Journal Description

Mathematics

Mathematics is a peer-reviewed, open access journal which provides an advanced forum for studies related to mathematics, and is published semimonthly online by MDPI. The European Society for Fuzzy Logic and Technology (EUSFLAT) and International Society for the Study of Information (IS4SI) are affiliated with Mathematics and their members receive a discount on article processing charges.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), RePEc, and other databases.
Journal Rank: JCR - Q1 (Mathematics) / CiteScore - Q1 (General Mathematics )
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.4 days after submission; acceptance to publication is undertaken in 2.4 days (median values for papers published in this journal in the first half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Sections: published in 17 topical sections.
Companion journals for Mathematics include: Foundations, Analytics, International Journal of Topology, Geometry and Logics.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 2.2 (2024); 5-Year Impact Factor: 2.0 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2227-7390

Latest Articles

41 pages, 5138 KB

Open AccessArticle

Improved Enterprise Development Optimization with Historical Trend Updating for High-Precision Photovoltaic Model Parameter Estimation

by Zhiping Li, Yi Liao and Haoxiang Zhou

Mathematics 2026, 14(1), 121; https://doi.org/10.3390/math14010121 (registering DOI) - 28 Dec 2025

Abstract

Accurate parameter estimation of photovoltaic (PV) models is fundamentally a challenging nonlinear optimization problem, characterized by strong nonlinearity, high dimensionality, and multiple local optima. These characteristics significantly hinder the convergence accuracy, stability, and efficiency of conventional metaheuristic algorithms when applied to PV parameter identification. Although the enterprise development (ED) optimization algorithm has shown promising performance in various optimization tasks, it still suffers from slow convergence, limited solution precision, and poor robustness in complex PV parameter estimation scenarios. To overcome these limitations, this paper proposes a multi-strategy enhanced enterprise development (MEED) optimization algorithm for high-precision PV model parameter estimation. In MEED, a hybrid initialization strategy combining chaotic mapping and adversarial learning is designed to enhance population diversity and improve the quality of initial solutions. Furthermore, a historical trend-guided position update mechanism is introduced to exploit accumulated search information and accelerate convergence toward the global optimum. In addition, a mirror-reflection boundary control strategy is employed to maintain population diversity and effectively prevent premature convergence. The proposed MEED algorithm is first evaluated on the IEEE CEC2017 benchmark suite, where it is compared with 11 state-of-the-art metaheuristic algorithms under 30-, 50-, and 100-dimensional settings. Quantitative experimental results demonstrate that MEED achieves superior solution accuracy, faster convergence speed, and stronger robustness, yielding lower mean fitness values and smaller standard deviations on the majority of test functions. Statistical analyses based on Wilcoxon rank-sum and Friedman tests further confirm the significant performance advantages of MEED. Moreover, MEED is applied to the parameter estimation of single-diode and double-diode PV models using real measurement data. The results show that MEED consistently attains lower root mean square error (RMSE) and integrated absolute error (IAE) than existing methods while exhibiting more stable convergence behavior. These findings demonstrate that MEED provides an efficient and reliable optimization framework for PV model parameter estimation and other complex engineering optimization problems. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

26 pages, 1804 KB

Open AccessArticle

A Dependent Bivariate Burr XII Inverse Weibull Model: Application to Diabetic Retinopathy and Dependent Competing Risks Data

by Ammar M. Sarhan, Ahlam H. Tolba, Dina A. Ramadan and Thamer Manshi

Mathematics 2026, 14(1), 120; https://doi.org/10.3390/math14010120 (registering DOI) - 28 Dec 2025

Abstract

This paper introduces a novel bivariate distribution, referred to as the Bivariate Burr XII Inverse Weibull (BBXII-IW) distribution, constructed via the Marshall–Olkin approach from the univariate Burr XII Inverse Weibull (BXII-IW) distribution. The proposed BBXII-IW model provides a flexible framework for modeling dependent bivariate data, including competing risk scenarios. The key statistical properties of the distribution are derived, and parameter estimation is conducted using the maximum likelihood method. The model’s performance is evaluated using two types of real-world datasets: (1) bivariate data and (2) dependent competing risk data related to diabetic retinopathy. The results demonstrate that the BBXII-IW distribution offers an improved fit compared to existing models, highlighting its flexibility and practical relevance in modeling complex dependent structures. Full article

(This article belongs to the Special Issue Advances in Statistical Approaches with Applications for Multivariate Data Analysis)

10 pages, 274 KB

Open AccessArticle

On the Analysis of a System of Equations Containing a Parameter n and Describing a Special State of a Certain Table of Numbers

by Dostonjon Numonjonovich Barotov

Mathematics 2026, 14(1), 119; https://doi.org/10.3390/math14010119 (registering DOI) - 28 Dec 2025

Abstract

In this paper, we study a system of 10 equations containing a parameter—a non-negative integer n—and associated with the problem of filling a special table with natural numbers. As a result, by developing a hybrid approach—first, by proving an a priori estimate for the solution of the system, which implies a finite set of solutions, thereby significantly narrowing the search space for a solution, and second, by performing a computer calculation of all remaining vectors satisfying the a priori estimate—we established that the system is not solvable for all values of the parameter n. We also prove a criterion for

(x_{0}, x_{1}, \dots, x_{9}) \in N^{10}

to be a solution to the system for some value of the parameter n. Furthermore, we prove a fact that, in particular, implies that the set of values of the parameter n for which the system has at least 10 solutions is countable. Full article

27 pages, 1312 KB

Open AccessArticle

Improved Doubly Robust Inference with Nonprobability Survey Samples Using Finite Mixture Models: Application to Health Monitoring SMS Survey Data

by Ziying Yang, Xu Wang, Wenjing Wu and Jing Gu

Mathematics 2026, 14(1), 118; https://doi.org/10.3390/math14010118 (registering DOI) - 28 Dec 2025

Abstract

Nonprobability sampling has been increasingly used in epidemiologic research, yet direct inference based on such samples is subject to selection bias. Current adjustment methods commonly rely on a reference probability-based survey sample that shares a set of covariates with the nonprobability sample. However, these common covariates are often limited and may bias estimates in the presence of population heterogeneity. Existing methods generally assume population homogeneity in models and fail to address such heterogeneity adequately. To overcome this limitation, we propose the Nonprobability Heterogeneity-adjusted Doubly Robust (NHDR) method, a novel inference framework that explicitly accounts for population heterogeneity during selection bias adjustment. NHDR proceeds in three stages: (1) identifying latent subpopulations via finite mixture modeling; (2) incorporating the resulting latent-class structure as a grouping variable into mixed-effects models for both the propensity score and outcome projection; and (3) constructing a doubly robust estimator that integrates these adjusted models. The key methodological contribution of NHDR is its formal integration of latent-class-based population structure into a doubly robust estimation framework, which enables more reliable inference under heterogeneous population settings. Simulation studies demonstrate that the proposed method control the coverage probabilities well in most scenarios. Under heterogeneous conditions, NHDR consistently outperforms existing methods achieving an average reduction in relative bias of approximately 1.8–4.5% and a corresponding decrease in mean squared error of about 5.1–15.5 compared to the benchmark method. We illustrate the practical utility of NHDR by applying it to estimate nine health indicators using data from the Health Monitoring SMS Survey in Guangzhou, China, with the seventh Guangdong Health Service Survey serving as the reference sample. Full article

(This article belongs to the Section D: Statistics and Operational Research)

25 pages, 915 KB

Open AccessArticle

Dynamic Behavior and Exponential Stability of the Modified Moore–Gibson–Thompson Thermoelastic Model with Frictional Damping

by Mouataz Billah Mesmouli, Houssem Eddine Khochemane, Loredana Florentina Iambor and Taher S. Hassan

Mathematics 2026, 14(1), 117; https://doi.org/10.3390/math14010117 (registering DOI) - 28 Dec 2025

Abstract

This paper investigates a modified one-dimensional Moore–Gibson–Thompson (MGT) thermoelasticity model that significantly extends the classical formulation by incorporating two key structural modifications: frictional damping and a novel cross-coupling structure. The system introduces a viscous frictional damping mechanism proportional to the velocity acting on the mechanical (elastic) field, enhancing dissipation, which is a common feature in models extending Green–Naghdi Type III thermoelasticity. The core novelty, however, lies in introducing an additional coupling structure that explicitly links the thermal relaxation effects with the mechanical dissipation effects. This modification moves beyond the standard MGT coupling and is rooted in an effort to model complex visco-thermal interactions, representing the primary contribution to the literature. The well posedness of this modified system is first established using semigroup theory. Through the construction of a new Lyapunov functional, sufficient conditions are then rigorously derived, ensuring the exponential stability of solutions under specific parameter regimes. Furthermore, a critical balance condition is identified between the thermal conductivity and the thermal relaxation time, beyond which the system’s energy decay ceases to be exponential. Finally, numerical experiments employing an explicit–implicit finite difference scheme validate the theoretical findings and illustrate the substantial influence of both the modified coupling and the frictional damping on the system’s long-term energy behavior. Full article

25 pages, 756 KB

Open AccessArticle

Hybrid Graph Convolutional-Recurrent Framework with Community Detection for Spatiotemporal Demand Prediction in Micromobility Systems

by Mayme Moon Zin, Karn Patanukhom, Merkebe Getachew Demissie and Santi Phithakkitnukoon

Mathematics 2026, 14(1), 116; https://doi.org/10.3390/math14010116 (registering DOI) - 28 Dec 2025

Abstract

The rapid growth of dockless electric scooter (e-scooter) sharing services has transformed short-distance urban mobility, offering convenience and sustainability benefits while amplifying challenges related to demand imbalance, fleet rebalancing, and spatial inequity. Accurate spatiotemporal demand prediction is therefore essential for optimizing resource allocation and supporting data-driven policy interventions. This study proposes a hybrid deep learning framework that integrates a Graph Convolutional Network (GCN) with a Gated Recurrent Unit (GRU) and community detection to enhance short-term prediction of e-scooter pick-up and drop-off demands. The Louvain algorithm is employed to partition urban areas into mobility-based communities, enabling the model to capture functional connectivity rather than relying solely on geographic proximity. Using real-world e-scooter trip data from Calgary, Canada, the model’s performance is evaluated against established baselines, including a Masked Fully Convolutional Network (MFCN) and conventional GRU architectures. Results show that the proposed approach achieves up to 11.8% improvement in mean absolute error (MAE) compared with the MFCN baseline and more robust generalization across temporal horizons. The findings demonstrate that integrating community structures into graph-based learning effectively captures complex urban dynamics, providing practical insights for sustainable micromobility operation and service deployment. Full article

(This article belongs to the Special Issue Theoretical and Applied Mathematics in Supply Chain Management)

29 pages, 2471 KB

Open AccessArticle

MISA-GMC: An Enhanced Multimodal Sentiment Analysis Framework with Gated Fusion and Momentum Contrastive Modality Relationship Modeling

by Zheng Du, Yapeng Wang, Xu Yang, Sio-Kei Im and Zhiwen Wang

Mathematics 2026, 14(1), 115; https://doi.org/10.3390/math14010115 (registering DOI) - 28 Dec 2025

Abstract

Multimodal sentiment analysis jointly exploits textual, acoustic, and visual signals to recognize human emotions more accurately than unimodal models. However, real-world data often contain noisy or partially missing modalities, and naive fusion may allow unreliable signals to degrade overall performance. To address this, we propose an enhanced framework named MISA-GMC, a lightweight extension of the widely used MISA backbone that explicitly accounts for modality reliability. The core idea is to adaptively reweight modalities at the sample level while regularizing cross-modal representations during training. Specifically, a reliability-aware gated fusion module down-weights unreliable modalities, and two auxiliary training-time regularizers (momentum contrastive learning and a lightweight correlation graph) help stabilize and refine multimodal representations without adding inference-time overhead. Experiments on three benchmark datasets—CMU-MOSI, CMU-MOSEI, and CH-SIMS—demonstrate the effectiveness of MISA-GMC. For instance, on CMU-MOSI, the proposed model improves 7-class accuracy from 43.29 to 45.92, reduces the mean absolute error (MAE) from 0.785 to 0.712, and increases the Pearson correlation coefficient (Corr) from 0.764 to 0.795. This indicates more accurate fine-grained sentiment prediction and better sentiment-intensity estimation. On CMU-MOSEI and CH-SIMS, MISA-GMC also achieves consistent gains over MISA and strong baselines such as LMF, ALMT, and MMIM across both classification and regression metrics. Ablation studies and missing-modality experiments further verify the contribution of each component and the robustness of MISA-GMC under partial-modality settings. Full article

(This article belongs to the Special Issue Applications of Machine Learning and Pattern Recognition)

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19 pages, 1104 KB

Open AccessArticle

V2G System Optimization for Photovoltaic and Wind Energy Utilization: Bilevel Programming with Dual Incentives of Real-Time Pricing and Carbon Quotas

by Junfeng Cui, Xue Feng, Hongbo Zhu and Zongyao Wang

Mathematics 2026, 14(1), 114; https://doi.org/10.3390/math14010114 (registering DOI) - 28 Dec 2025

Abstract

Considering the global objective of carbon emission reduction, this paper focuses on optimizing the operational efficiency of grid-connected electric vehicles (EVs) and promoting sustainable energy integration and thus proposes a novel dual-incentive mechanism combining real-time pricing (RTP) and carbon quotas. A core of this study is the development of a bilevel programming model that effectively captures the strategic interaction between power suppliers (PS) and microgrid (MG) users. At the upper level, the model enables the PS to optimize electricity prices, achieving both revenue maximization and grid balance maintenance; at the lower level, it supports MGs in rational scheduling of EV charging/discharging, photovoltaic and wind energy (PWE) utilization, and load consumption, ensuring the fulfillment of user demands while maximizing MG profits. To address the non-convex factors in the model that hinder an efficient solution, another key is the design of a bilevel distributed genetic algorithm, which realizes efficient decentralized decision making and provides technical support for the practical application of the model. Through comprehensive simulations, the study verifies significant quantitative outcomes. The proposed algorithm converges after only 61 iterations, ensuring efficient solution performance. The average purchase price of electricity from the PS for the MG is USD 1.1, while the selling price of PWE sources from MG for the PS is USD 0.6. This effectively promotes the MG to prioritize the consumption of PWE sources and encourages the PS to repurchase the electricity generated by PWE sources. On average, carbon emissions decreased by approximately 300 g each time slot, and the average amount of carbon trading was around USD 8. Ultimately, this research delivers a practical and impactful solution for the development of MGs and the advancement of carbon reduction goals. Full article

(This article belongs to the Special Issue Applied Machine Learning and Soft Computing)

19 pages, 2018 KB

Open AccessArticle

Effective Target Privacy Protection Against Dynamic-Link-Prediction-Based Attacks via Adversarial Learning

by Mengdi Sun and Minghu Tang

Mathematics 2026, 14(1), 113; https://doi.org/10.3390/math14010113 (registering DOI) - 28 Dec 2025

Abstract

Graph data mining has emerged as a prominent area of research in both academic and industrial domains. Dynamic link prediction, a critical subfield within graph data mining, offers a more realistic representation of real-world networks compared to static link prediction, making dynamic link prediction attacks particularly threatening to privacy. While privacy protection in dynamic networks can be achieved by removing certain sensitive links, attackers can still infer hidden sensitive connections from observable network data. Moreover, existing studies seldom address target-level defense against dynamic link prediction attacks. To address these challenges, this paper proposes a Target-Level Privacy protection method against Dynamic Link Prediction attacks (TP-DLP). The method leverages temporal information in dynamic networks to implement targeted protection based on link gradient information, operating within a perturbation range that preserves the inherent characteristics of dynamic networks. Using dynamic social networks as a case study, the approach distinguishes the privacy levels of dynamic links to achieve target-level privacy protection. Extensive experimental results demonstrate that TP-DLP significantly enhances privacy protection while preserving network utility, making it well-suited for targeted defense against dynamic network link prediction. It can be concluded that our method achieves a balanced trade-off between privacy protection effectiveness and network structural fidelity. Full article

24 pages, 11970 KB

Open AccessArticle

Data-Driven Probabilistic Wind Power Forecasting and Dispatch with Alternating Direction Method of Multipliers over Complex Networks

by Lina Sheng, Nan Fu, Juntao Mou, Linglong Zhu and Jinan Zhou

Mathematics 2026, 14(1), 112; https://doi.org/10.3390/math14010112 (registering DOI) - 28 Dec 2025

Abstract

This paper proposes a privacy-preserving framework that couples probabilistic wind power forecasting with decentralized anomaly detection in complex power networks. We first design an adaptive federated learning (FL) scheme to produce probabilistic forecasts for multiple geographically distributed wind farms while keeping their raw data local. In this scheme, an artificial neural network with quantile regression is trained collaboratively across sites to provide calibrated prediction intervals for wind power outputs. These forecasts are then embedded into an alternating direction method of multipliers (ADMM)-based load-side dispatch and anomaly detection model for decentralized power systems with plug-and-play industrial users. Each monitoring node uses local measurements and neighbor communication to solve a distributed economic dispatch problem, detect abnormal load behaviors, and maintain network consistency without a central coordinator. Experiments on the GEFCom 2014 wind power dataset show that the proposed FL-based probabilistic forecasting method outperforms persistence, local training, and standard FL in RMSE and MAE across multiple horizons. Simulations on IEEE 14-bus and 30-bus systems further verify fast convergence, accurate anomaly localization, and robust operation, indicating the effectiveness of the integrated forecasting–dispatch framework for smart industrial grids with high wind penetration. Full article

(This article belongs to the Special Issue Advanced Machine Learning Research in Complex System)

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14 pages, 4855 KB

Open AccessArticle

Generalized Synchronization of a Novel Hyperchaotic System and Application in Secure Communication

by Mohamed M. El-Dessoky, Nehad Almohammadi and Mansoor Alsulami

Mathematics 2026, 14(1), 111; https://doi.org/10.3390/math14010111 (registering DOI) - 28 Dec 2025

Abstract

In this paper, a generalized synchronization (GS) framework for identical hyperchaotic systems is presented. The main objective is to achieve generalized synchronization with guaranteed global stability and effective convergence, which remains a key challenge in synchronization-based secure communication systems. The proposed controller is systematically derived to ensure global asymptotic convergence of the synchronization errors for arbitrary initial conditions and distinct scaling factors. This formulation unifies complete, anti-, and generalized synchronization within a single control structure. To demonstrate the applicability of the proposed method, it is integrated into an image encryption algorithm, where the hyperchaotic trajectories of the drive system generate highly random permutation and diffusion sequences. Simulation results verify that the designed controller achieves effective generalized synchronization and that the encrypted images exhibit uniform histograms and low pixel correlation, indicating strong security and resistance to statistical attacks. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)

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19 pages, 3910 KB

Open AccessArticle

Defect Detection Algorithm of Galvanized Sheet Based on S-C-B-YOLO

by Yicheng Liu, Gaoxia Fan, Hanquan Zhang and Dong Xiao

Mathematics 2026, 14(1), 110; https://doi.org/10.3390/math14010110 (registering DOI) - 28 Dec 2025

Abstract

Galvanized steel sheets are vital anti-corrosion materials, yet their surface quality is prone to defects that impact performance. Manual inspection is inefficient, while conventional machine vision struggles with complex, small-scale defects in industrial settings. Although deep learning offers promising solutions, standard object detection models like YOLOv5 (which is short for ‘You Only Look Once’) exhibit limitations in handling the subtle textures, scale variations, and reflective surfaces characteristic of galvanized sheet defects. To address these challenges, this paper proposes S-C-B-YOLO, an enhanced detection model based on YOLOv5. First, a Squeeze-and-Excitation (SE) attention mechanism is integrated into the deep layers of the backbone network to adaptively recalibrate channel-wise features, improving focus on defect-relevant information. Second, a Transformer block is combined with a C3 module to form a C3TR module, enhancing the model’s ability to capture global contextual relationships for irregular defects. Finally, the original path aggregation network (PANet) is replaced with a bidirectional feature pyramid network (Bi-FPN) to facilitate more efficient multi-scale feature fusion, significantly boosting sensitivity to small defects. Extensive experiments on a dedicated galvanized sheet defect dataset show that S-C-B-YOLO achieves a mean average precision (mAP@0.5) of 92.6% and an inference speed of 62 FPS, outperforming several baseline models including YOLOv3, YOLOv7, and Faster R-CNN. The proposed model demonstrates a favorable balance between accuracy and speed, offering a robust and practical solution for automated, real-time defect inspection in galvanized steel production. Full article

(This article belongs to the Special Issue Advance in Neural Networks and Visual Learning)

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22 pages, 1273 KB

Open AccessArticle

Reaction–Diffusion on the Sphere with a Nonlinear Source Term: Symmetry Analysis, Group Classification, and Similarity Solutions

by Khalid Ali Alanezy

Mathematics 2026, 14(1), 109; https://doi.org/10.3390/math14010109 (registering DOI) - 28 Dec 2025

Abstract

We consider the nonlinear reaction–diffusion equation on the unit sphere

u_{t} = Δ_{S^{2}} u + f (u)

f_{u u} \neq 0

, and carry out a complete Lie point symmetry analysis. Solving the associated determining system [...] Read more.

We consider the nonlinear reaction–diffusion equation on the unit sphere

u_{t} = Δ_{S^{2}} u + f (u)

f_{u u} \neq 0

, and carry out a complete Lie point symmetry analysis. Solving the associated determining system yields a rigidity theorem: for every genuinely nonlinear

f (u)

, the admitted symmetry algebra is

so (3) \oplus ⟨ \partial_{t} ⟩

, generated by the rotational Killing fields and time translation. We further show through a group classification that the source families that enlarge symmetries in Euclidean space do not produce any additional point symmetries on

S^{2}

. From an optimal system of subalgebras, we derive curvature-adapted reductions in which the Laplace–Beltrami operator becomes a Legendre-type operator in intrinsic invariants. For the specific nonlinear source

f (u) = - e^{u} - 2

, specific reduced ODEs admit a hidden one-parameter symmetry, yielding a first integral and explicit steady states on

S^{2}

. Full article

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11 pages, 267 KB

Open AccessArticle

On Bassian Modules

by Askar Tuganbaev

Mathematics 2026, 14(1), 108; https://doi.org/10.3390/math14010108 (registering DOI) - 28 Dec 2025

Abstract

A familiar description of torsion Bassian Abelian groups is extended to Bassian modules over non-primitive Dedekind prime rings. Full article

(This article belongs to the Section A: Algebra and Logic)

14 pages, 359 KB

Open AccessFeature PaperArticle

Assessing Dominance in Survival Functions: A Test for Right-Censored Data

by Félix Belzunce, Carolina Martínez-Riquelme and Jaime Valenciano

Mathematics 2026, 14(1), 107; https://doi.org/10.3390/math14010107 (registering DOI) - 27 Dec 2025

Abstract

This paper proposes a new statistical test to assess the dominance of survival functions in the presence of right-censored data. Traditional methods, such as the Log-Rank test, are inadequate for determining whether one survival function consistently dominates another, especially when survival curves cross. The proposed test is based on the supremum of the difference between Kaplan–Meier estimators and allows for distinguishing between dominance and crossing survival curves. The paper presents the test’s asymptotic properties, along with simulations and applications to real datasets. The results demonstrate that the test has high sensitivity for detecting crossings and dominance compared to conventional methods. Full article

(This article belongs to the Special Issue Probability, Statistics & Symmetry, 2nd edition)

22 pages, 1166 KB

Open AccessArticle

Establishment and Analysis of a General Mass Model for Solenoid Valves Used in Space Propulsion Systems

by Yezhen Sun, Sen Hu and Guozhu Liang

Mathematics 2026, 14(1), 106; https://doi.org/10.3390/math14010106 (registering DOI) - 27 Dec 2025

Abstract

The solenoid valve component is the core part affecting the total mass of space propulsion system, and the accuracy of the solenoid valve mass model directly impacts the accuracy of the system mass estimation and optimization design. This study focuses on the solenoid valves used in gas path control for cold gas propulsion systems. The relationship between the gas flow rate and volume flow rate of the solenoid valve is derived. By analyzing the parameters affecting the mass of the solenoid valves, a general calculation mass model of the gas solenoid valve used in cold gas propulsion is proposed based on strength theory. Combining with the existing general calculation mass model for liquid solenoid valves and collecting mass data of 16 gas solenoid valves and 33 liquid solenoid valves used in space propulsion system, the mass calculation formulas of the gas and liquid solenoid valves are obtained by employing several mathematical fitting methods, including quadratic polynomial surface, Manski formula, bivariate power function, and pressure-corrected polynomial. The accuracy of different mass model formulas is compared to assess their performance in calculating the solenoid valve mass. The results show that the quadratic surface formula can better reflect the relationship between the mass of the gas solenoid valves and the valve parameters within the medium volume flow range of 1 × 10⁻⁹ to 3.9 × 10⁻³ m³/s and the proof pressure range of 0.4 to 49.74 MPa. For the calculation of liquid solenoid valve mass, the accuracy of quadratic polynomial surface fitting, bivariate power function equation, and univariate polynomial equation with pressure correction is comparable within the liquid volume flow range of 1.8 × 10⁻⁷ to 1.28 × 10⁻⁴ m³/s and the inlet pressure range of 0.99 to 4.24 MPa; the appropriate calculation formula can be selected based on the pressure conditions in the liquid solenoid valve chamber in practical applications. Sensitivity analysis shows a consistent trend for gas and liquid solenoid valves: proof pressure (gas valves) or inlet working pressure (liquid valves) are the dominant factors affecting valve mass, while volume flow rate has a moderate impact. The proposed solenoid valve mass model in this study can be used to calculate the mass of gas solenoid valves for space cold gas propulsion systems and liquid solenoid valves for liquid rocket thrusters with thrust below 1000 N, providing an important reference for the mass modeling and optimization design of the space propulsion systems. Full article

(This article belongs to the Special Issue Dynamic Modeling and Simulation for Control Systems, 3rd Edition)

26 pages, 1457 KB

Open AccessArticle

Calculating the Projective Norm of Higher-Order Tensors Using a Gradient Descent Algorithm

by Aaditya Rudra and Maria Anastasia Jivulescu

Mathematics 2026, 14(1), 105; https://doi.org/10.3390/math14010105 (registering DOI) - 27 Dec 2025

Abstract

Projective Norms are a class of tensor norms that map from the input to the output spaces. These norms are useful for providing a measure of entanglement. Calculating the projective norms is an NP-hard problem, which creates challenges in computing due to the complexity of the exponentially growing parameter space for higher-order tensors. We develop a novel gradient descent algorithm to estimate the projective norm of higher-order tensors. The algorithm exhibits stable convergence to a minimum nuclear-rank decomposition of the given tensor in all our numerical experiments. We further extend the algorithm to symmetric tensors and to density matrices. We demonstrate the performance of our algorithm by computing the nuclear rank and the projective norm for both pure and mixed states and provide numerical evidence supporting these results. Full article

(This article belongs to the Special Issue Recent Advances in Scientific Computing & Applications)

37 pages, 1452 KB

Open AccessFeature PaperArticle

Optimizing Low-Carbon Supply Chain Decisions Considering Carbon Trading Mechanisms and Data-Driven Marketing: A Fairness Concern Perspective

by Tao Yang, Yueyang Zhan and Huajun Tang

Mathematics 2026, 14(1), 104; https://doi.org/10.3390/math14010104 (registering DOI) - 27 Dec 2025

Abstract

As low-carbon supply chains increasingly integrate green transition strategies with digital transformation, coordinating high-cost green technology investments with data-driven marketing (DDM) becomes a complex managerial task. While these dual investments are essential for market growth, the inherent tension between economic efficiency and fairness concerns often triggers strategic friction phenomenon whose impact under cap-and-trade regulations remains insufficiently explored. This paper investigates the strategic implications of fairness concerns in a low-carbon supply chain in which a manufacturer invests in carbon emission reduction and a retailer engages in data-driven marketing (DDM), under a cap-and-trade regulation. We formulate four Stackelberg game models—Neutral Benchmark (NF), Retailer Fairness (RF), Manufacturer Fairness (MF), and Bilateral Fairness (BF)—to analyze the interplay between behavioral equity and economic efficiency. The main analytical results indicate that (1) fairness concerns universally function as an “efficiency tax” on the supply chain system, where the rational benchmark consistently yields the highest system efficiency. In contrast, bilateral fairness concerns lead to the worst performance due to double friction effects. (2) Counter-intuitively, the retailer can “weaponize” fairness concerns to extract surplus from the leader. Specifically, in environments with high carbon emission reduction costs, a fairness-concerned retailer compels the manufacturer to grant significant wholesale price concessions, thereby achieving higher profits than in a purely rational setting. (3) The manufacturer’s fairness creates a “Benevolence Trap” for the follower; to balance equity, a fair manufacturer tends to underinvest in green technologies, which severely contracts market demand and, unlike the retailer fairness scenario, fails to yield economic benefits for the retailer. (4) A critical “regime-switching” dynamic exists regarding the carbon trading price. While the retailer benefits from fairness strategies in nascent carbon markets, a pivot to rationality becomes optimal as carbon prices surge and efficiency dividends dominate. These findings offer novel managerial insights for supply chain members to navigate behavioral complexities and for policymakers to align incentive mechanisms. Full article

(This article belongs to the Special Issue Operations and Supply Chain Management with the Application of Mathematical Models, 2nd Edition)

16 pages, 3705 KB

Open AccessArticle

Benchmarking Adversarial Patch Selection and Location

by Shai Kimhi, Moshe Kimhi and Avi Mendelson

Mathematics 2026, 14(1), 103; https://doi.org/10.3390/math14010103 (registering DOI) - 27 Dec 2025

Abstract

Adversarial patch attacks threaten the reliability of modern vision models. We present PatchMap, the first spatially exhaustive benchmark of patch placement, built by evaluating over

1.5 \times 10^{8}

forward passes on ImageNet validation images. PatchMap reveals systematic “hot-spots” where small patches (as [...] Read more.

Adversarial patch attacks threaten the reliability of modern vision models. We present PatchMap, the first spatially exhaustive benchmark of patch placement, built by evaluating over

1.5 \times 10^{8}

forward passes on ImageNet validation images. PatchMap reveals systematic “hot-spots” where small patches (as little as 2% of the image) induce confident misclassifications and large drops in model confidence. To demonstrate its utility, we propose a simple segmentation-guided placement heuristic that leverages off-the-shelf masks to identify vulnerable regions without any gradient queries. Across five architectures-including adversarially trained ResNet-50-our method boosts attack success rates by 8–13 percentage points compared to random or fixed placements. Full article

(This article belongs to the Special Issue AI Security and Edge Computing in Distributed Edge Systems)

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24 pages, 1271 KB

Open AccessArticle

Design of Decoupling Control Based TSK Fuzzy Brain-Imitated Neural Network for Underactuated Systems with Uncertainty

by Duc Hung Pham and Vu The Mai

Mathematics 2026, 14(1), 102; https://doi.org/10.3390/math14010102 (registering DOI) - 26 Dec 2025

Abstract

This paper proposes a Takagi–Sugeno–Kang Elliptic Type-2 Fuzzy Brain-Imitated Neural Network (TET2FNN)-based decoupling control strategy for nonlinear underactuated mechanical systems subject to uncertainties. A sliding-mode framework is employed to construct a decoupled control architecture, in which an intermediate variable is introduced to separate two second-order sliding surfaces, thereby forming a decoupled slip surface. The TET2FNN acts as the main controller and approximates the ideal control law online, while a robust compensator is incorporated to suppress approximation errors and guarantee closed-loop stability. Simulation studies conducted on a double inverted pendulum system demonstrate that the proposed method achieves improved tracking accuracy and disturbance rejection compared with representative state-of-the-art controllers. Furthermore, the computational burden remains reasonable, indicating that the proposed scheme is suitable for real-time implementation and practical nonlinear control applications. Full article

(This article belongs to the Special Issue Intelligent Control and Applications of Nonlinear Dynamic System)

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Meet Us at the 2026 Joint Mathematics Meeting, 4–7 January 2026, Washington D.C., USA

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Topic in Applied Sciences, Energies, Fluids, Materials, Mathematics, Micromachines

Fluid Mechanics, 2nd Edition Topic Editors: Sihui Hong, Chaobin Dang, Shuangfeng Wang
Deadline: 31 December 2025

Topic in Applied Sciences, Electronics, Entropy, Mathematics, Symmetry, Technologies, Chips

Quantum Information and Quantum Computing, 2nd Volume Topic Editors: Durdu Guney, David Petrosyan
Deadline: 6 January 2026

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A Real-World Application of Chaos Theory Topic Editors: Adil Jhangeer, Mudassar Imran
Deadline: 28 February 2026

Topic in Entropy, Fluids, Mathematics, Dynamics, Applied Sciences, Physics

Advanced Multiscale Techniques and Wavelet Analysis in Turbulent Flow Studies Topic Editors: Xiaojing Zheng, Youhe Zhou, Jizeng Wang
Deadline: 31 March 2026

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13–15 April 2026 The 1st International Online Conference on Fractal and Fractional: Theoretical Foundations and Interdisciplinary Applications

20–22 October 2026 The 1st International Online Conference on Symmetry (IOCSYM 2026)

10–12 June 2026 The 2nd International Online Conference on Mathematics and Applications (IOCMA2026)

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Special Issue in Mathematics

Advances in Control Theory and Optimizations Guest Editors: Francisco Miranda, Helena Sofia Rodrigues
Deadline: 30 December 2025

Special Issue in Mathematics

Recent Advances in Mathematical Optics Guest Editor: Christian Cuadrado-Laborde
Deadline: 30 December 2025

Special Issue in Mathematics

Stochastic Processes and Its Applications Guest Editors: Elvira Di Nardo, Luis Alberiko Gil-Alana
Deadline: 30 December 2025

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Evolutionary Computation and Multi-Objective Optimization Guest Editors: Alejandro Rodríguez-Molina, Miguel Gabriel Villarreal-Cervantes, Efrén Mezura-Montes
Deadline: 30 December 2025

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Topical Collection in Mathematics

Topology and Foundations Collection Editors: Lorentz Jäntschi, Dušanka Janežič

Topical Collection in Mathematics

Theoretical and Mathematical Ecology Collection Editor: Yuri V. Tyutyunov

Topical Collection in Mathematics

Knowledge-Driven Artificial Intelligence: Models, Optimization and Algorithms Collection Editors: Huawen Liu, Qing Li

Topical Collection in Mathematics

Applied Mathematics for Emerging Trends in Mechatronic Systems Collection Editors: Paolo Mercorelli, Aydin Azizi

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