Mathematics

11 pages, 287 KB

Open AccessFeature PaperArticle

Gallai’s Theorem: A Detailed Exposition of Witt’s Proof

by Roger D. Maddux

Mathematics 2026, 14(16), 2924; https://doi.org/10.3390/math14162924 (registering DOI) - 13 Aug 2026

Gallai’s theorem states that if the Euclidean plane is colored with finitely many colors then every finite configuration of points admits a monochromatic homothetic copy. This paper presents a detailed exposition of a proof originally published by Ernst Witt in 1952 and subsequently [...] Read more.

Gallai’s theorem states that if the Euclidean plane is colored with finitely many colors then every finite configuration of points admits a monochromatic homothetic copy. This paper presents a detailed exposition of a proof originally published by Ernst Witt in 1952 and subsequently expanded by Alexander Soifer. Additional intermediate steps are supplied throughout, yielding a self-contained and easily verified proof. The argument is formulated recursively through finite configurations associated with a double induction and thereby makes explicit the finite structures underlying the theorem. Full article

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

28 pages, 1356 KB

Open AccessArticle

A Dual-Quaternion Framework for Bennett-Limit Diagnostics in Rigid Kresling Origami FOLD–TWIST–FOLD Robots

by Bogdan Fustei, Monica Leba and Andreea Ionica

Mathematics 2026, 14(16), 2923; https://doi.org/10.3390/math14162923 (registering DOI) - 12 Aug 2026

Abstract

This paper presents a dual-quaternion (DQ) framework for the rigid-kinematic modeling and validation of rigid hexagonal Kresling origami robots executing a prescribed FOLD–TWIST–FOLD motion. Triangular panels are modeled as rigid bodies, and crease lines are modeled as fixed revolute axes. Exact spatial 4R [...] Read more.

This paper presents a dual-quaternion (DQ) framework for the rigid-kinematic modeling and validation of rigid hexagonal Kresling origami robots executing a prescribed FOLD–TWIST–FOLD motion. Triangular panels are modeled as rigid bodies, and crease lines are modeled as fixed revolute axes. Exact spatial 4R closure is decomposed into a primal orientation closure and a dual transported-translation closure. Under explicit non-degeneracy, paired-normal transport, and branch assumptions, the projection of the dual closure recovers a Bennett-type axis ratio. For Kresling, this result is applied only in the intersecting-axis limit d = 0, serving as a local axis-geometry diagnostic rather than a sufficient global closure criterion. The baseline three-cell module (N_s = 6, R = 48 mm, H₀ = 60 mm, initial twist 30°) executes a (−2 mm, +5°, −2 mm) actuation command over 37 states, preserving rigid-edge/panel and DQ consistency within the prescribed tolerances. Comparative benchmarks show that DQ and homogeneous-transform mappings achieve practically identical accuracy, although homogeneous transforms run faster in the tested MATLAB 2025b workload; Direct LM is faster for computing the endpoint solution, whereas DQ-parametric homotopy provides state-by-state path certification. A generalized implementation evaluates 27 configurations (21 accepted, six rejected), with all six controlled actuation-order permutations accepted. The framework serves as a pre-prototyping rigid-kinematic qualification tool that separates local axis compatibility from global rigid-origami feasibility. Full article

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

27 pages, 978 KB

Open AccessArticle

A Robust High-Dimensional MANOVA Test Based on Weighted MRCD Estimation

by Hasan Bulut

Mathematics 2026, 14(16), 2922; https://doi.org/10.3390/math14162922 - 12 Aug 2026

Abstract

Classical MANOVA procedures are not directly applicable in high-dimensional settings where the number of variables is comparable to, or exceeds, the sample size, and many existing high-dimensional MANOVA tests remain sensitive to outlying observations. This study proposes a weighted minimum regularized covariance determinant [...] Read more.

Classical MANOVA procedures are not directly applicable in high-dimensional settings where the number of variables is comparable to, or exceeds, the sample size, and many existing high-dimensional MANOVA tests remain sensitive to outlying observations. This study proposes a weighted minimum regularized covariance determinant (MRCD)-based robust Wilks’ Lambda test for one-way high-dimensional MANOVA. The proposed method combines MRCD-based robust location and scatter estimation with a robust distance-based reweighting step and uses permutation calibration to obtain p-values. Through extensive Monte Carlo simulations, the method is evaluated in terms of Type-I error control, power, and robustness under structured contamination. Under clean data, the proposed test maintains empirical Type-I error near the nominal level, with only modest aggregate differences from Cheng-GM; Schott’s test can have higher power under weak signals. Under contaminated null scenarios where outliers create artificial group separation, the proposed method yields lower false-rejection rates than the competitors considered. A controlled sensitivity illustration using breast-cancer gene-expression data shows the same qualitative behavior after imposed contamination. The method is therefore positioned as a robustness-oriented option for contamination-prone high-dimensional MANOVA, at the cost of additional computation. Full article

(This article belongs to the Special Issue Statistical Inference: Methods and Applications)

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26 pages, 555 KB

Open AccessArticle

The Optimal and Robust Siting and Ranking Framework for ESS and STATCOM Under 765 kV Double-Circuit N-2 Contingencies

by Minsoo Kim, Minkyu Jung, Hyeonjun Im, Minyoung Lee and Duehee Lee

Mathematics 2026, 14(16), 2921; https://doi.org/10.3390/math14162921 - 12 Aug 2026

Abstract

We propose an optimal and robust siting and ranking framework for energy storage systems (ESSs) and static synchronous compensators (STATCOMs) under 765 kV double-circuit N-2 contingencies in the Republic of Korea transmission lines. Each N-2 contingency is constructed by pairing two parallel circuits [...] Read more.

We propose an optimal and robust siting and ranking framework for energy storage systems (ESSs) and static synchronous compensators (STATCOMs) under 765 kV double-circuit N-2 contingencies in the Republic of Korea transmission lines. Each N-2 contingency is constructed by pairing two parallel circuits that share the same 765 kV sending and receiving substations, so the contingency set is enumerated directly from the network topology. The framework screens 765 kV candidate buses at a fixed reference capacity, and it sizes ESS and STATCOM supports through independent active- and reactive-power capacity sweeps of repeated static post-contingency AC power flows. Then, it ranks candidates by a rank-aggregated robust score that combines scenario-averaged severity, scenario-averaged improvement, and evaluation reliability. Each sweep exercises only the non-negative support direction of its resource, namely ESS discharge and capacitive STATCOM operation, which is stated as an explicit modeling assumption. Here, robust means averaged over the multi-scenario set of the study year, load level, planning condition, and HVDC state rather than the worst-case in the min–max sense. The framework shows that the bus, which performs best on individual contingency pairs, does not coincide with the bus that is robust across scenarios, and it separately identifies the most frequent worst pair and the most difficult pair without relying on substation names. It therefore recommends the scenario-robust bus rather than the single-pair winner, orders candidates by the rank-aggregated robust score, and gives a STATCOM-leaning ESS–STATCOM mix at the top of the ranking under a transparent cost proxy, weight, cost-ratio, and coverage assumption. Therefore, the recommendation stays reproducible under alternative planning inputs. Full article

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

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8 pages, 240 KB

Open AccessArticle

On the “Exclusion Principle” in All-Pay Auctions with Incomplete Information

by Nicola Dimitri

Mathematics 2026, 14(16), 2920; https://doi.org/10.3390/math14162920 - 12 Aug 2026

Abstract

The number, and types, of participants in an auction can meaningfully affect the outcome of the competition. In a complete-information, all-pay context, previous studies proved that in first-price auctions, if the auctioneer maximizes the total bid received, then under appropriate conditions on the [...] Read more.

The number, and types, of participants in an auction can meaningfully affect the outcome of the competition. In a complete-information, all-pay context, previous studies proved that in first-price auctions, if the auctioneer maximizes the total bid received, then under appropriate conditions on the bidders’ values, it may be optimal for them to exclude some participants, in particular the best ones. In this paper, we investigate whether the same principle can still hold with incomplete information. In the simplest context of independent and identically distributed values, by considering the symmetric Bayes–Nash Bidding Equilibrium, we show that this principle does not hold in all-pay auctions, whether in first- or second-price auctions, as the auctioneer would want to admit as many bidders as possible when maximizing the total bid. Exclusion of some bidders instead may be optimal if, in a first-price all-pay auction, the auctioneer maximizes the highest bid. Such findings exhibit an interesting duality with the complete-information case, since the opposite conclusions follow. Moreover, in a first-price, winner-only pays auction, including as many bidders as possible would be convenient for the auctioneer in terms of both maximizing the total bid and the first bid. Finally, a comparison of first-price all-pay auctions with winner-only pays auctions with incomplete information shows the following interesting relation: The total bid in all-pay auctions is equal to the highest bid in winner-only pays auctions and, by Revenue Equivalence, is also equal to the second-highest bid. Therefore, whether or not exclusion is profitable for the auctioneer depends upon the type of the auction, the goal of the auctioneer, and the information distribution among bidders and the auctioneer. Full article

(This article belongs to the Special Issue Mathematical Methods and Statistics for Economics, Actuarial Science and Finance)

28 pages, 2147 KB

Open AccessArticle

Research on Improved Methods for LightGCN Based on Diffusion Models

by Weiqing Zhuang and Zhikai Chen

Mathematics 2026, 14(16), 2919; https://doi.org/10.3390/math14162919 - 12 Aug 2026

Abstract

Recommendation systems have become crucial tools for mitigating information overload and enhancing user experience. Among them, LightGCN has achieved remarkable results in both efficiency and performance by simplifying graph convolution operations. However, in scenarios with sparse training data, LightGCN tends to produce homogeneous [...] Read more.

Recommendation systems have become crucial tools for mitigating information overload and enhancing user experience. Among them, LightGCN has achieved remarkable results in both efficiency and performance by simplifying graph convolution operations. However, in scenarios with sparse training data, LightGCN tends to produce homogeneous recommendation results, failing to reflect users’ genuine and diverse interest preferences. This study proposes an improved method for LightGCN based on the diffusion model, named Diffusion_LightGCN, which aims to enhance the performance of the LightGCN recommendation system by generating high-quality pseudo-user embeddings. Different from existing diffusion-based recommendation methods that use diffusion models to directly generate items, item distributions, or pseudo-users (e.g., DiffuRec, C-DiffRec, and CDiff4Rec), Diffusion_LightGCN generates pseudo-user embeddings that serve as auxiliary training signals for graph-based collaborative filtering, thereby organically coupling the generative capability of diffusion models with the propagation mechanism of graph neural networks. To address the issue of embedding quality degradation in traditional LightGCN under sparse data, this study combines the generative capability of diffusion models with the efficient propagation mechanism of graph neural networks. By integrating original embeddings with pseudo-embeddings through an embedding dynamic fusion mechanism and optimizing the loss function, the training effectiveness of LightGCN is ultimately improved. The results demonstrate that this method outperforms LightGCN in sparse data scenarios. Full article

(This article belongs to the Special Issue Machine Learning and Artificial Intelligence in Recommender Systems)

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25 pages, 4631 KB

Open AccessArticle

VEX: Chroma-Stable Virtual Exposure Routing for Low-Light Image Enhancement

by Yuting Liu and Xinying Liu

Mathematics 2026, 14(16), 2918; https://doi.org/10.3390/math14162918 - 12 Aug 2026

Abstract

Low-light image enhancement (LLIE) seeks to improve visibility while suppressing amplified noise, preserving color, and preventing highlight over-enhancement. Most existing methods infer a normally exposed image from a single observed representation, requiring one feature stream to reconcile shadow brightening, highlight protection, denoising, and [...] Read more.

Low-light image enhancement (LLIE) seeks to improve visibility while suppressing amplified noise, preserving color, and preventing highlight over-enhancement. Most existing methods infer a normally exposed image from a single observed representation, requiring one feature stream to reconcile shadow brightening, highlight protection, denoising, and chromatic correction. We propose VEX, a chroma-stable virtual exposure routing network for single-image LLIE. A Chroma-Stable Virtual Exposure Generator (CVEG) decomposes the input into luminance and log-chroma components and produces five learnable virtual exposure states by perturbing luminance in a bounded space under a shared log-chroma constraint. A shared multi-scale encoder extracts comparable features from all states. At each of four scales, a Noise-Saturation-aware Exposure Router (NSER) combines learned feature evidence with explicit luminance, mid-tone, saturation, and detail-variation priors to predict pixel-wise softmax weights over the exposure states. An Exposure State Mixer (ESM) then performs gated multi-view recalibration of the routed bottleneck, after which a routed-skip decoder predicts a residual correction. VEX therefore casts LLIE as spatially adaptive selection among internal exposure hypotheses rather than as direct single-state regression. Extensive experiments on CDD-11 and LOL demonstrate that VEX consistently outperforms representative traditional and learning-based methods across fidelity, structural, and perceptual criteria. Full article

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

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13 pages, 299 KB

Open AccessArticle

On Singleton 3-Rainbow Domination of Cylindrical Graphs

by Janez Žerovnik

Mathematics 2026, 14(16), 2917; https://doi.org/10.3390/math14162917 - 12 Aug 2026

Abstract

Domination is among the most popular topics in graph theory, both due to its combinatorial appeal and a number of practical applications. Cylindrical graphs are Cartesian products of a path and a cycle. Usually, for some special subfamilies, natural constructions exist that give [...] Read more.

Domination is among the most popular topics in graph theory, both due to its combinatorial appeal and a number of practical applications. Cylindrical graphs are Cartesian products of a path and a cycle. Usually, for some special subfamilies, natural constructions exist that give exact solutions. On the other hand, in general, exact values of invariants on cylindrical graphs are hard to compute. In this paper, tight lower and upper bounds for singleton three-rainbow domination are proven and a subfamily of cylindrical graphs is found for which exact values are provided. The results complement recent findings for singleton two- and four-rainbow domination. Full article

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21 pages, 2061 KB

Open AccessArticle

Estimation of Sensitive Study Variable Using Randomized Response Model Under Systematic Sampling

by Muhammad Nouman Qureshi, Sajida Balqees, Muhammad Hanif and Magdy Nagy

Mathematics 2026, 14(16), 2916; https://doi.org/10.3390/math14162916 - 12 Aug 2026

Abstract

This study addresses the important problem of estimating sensitive study variables when respondents may be reluctant to provide truthful answers due to privacy concerns. In such situations, scrambled response techniques are widely used because they protect the respondent’s confidentiality while still allowing reliable [...] Read more.

This study addresses the important problem of estimating sensitive study variables when respondents may be reluctant to provide truthful answers due to privacy concerns. In such situations, scrambled response techniques are widely used because they protect the respondent’s confidentiality while still allowing reliable estimation of population characteristics. In this paper, we propose a generalized exponential estimator for estimating the population mean of a sensitive study variable using systematic sampling. A distinguishing feature of this research is the incorporation of the randomized response technique (RRT) into the systematic sampling framework, together with the inclusion of the logarithmic function in the proposed estimator, to improve the estimation of sensitive characteristics while making effective use of auxiliary information. The expressions for the approximate bias and mean square error (MSE) of the proposed estimator are derived using the Taylor and exponential series expansions. An efficient class of exponential-type estimators is obtained by employing different choices of robust parameters of the auxiliary variable. Theoretical comparisons of the proposed estimator are made with the sample mean, modified ratio, and modified exponential ratio estimators. A simulation study is conducted to evaluate the performance of the proposed estimator using five simulated populations at different levels of correlation. The theoretical results, simulation study and real data application show that the proposed estimator provides lower MSE and higher efficiency than the competing estimators under a wide range of conditions. Full article

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31 pages, 404 KB

Open AccessArticle

Sharp Growth Rate of Linear Rayleigh–Taylor Instability of Primitive Equations

by Yanhua Wang, Yantao Chen and Quan Wang

Mathematics 2026, 14(16), 2915; https://doi.org/10.3390/math14162915 - 12 Aug 2026

Abstract

We investigate the linear Rayleigh–Taylor instability within the framework of two-dimensional viscous primitive equations (VPE) under hydrostatic approximation, which govern large-scale oceanic and atmospheric flows. For steady quiescent background states with arbitrary density profiles satisfying the Rayleigh–Taylor instability criterion (i.e., heavier fluid overlying [...] Read more.

We investigate the linear Rayleigh–Taylor instability within the framework of two-dimensional viscous primitive equations (VPE) under hydrostatic approximation, which govern large-scale oceanic and atmospheric flows. For steady quiescent background states with arbitrary density profiles satisfying the Rayleigh–Taylor instability criterion (i.e., heavier fluid overlying lighter fluid), we conduct a rigorous spectral analysis of the linearized perturbation system. By eliminating the vertical velocity and pressure fields, we derive a nonlocal eigenvalue problem governing the horizontal velocity perturbations. To address the intrinsic non-self-adjoint nature of the system, we introduce an auxiliary variational problem with a frozen parameter, which enables us to establish the existence and uniqueness of a positive eigenvalue

λ_{k}

for each admissible horizontal wavenumber k. We further prove that

λ_{k}

is bounded above by a wavenumber-dependent critical value

s_{k}

, and verify the sharpness of the maximal growth rate defined as

Λ : = \sup_{k \neq 0} λ_{k}

. In particular, we show that no small perturbation can grow faster than the exponential rate

e^{Λ t}

, and construct explicit normal-mode solutions that achieve growth rates arbitrarily close to

Λ

. We also derive complete energy estimates for the linearized system, establishing uniform temporal control of solutions via the bound

e^{2 Λ t}

. Additionally, we demonstrate that positive viscosity dissipation uniformly bounds all unstable eigenvalues, whereas the zero-dissipation limit gives rise to an unbounded unstable spectrum where eigenvalues grow without bound as the horizontal wavenumber increases. This spectral behavior stands in stark contrast to that of the classical Euler equations, for which all unstable eigenvalues remain uniformly bounded. This fundamental discrepancy reveals that the hydrostatic approximation essentially distorts the intrinsic instability properties of the original fluid system, thereby invalidating its physical validity for numerical simulations of Rayleigh–Taylor instability. Full article

(This article belongs to the Special Issue Recent Studies on Partial Differential Equations and Its Applications)

17 pages, 6589 KB

Open AccessArticle

Assessment of Numerical Models for Unsteady Cloud Cavitation and Erosion Potential Around Different Hydrofoils

by Yilong Wang, Zhihua Zhou, Wenfei Yu, Yuanding Wang, Jiaqiong Wang and Linlin Geng

Mathematics 2026, 14(16), 2914; https://doi.org/10.3390/math14162914 - 12 Aug 2026

Abstract

The accuracy of the numerical simulation of unsteady cloud cavitation around a hydrofoil depends on the combination of the cavitation model, the Reynolds-Averaged Navier–Stokes (RANS) turbulence model and the exponential coefficient n of the Reboud’s correction. To assess the influence of such choices, [...] Read more.

The accuracy of the numerical simulation of unsteady cloud cavitation around a hydrofoil depends on the combination of the cavitation model, the Reynolds-Averaged Navier–Stokes (RANS) turbulence model and the exponential coefficient n of the Reboud’s correction. To assess the influence of such choices, three turbulence models, three cavitation models and two values of n have been combined to predict the shedding frequency and the morphology of cloud cavitation around the NACA65012 and NACA0009 hydrofoils. The comparison with analogous experimental results obtained in a cavitation tunnel indicates, firstly, that the same numerical set-up differs in accuracy depending on the hydrofoil geometry. Secondly, within the scope of the two tested hydrofoils and corresponding flow conditions, the Shear Stress Transport (SST) turbulence model and the value of n = 10 appear to be more accurate and more robust for all tested cases. And finally, the predicted shedding frequency is more sensitive to the selection of the turbulence model than to the cavitation model. If an erosion model is implemented, then it is found that the predicted potential energy distribution of the cavitating flow is sensitive to the selected cavitation model. In our case, the Sauer model gives a more accurate distribution and intensity of erosion power than the rest of the cavitation models. Full article

(This article belongs to the Special Issue Advances in Numerical Modeling and Simulation for Computational Fluid Dynamics)

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22 pages, 727 KB

Open AccessArticle

Elite Guided Jaya Algorithm with Mutation Strategy for Global Optimization and Parameter Extraction

by Yiying Zhang

Mathematics 2026, 14(16), 2913; https://doi.org/10.3390/math14162913 - 12 Aug 2026

Abstract

During the development of metaheuristic algorithms, many classic algorithms have emerged, and the Jaya algorithm (JAYA) is one of them. The inspiration of JAYA comes from the idea that the population should stay away from the worst solution already discovered during the search [...] Read more.

During the development of metaheuristic algorithms, many classic algorithms have emerged, and the Jaya algorithm (JAYA) is one of them. The inspiration of JAYA comes from the idea that the population should stay away from the worst solution already discovered during the search process, and should approach the best solution already discovered. Although JAYA demonstrates excellent performance in solving simple problems, it is prone to getting stuck in local optima when solving complex problems due to its reliance on the worst and best solutions already discovered. In order to improve the performance of JAYA, this paper proposes an improved JAYA, namely an elite guided Jaya algorithm with mutation strategy (EDJAYA). EDJAYA adopts an elite population driven mechanism to optimize its search strategy, and designs a mutation strategy to enhance its ability to escape from local optima. The excellent optimization performance of EDJAYA has been verified through solving numerical problems and practical problems of photovoltaic model parameter estimation. Full article

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

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29 pages, 1260 KB

Open AccessFeature PaperArticle

Platform Promotional Subsidy Strategies Under Price Regulation: A Tripartite Interactive Game Analysis of Regulator, Platform and Consumers

by Zeyu Zhang and Zonghuo Li

Mathematics 2026, 14(16), 2912; https://doi.org/10.3390/math14162912 - 12 Aug 2026

Abstract

This paper investigates platform promotional subsidy strategies under price regulation. We construct a dynamic game model of incomplete information. The model involves a regulator, a monopoly platform, and heterogeneous consumers. The regulator sets a subsidy cap. The platform chooses the price and subsidy [...] Read more.

This paper investigates platform promotional subsidy strategies under price regulation. We construct a dynamic game model of incomplete information. The model involves a regulator, a monopoly platform, and heterogeneous consumers. The regulator sets a subsidy cap. The platform chooses the price and subsidy after observing its cost type. Consumers decide whether to purchase based on their valuation and the perceived subsidy value. We solve the game by backward induction. We characterize the perfect Bayesian equilibria. We derive closed-form solutions for the critical subsidy threshold, separating/self-selection equilibrium conditions, and optimal regulatory policies. The analysis yields three main findings. First, the sign of the net social benefit of the promotional subsidy determines the optimal policy. A positive sign calls for a high subsidy cap. This induces full market coverage. A negative sign calls for a ban on subsidies. Second, when the subsidy cap lies between the two platform types’ critical thresholds, a natural separating/self-selection equilibrium emerges. It operates at no cost. The feasible interval widens linearly in the cost gap. Third, diseconomies of scale have a stronger marginal effect on the critical subsidy than marginal cost. There exists an endogenously determined critical proportion of high-value consumers, which depends on the model parameters, such as

v_{H}, v_{L}, c_{I}, η_{I}

. This proportion divides the policy space into two regimes. These results provide a basis for low-cost information screening and differentiated regulation. Full article

(This article belongs to the Special Issue Intelligent Operations, Game Analysis and Decision Making in Platform-Based Supply Chains)

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31 pages, 9074 KB

Open AccessArticle

Network Structural Characteristics of Insurance Institutions and Systemic Risk Contagion

by Yue Wang, Jiyun Qiu and Rui Liu

Mathematics 2026, 14(16), 2911; https://doi.org/10.3390/math14162911 - 12 Aug 2026

Abstract

The increasing systemic importance of the insurance sector implies that risks originating within it can propagate across industries through complex interconnections, potentially exerting significant impacts on China’s broader economy. This study investigates the transmission and spillover mechanisms of intra-industry risk among insurance-related institutions [...] Read more.

The increasing systemic importance of the insurance sector implies that risks originating within it can propagate across industries through complex interconnections, potentially exerting significant impacts on China’s broader economy. This study investigates the transmission and spillover mechanisms of intra-industry risk among insurance-related institutions from a network perspective. Specifically, a DCC-GARCH-CoVaR framework is employed to quantify risk spillovers among six listed financial institutions engaged in insurance-related businesses. Based on forecast error variance decomposition, an insurance network is constructed and further examined using network topology analysis to characterize its structural properties. The findings indicate that, despite gradual improvements in resilience and stability across institutions, network structural characteristics remain critical drivers of the formation and diffusion of systemic risk. Highly connected and central institutions are more likely to act as contagion hubs, facilitating rapid outward risk transmission, particularly in densely connected networks with shorter path lengths. Moreover, a nonlinear relationship exists between network structure and systemic risk, with leverage growth playing a moderating role. Elevated leverage amplifies the risk-enhancing effect of network centrality, while under certain conditions, mitigating adverse effects associated with passive risk exposure. Additionally, capital adequacy conditions shape risk dynamics: well-capitalized institutions can utilize network connections for risk diversification, whereas undercapitalized institutions tend to intensify contagion. Significant heterogeneity is also observed in the effects of coreness, density, and path length across institutional groups. Full article

(This article belongs to the Special Issue Applied Mathematics in Quantitative Finance and Risk Management: Theory, Evidence, and Applications)

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26 pages, 675 KB

Open AccessArticle

Reinforcement Learning-Based Day-to-Day Route Choice Models Under Full and Partial Information Conditions: A Theoretical and Experimental Research

by Yuance Yang, Ning Jia, Nianlu Ren, Hongye Fan and Zhanghao Lei

Mathematics 2026, 14(16), 2910; https://doi.org/10.3390/math14162910 - 12 Aug 2026

Abstract

Learning behavior plays an important role in the day-to-day route choice process, which is essentially a repeated decision-making process. This paper proposes two learning models for the day-to-day route choice process, one for the full-information (FI) condition and one for the partial-information (PI) [...] Read more.

Learning behavior plays an important role in the day-to-day route choice process, which is essentially a repeated decision-making process. This paper proposes two learning models for the day-to-day route choice process, one for the full-information (FI) condition and one for the partial-information (PI) condition. Both models are developed based on attraction-based reinforcement learning theory but with different learning mechanisms. In our FI model, travelers learn by comparing different paths, while in our PI model, travelers’ route choice decision is modeled by long-term cost minimization and a policy-based reinforcement algorithm. Both models are rooted in individual strategy updating and are theoretically linked to classical network flow assignment theory: we establish that individual-level invariance of choice probabilities is sufficient for aggregate stationarity consistent with Wardrop equilibrium, and provide an explicit stability condition for the FI dynamics. To validate the models, laboratory experiments under FI and PI conditions were designed and conducted. The experimental data confirm the presence of learning behavior and show that the proposed models achieve better predictive performance than several behavioral benchmarks, including EWA, Q-learning, mixed logit, and a Selten variant. Moreover, an interesting phenomenon was found in the FI experimental dataset: even though FI was provided, some subjects still made decisions hinging on their own experience. Our findings would benefit traffic information services and traffic administration. Full article

(This article belongs to the Special Issue AI, Machine Learning and Optimization)

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24 pages, 622 KB

Open AccessArticle

Structure and Explicit of Solutions for a Third-Order System of Rational Difference Equations

by Burak Oğul

Mathematics 2026, 14(16), 2909; https://doi.org/10.3390/math14162909 - 12 Aug 2026

Abstract

This paper investigates the qualitative behavior, global dynamics, and explicit solution forms of four distinct coupled systems of third-order rational difference equations. By employing an analytical approach based on mathematical induction and recursive substitution, we derive the general closed-form expressions for the solutions [...] Read more.

This paper investigates the qualitative behavior, global dynamics, and explicit solution forms of four distinct coupled systems of third-order rational difference equations. By employing an analytical approach based on mathematical induction and recursive substitution, we derive the general closed-form expressions for the solutions in each of the four cases under arbitrary non-zero real initial conditions. Furthermore, we provide rigorous proofs for the asymptotic behavior and the exact periodicity of these solutions. We completely characterize the structural conditions under which the trajectories exhibit periodic persistence or generate potential singularities, thereby determining the corresponding forbidden sets. These results provide a complete classification of the global behavior for these higher-order coupled systems, resolving open questions regarding their long-term dynamics and significantly extending the existing literature on non-linear rational difference equations. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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34 pages, 4238 KB

Open AccessArticle

Impedance Reshaping Control for Stability Enhancement of Grid-Following Inverters in Weak Grids

by Bogu Huang, Yibo Wang, Yuhan Guo, Wentao Yang, Yuxuan Wu and Yanxin Hu

Mathematics 2026, 14(16), 2908; https://doi.org/10.3390/math14162908 - 12 Aug 2026

Abstract

Although the weak-grid stability of grid-following inverters has been widely studied, traditional SCR-based assessment mainly describes grid strength and does not directly quantify the minimum SCR required for stable operation under a specified active-power command. This paper uses a critical-SCR-oriented operating-boundary evaluation to [...] Read more.

Although the weak-grid stability of grid-following inverters has been widely studied, traditional SCR-based assessment mainly describes grid strength and does not directly quantify the minimum SCR required for stable operation under a specified active-power command. This paper uses a critical-SCR-oriented operating-boundary evaluation to compare the control-dependent dynamic stability requirement with the physical steady-state feasibility limit. It proposes a channel-specific impedance-reshaping extension that retains the PLL-related q-q and d-q compensation paths and adds a voltage-loop-related q-d compensation path. The added path reduces the control-induced dynamic critical SCR, thereby allowing stable operation at lower SCR than with the original control structure. Firstly, the steady-state critical SCR is derived from the steady-state power-transfer relation under a specified active-power command to define the physical feasibility lower bound, and the dynamic critical SCR is identified using a scanning procedure based on the generalized Nyquist criterion. The resulting control-induced SCR gap quantifies the additional grid-strength requirement introduced by the control dynamics. Secondly, a simplified d-q admittance representation is used to relate established PLL-related and voltage-loop-related coupling effects to the corresponding compensation paths. Thirdly, the proposed channel-specific impedance-reshaping design is used to mitigate these effects while retaining the selected voltage-loop bandwidth, and an anti-drift correction is further introduced to improve implementation robustness. Frequency-domain analysis and time-domain simulations validate the dynamic critical-SCR evaluation, while hardware-in-the-loop experiments demonstrate the real-time operation of the proposed controller. The results show that it moves the dynamic critical-SCR boundary close to the steady-state physical boundary and improves the weak-grid stability margin. The hardware-in-the-loop results further confirm bounded operation under SCR and frequency steps. Full article

(This article belongs to the Topic Sustainable Industrial Processes: Mathematical Methodologies, Modelling, AI and Energy Efficiency)

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22 pages, 1776 KB

Open AccessArticle

AGTA: Topology-Aware Sequential Decision-Making in Multi-Agent Reinforcement Learning

by Kun Hu, Shanghua Wen, Wendi Wu, Xiang Zhang and Minglong Li

Mathematics 2026, 14(16), 2907; https://doi.org/10.3390/math14162907 - 12 Aug 2026

Abstract

Multi-agent reinforcement learning (MARL) has long grappled with fundamental coordination challenges due to the existence of complex inter-agent correlations that are inherent in multi-agent systems. While the recent advancement of the sequential decision-making paradigm provides fine-grained supervision for the multi-agent decision-making process, the [...] Read more.

Multi-agent reinforcement learning (MARL) has long grappled with fundamental coordination challenges due to the existence of complex inter-agent correlations that are inherent in multi-agent systems. While the recent advancement of the sequential decision-making paradigm provides fine-grained supervision for the multi-agent decision-making process, the absence of computationally tractable solutions for inter-agent correlation management remains a critical challenge and substantially constrains the impact of this paradigm. To tackle this challenge, in this paper, we introduce Action Generation with Topology Awareness (AGTA), a topology-aware sequential decision-making framework in MARL that integrates inter-agent correlation modeling with topology-guided decision-order optimization. AGTA extracts inter-agent mutual attention via multi-agent transformer during the learning dynamics. Subsequently, it captures directed acyclic graphs(DAGs) directly from the extracted attention matrices to model inter-agent correlations. Finally, it refines the action generation order by analyzing and solving topological constraints, thus realizing inter-agent correlation management. Extensive experiments demonstrate that the proposed method outperforms the state-of-the-art counterparts. Full article

(This article belongs to the Special Issue Mathematics and Machine Learning for Intelligent Perception, Decision-Making, and Control)

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27 pages, 13065 KB

Open AccessArticle

Parameter Estimation in a Chaotic Supply Chain System Using Extended Kalman Filter

by Neslihan Açıkgöz, Gültekin Çağıl, Yılmaz Uyaroğlu, Ufuk Kula, Serap Ercan Cömert and Ayşe Ünlü

Mathematics 2026, 14(16), 2906; https://doi.org/10.3390/math14162906 - 11 Aug 2026

Abstract

Demand, inventory, and production continuously influence one another in supply chains. Sudden demand changes, inaccurate records, production disruptions, and external factors can create irregular fluctuations. Although demand or sales data are often available, inventory and production information may be incomplete or delayed. This [...] Read more.

Demand, inventory, and production continuously influence one another in supply chains. Sudden demand changes, inaccurate records, production disruptions, and external factors can create irregular fluctuations. Although demand or sales data are often available, inventory and production information may be incomplete or delayed. This study applies the Extended Kalman Filter (EKF) to jointly estimate inventory, production quantity, and three model parameters using demand as the only measured variable. Its main contribution is a recursive structure that updates the state and parameter estimates with each new measurement. The method is tested on a three-variable supply chain model exhibiting chaotic behavior. Performance is assessed through numerical errors, convergence, time series, phase portraits, and bifurcation diagrams. Robustness to different initial estimates, sensitivity to process- and measurement-noise covariance settings, and performance under chaotic, semi-chaotic, and ordered regimes are also examined. Parameter errors range from 1.5% to 2.4%, while normalized errors for demand, inventory, and production remain below 3%. A larger assumed measurement-noise covariance slows convergence. Under the selected conditions, the closest agreement occurs in the chaotic regime, but this is not interpreted as a general superiority of the EKF. With real-data validation, the framework could support estimation of missing or delayed information, short-term monitoring, early warning, and decision support. Full article

(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)

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