a variety of earlier contributions but also demonstrate the effectiveness of distribution-theoretic methods in the analysis of multivalent functions. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions, 2nd Edition)

17 pages, 1456 KB

Open AccessArticle

A Unified Constant-Time Switch Rule for Constructing Edge-Disjoint Hamiltonian Cycles in Gaussian Networks

by Bader Albader

Mathematics 2026, 14(12), 2211; https://doi.org/10.3390/math14122211 (registering DOI) - 19 Jun 2026

Viewed by 101

Abstract

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that, when the generator

α = a + b i

satisfies

\gcd (a, b) = 1

, the real and imaginary dimensions directly [...] Read more.

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that, when the generator

α = a + b i

satisfies

\gcd (a, b) = 1

, the real and imaginary dimensions directly form two edge-disjoint Hamiltonian cycles. A later construction extended the result to the non-coprime case

\gcd (a, b) = d > 1

, but its proof relied on long node-sequence tables and separate odd/even cases for d. This paper presents a unified closed-form construction that covers both

d = 1

and

d > 1

, and both odd and even d, without separate case tables. In the rectangular representation with d rows and

r = (a^{2} + b^{2}) / d

columns, the construction uses a constant-time local switch rule, meaning constant time per individual switch, for each

q = 1, 2, \dots, d - 1

at column

a_{q} = q - 1

. Each switch removes two horizontal edges and inserts two vertical edges. The switched horizontal structure forms the first Hamiltonian cycle, while its edge-complement in the Gaussian network forms the second Hamiltonian cycle. Thus, the full edge set is partitioned into two edge-disjoint Hamiltonian cycles. The construction requires

O (d)

switch-generation time and

O (N)

time to list the two cycles, where

N = a^{2} + b^{2}

. Exhaustive validation for all

1 \leq a \leq b \leq 100

, excluding only the degenerate

N = 2

network, and large-scale validation up to

N = 3,250,000

confirm implementation correctness and demonstrate practical scalability. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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32 pages, 3105 KB

Open AccessReview

A Review on Deep State Space Models for Sequential Healthcare Data Prediction

by Wenjie Li, Yongming Xie and Yinglong Dai

Mathematics 2026, 14(12), 2210; https://doi.org/10.3390/math14122210 (registering DOI) - 19 Jun 2026

Viewed by 126

Abstract

Sequential data prediction is a crucial area in healthcare. Healthcare data have the characteristics of non-stationarity, long-range dependence (LRD), and irregular sampling. Modeling these complex temporal features is highly challenging. Recurrent Neural Networks (RNNs) and their variants are limited in learning long-range dependencies [...] Read more.

Sequential data prediction is a crucial area in healthcare. Healthcare data have the characteristics of non-stationarity, long-range dependence (LRD), and irregular sampling. Modeling these complex temporal features is highly challenging. Recurrent Neural Networks (RNNs) and their variants are limited in learning long-range dependencies (LRDs) due to the inherent issues of vanishing and exploding gradients. Transformers alleviate this limitation by using the self-attention mechanism. Its quadratic computational complexity and memory bottleneck limit its scalability in long-range healthcare data. In this context, Structured State Space Models (SSMs) have emerged as a promising alternative. Compared with conventional RNNs, they can alleviate the difficulty of modeling LRDs more efficiently, and many modern SSM variants achieve linear time sequence modeling while reducing the computational burden associated with Transformers. In this review, we provide a formal definition of Healthcare Process Modeling, compare the core theoretical frameworks of RNNs, Transformers, and SSMs, trace the architectural evolution of SSM architectures, and provide a comprehensive review of healthcare applications and open challenges, including LSSL, S4, S5, Mamba, and their related variants. Existing studies suggest that structured SSMs are promising for selected long-sequence healthcare prediction tasks, particularly when computational efficiency and long-context retention are important. With these advantages, they may help alleviate the computational burden in certain healthcare tasks and provide a basis for further exploring the practical application of data-driven healthcare systems in clinical practice. Full article

(This article belongs to the Topic The Future of Artificial Intelligence: Trends, Challenges, and Developments)

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31 pages, 452 KB

Open AccessArticle

A Banach-Space Framework for Proposed (v,w)–s–Convex Response-Curve Certification in Machine Learning

by Ahad Hamoud Alotaibi, Muhammad Saeed Ahmad, Muhammad Waseem Asghar and Mujahid Abbas

Mathematics 2026, 14(12), 2209; https://doi.org/10.3390/math14122209 (registering DOI) - 19 Jun 2026

Viewed by 95

Abstract

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed

(v, w)

–s [...] Read more.

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed

(v, w)

–s–convex response-curve certification. The response curve is treated as an element of the Banach space of continuous functions under the supremum norm, while derivative-based certificates are handled in a Lipschitz and Sobolev-type norm when required. Generalized convexity is represented through a bounded structural operator, whose order condition defines a closed convex acceptance set. The violation score is measured by the positive part of the operator residual, and the Hermite–Hadamard, Fejér, and Ostrowski quantities are interpreted as bounded certificate functionals. The auxiliary profiles are constructed from validation-curve residuals through a split-calibrated procedure and then tested on held-out triples. The framework certifies only scalar response-curve summaries under explicit structural and empirical assumptions; it does not certify a full learning system, guarantee generalization, or replace dense sampling when the structural gate fails. Full article

38 pages, 2010 KB

Open AccessReview

Beyond Neural Solvers: A Critical Review of Machine Learning for Combinatorial Optimization

by Mostafa E. A. Ibrahim, Alaa E. S. Ahmed and Yassine Daadaa

Mathematics 2026, 14(12), 2208; https://doi.org/10.3390/math14122208 (registering DOI) - 19 Jun 2026

Viewed by 247

Abstract

Combinatorial optimization is a key component in critical decision problems such as routing, scheduling, network design, and graph optimization. Although combinatorial optimization methods, including exact algorithms, approximation methods, constraint programming, mixed integer programming, and metaheuristics, are widely available, they often face obstacles, such [...] Read more.

Combinatorial optimization is a key component in critical decision problems such as routing, scheduling, network design, and graph optimization. Although combinatorial optimization methods, including exact algorithms, approximation methods, constraint programming, mixed integer programming, and metaheuristics, are widely available, they often face obstacles, such as limited scalability and adaptability in various applications. In this study, a systematic critical review of machine learning for combinatorial optimization is provided to characterize the usage and evaluation of learning-based approaches. A detailed analysis is used to infer and determine findings and limitations. The paper emphasizes how machine learning for computational optimization has changed over time, moving from end-to-end neural solvers to hybrid systems. Learning components are essential for directing, speeding up, or enhancing traditional solver backbones such as constraint programming and metaheuristics in hybrid systems. The review also critically examines current limits that impact performance in general, including scalability, deployment readiness, generalization, and benchmark consistency. Even though using large language models for problem formulation and heuristic synthesis has potential, more work needs to be done to ensure reliable validation. As a conclusion, this article examines recent studies’ findings, emphasizes the growing trend toward hybrid learning-driven optimization frameworks, and underlines important methodological limits and unresolved issues. Full article

(This article belongs to the Special Issue Mathematical Methods and Applications in Signal Analysis, Machine Learning, and Artificial Intelligence)

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26 pages, 1991 KB

Open AccessArticle

The Maximal Almost Sure Lyapunov Exponent of Three-Dimensional Linear Stratonovich Stochastic Differential Equations

by Jianyue Su and Ziying He

Mathematics 2026, 14(12), 2207; https://doi.org/10.3390/math14122207 (registering DOI) - 19 Jun 2026

Viewed by 212

Abstract

The sign of the maximal almost sure Lyapunov exponent determines the stability of stochastic systems, while its numerical computation for three-dimensional linear Stratonovich stochastic differential equations remains challenging due to the failure of classical two-dimensional strategies. The spherical angular motion of 3D systems [...] Read more.

The sign of the maximal almost sure Lyapunov exponent determines the stability of stochastic systems, while its numerical computation for three-dimensional linear Stratonovich stochastic differential equations remains challenging due to the failure of classical two-dimensional strategies. The spherical angular motion of 3D systems produces a Fokker–Planck equation with intractable mixed partial derivatives, preventing conventional analytical solutions. This paper develops a unified computational framework for three-dimensional linear Stratonovich stochastic systems using analytical derivation for degenerate cases and physics-informed neural network (PINN) approximation for general non-degenerate scenarios. For degenerate systems, we reduce the coefficient matrix to a lower triangular form via orthogonal transformation and establish tight upper bounds based on the logarithmic growth property of the Wiener process, yielding closed-form expressions for the maximal almost sure Lyapunov exponent under all parameter sign configurations. For non-degenerate systems, we reformulate the Fokker–Planck equation in spherical coordinates and construct a customized PINN with trigonometric encoding to enforce periodic boundary conditions. The network is trained by joint loss functions of equation residuals, boundary constraints and normalization consistency, and the converged stationary density is substituted into the Furstenberg–Khasminskii formula to calculate the exponent via Gauss–Legendre quadrature. Monte Carlo simulations confirm the accuracy and robustness of the proposed method, which reliably identifies the sign of the maximal almost sure Lyapunov exponent even in near-critical regimes. Numerical experiments on a 3D stochastic Hopf bifurcation model show that noise negatively shifts the bifurcation point, with the offset linearly proportional to the squared noise intensity. This work extends Lyapunov stability analysis from two-dimensional to three-dimensional linear Stratonovich stochastic systems, offering an effective tool for stability evaluation of general three-dimensional stochastic dynamical models. Full article

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