Mathematics

18 pages, 1332 KB

Open AccessArticle

Method and Algorithms for Computing Fuzzy Fréchet and Hausdorff Distance

by Mykhailo Zarichnyi, Oleh Berezsky, Mykola Berezkyi and Vasyl Teslyuk

Mathematics 2026, 14(5), 892; https://doi.org/10.3390/math14050892 (registering DOI) - 5 Mar 2026

Accurate image similarity assessment is a key problem in computer vision, particularly in segmentation and classification problems. Classical Hausdorff and Fréchet metrics provide pointwise distance values and do not allow similarity to be evaluated in the form of intervals, which limits their applicability [...] Read more.

Accurate image similarity assessment is a key problem in computer vision, particularly in segmentation and classification problems. Classical Hausdorff and Fréchet metrics provide pointwise distance values and do not allow similarity to be evaluated in the form of intervals, which limits their applicability in problems where uncertainty plays a significant role. In this study, a combined approach to computing distances between images based on fuzzy Fréchet and Hausdorff metrics is developed. Two theorems are proved demonstrating that, for convex polygonal contours, the fuzzy Hausdorff distance coincides with the fuzzy Fréchet distance. This result makes it possible to replace the computation of the fuzzy Hausdorff metric with the simpler fuzzy discrete Fréchet metric. A method and algorithms for determining the fuzzy discrete Fréchet distance and a combined distance between convex polygons are proposed; their computational complexity is evaluated, and an application example is provided. The results show that the combined fuzzy metric reduces computation time by at least a factor of two compared to the direct computation of the fuzzy Hausdorff metric, while preserving similarity assessment accuracy. The proposed approach can be applied to shape analysis, segmentation evaluation, and similarity modeling in image classification systems. Future research directions include extending the method to non-convex polygons and arbitrary geometric objects. Full article

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

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52 pages, 657 KB

Open AccessArticle

Graph-Theoretic Idealization of Semigroups via Bruck-Reilly Extensions

by Suha Wazzan and David A. Oluyori

Mathematics 2026, 14(5), 891; https://doi.org/10.3390/math14050891 (registering DOI) - 5 Mar 2026

Abstract

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—

Γ_{E}

,

Γ_{0}

,

Γ_{Cay}

,

Γ_{K}

, and [...] Read more.

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—

Γ_{E}

,

Γ_{0}

,

Γ_{Cay}

,

Γ_{K}

, and

Γ (G_{k})

—each encoding a distinct algebraic facet of

S B_{i} (⋈) B

. We prove explicit correspondences linking combinatorial invariants to algebraic structure: diameter captures generating efficiency and semilattice height; girth signals short relations; chromatic number bounds idempotent cardinalities and

D

-class counts; clique number measures maximal commuting subsets; and Laplacian spectra encode ideal size and Schützenberger groups. Our central result demonstrates that Green’s relations are combinatorially recoverable from graph pairs. For commutative

S B_{i} (⋈) B

,

(Γ_{E}, Γ_{K})

uniquely determines

J

-order,

D

-classes, and

H

-classes via neighborhood inclusions, bipartite components, and automorphism orbits, yielding the first algorithmic reconstruction of ideal-theoretic structure from graph data. The framework is implemented in SageMath as a reproducible open-source toolkit validated on concrete examples. This work synthesizes algebraic graph theory, semigroup theory, and computational mathematics into a unified algebraic-combinatorial dictionary, providing both new analytical tools and a methodological template for studying algebraic constructions via graph invariants. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

19 pages, 2053 KB

Open AccessArticle

A Methodological Approach for Risk Assessment of Municipal Water Supply Projects

by Maria D. Gracia, Julio Mar-Ortiz, Carlos Roberto Luna-Domínguez and Jannya Pancardo

Mathematics 2026, 14(5), 890; https://doi.org/10.3390/math14050890 (registering DOI) - 5 Mar 2026

Abstract

Critical infrastructure projects are essential for societal well-being and economic prosperity. These projects are characterized by significant complexities and uncertainties, including weather conditions, design challenges, and regulatory constraints, which can hinder their ability to meet time, cost, and quality goals. To address the [...] Read more.

Critical infrastructure projects are essential for societal well-being and economic prosperity. These projects are characterized by significant complexities and uncertainties, including weather conditions, design challenges, and regulatory constraints, which can hinder their ability to meet time, cost, and quality goals. To address the risks that can disrupt such projects, comprehensive risk assessment frameworks are necessary to support decision-makers in managing them. This paper introduces a novel methodological framework for risk assessment in municipal water supply projects. The framework decomposes overall project risk into specific risk criteria to evaluate the impact of each activity on the project’s global risk. It integrates the fuzzy analytical hierarchy process with a fuzzy multi-objective resource-constrained scheduling problem to enhance risk assessment. The practical applicability of the framework is demonstrated through a case study conducted with the Municipal Department of Water Systems in a metropolitan area in southern Tamaulipas, Mexico. The results confirm the effectiveness of the proposed methodology as a robust tool for assessing risks in municipal water supply projects. Full article

(This article belongs to the Special Issue Recent Advances in Optimization Algorithms for Scheduling and Operations Research)

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31 pages, 551 KB

Open AccessArticle

El Clásico Revisited: Discriminant Analysis Versus Logistic Regression for Bankruptcy Prediction in the Accommodation and Food Service Industry Across B9 Countries

by Simona Vojtekova, Katarina Kramarova, Veronika Labosova and Pavol Durana

Mathematics 2026, 14(5), 889; https://doi.org/10.3390/math14050889 - 5 Mar 2026

Abstract

Despite the rapid expansion of AI and machine-learning techniques in bankruptcy prediction, classical statistical methods such as discriminant analysis and logistic regression remain relevant because of their transparency and interpretability. These characteristics are crucial for stakeholders who require understandable decision-making tools, especially in [...] Read more.

Despite the rapid expansion of AI and machine-learning techniques in bankruptcy prediction, classical statistical methods such as discriminant analysis and logistic regression remain relevant because of their transparency and interpretability. These characteristics are crucial for stakeholders who require understandable decision-making tools, especially in NACE Rev. 2 Section I—Accommodation and Food Service Activities, a sector characterized by high operating leverage, vulnerability to economic shocks, and strong macroeconomic importance. The study aims to evaluate and compare the predictive performance of discriminant analysis and logistic regression for bankruptcy prediction and to identify key predictors that can serve as managerial early-warning signals for companies in crisis across B9 countries. The sample of 4395 companies was used. The classification ability of all models is assessed using multiple performance metrics, including overall accuracy, sensitivity, specificity, precision, the F1-score, the F2-score, the Matthews correlation coefficient, and the area under the receiver operating characteristic curve. The results show that both approaches achieve consistently high predictive performance, with all major metrics exceeding 0.92 on the test sample of prosperous and non-prosperous enterprises. Six significant bankruptcy predictors are identified for each method, with three common indicators: financial leverage, total liabilities to assets, and return on costs. The comparative analysis results in a methodological “draw,” confirming comparable predictive power. These findings reaffirm the relevance of classical prediction models and identify key financial indicators that can be used as practical early-warning signals by managers in the sector. Full article

(This article belongs to the Special Issue Advanced Methods in the Mathematical Modeling of Economics, Econometrics, and Financial Management, 2nd Edition)

31 pages, 810 KB

Open AccessArticle

Maximizing Consumer Surplus via Return Freight Insurance: Single Insurer Monopoly Versus Competitive Provision

by Jinyi Qin, Liang Huang and Yan Chen

Mathematics 2026, 14(5), 888; https://doi.org/10.3390/math14050888 - 5 Mar 2026

Abstract

This paper examines how e-commerce retailers should structure return freight insurance (RFI) partnerships—exclusive single-insurer or competitive multi-insurer—to maximize consumer surplus. Using a game-theoretic model with heterogeneous consumers and vertically differentiated insurance products, we find that monopolistic RFI provision can paradoxically enhance consumer welfare [...] Read more.

This paper examines how e-commerce retailers should structure return freight insurance (RFI) partnerships—exclusive single-insurer or competitive multi-insurer—to maximize consumer surplus. Using a game-theoretic model with heterogeneous consumers and vertically differentiated insurance products, we find that monopolistic RFI provision can paradoxically enhance consumer welfare over competition when markets exhibit high heterogeneity, limited loss aversion, and low compensation levels. The monopoly’s advantage stems from unified risk pooling, preventing adverse selection, cross-subsidization maintaining universal coverage, and quality preservation avoiding competitive price degradation. Optimal premiums show non-monotonic relationships with market parameters, creating distinct pricing regimes. Our results suggest retailers should match market structure to consumer characteristics: exclusive arrangements for heterogeneous, price-sensitive markets, while competitive provision for homogeneous, loss-averse segments. These insights advance our understanding of platform economics and inform tailored market design for internet insurance. Full article

(This article belongs to the Special Issue Mathematical Methods and Operations Research for Sustainable Planning, Scheduling and Supply Chain)

16 pages, 1083 KB

Open AccessArticle

On Non-Commensurate Fractional-Order System Control

by Mircea Ivanescu and Decebal Popescu

Mathematics 2026, 14(5), 887; https://doi.org/10.3390/math14050887 - 5 Mar 2026

Abstract

The control systems for models described by non-commensurate fractional-order differential equations are based on their transformation into large commensurate-order systems, which impose difficulties in determining control laws. In this context, in this paper, a new control method for this class of systems is [...] Read more.

The control systems for models described by non-commensurate fractional-order differential equations are based on their transformation into large commensurate-order systems, which impose difficulties in determining control laws. In this context, in this paper, a new control method for this class of systems is proposed. The results obtained are based on Lyapunov methods for differential equations with fractional exponents and on the application of the Yakubovich–Kalman–Popov lemma adapted for this class of systems. The stability criterion is presented as a frequency criterion and represented graphically by familiar frequency plots similar to those of the Nyquist or Popov type. If the parameters that define the model can be defined in a closed domain, the frequency criterion can be interpreted as “Popov’s circle criterion”. The two numerical applications present two important cases. The first studies the stability criterion in the case where the viscosity coefficients determine non-commensurate fractional-order exponents in the dynamic model of the system. The second example studies the complex problem of the human–machine system in which the human model imposes dynamics determined by non-commensurate fractional-order systems. The proposed investigation methods allow for a reduction in computational effort by several orders of magnitude for non-commensurate fractional-order systems, eliminate stability conditions that use matrix-based criteria for large-scale systems, and introduce standard frequency-domain criteria. Full article

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

18 pages, 2203 KB

Open AccessArticle

Diverse Jacobi Elliptic Function Solutions and Dynamical Behaviors for a High-Order KdV Type Wave Equation via Extended F-Expansion Method

by Jiayi Fu, Weixu Ni and Wenxia Chen

Mathematics 2026, 14(5), 886; https://doi.org/10.3390/math14050886 - 5 Mar 2026

Abstract

This paper focuses on a high-order Korteweg–de Vries wave equation. The extended F-expansion method, a modified form of Kudryashov’s auxiliary equation approach, is employed to construct Jacobi elliptic function solutions for this equation. Three distinct families of solutions are obtained, including solitary waves, [...] Read more.

This paper focuses on a high-order Korteweg–de Vries wave equation. The extended F-expansion method, a modified form of Kudryashov’s auxiliary equation approach, is employed to construct Jacobi elliptic function solutions for this equation. Three distinct families of solutions are obtained, including solitary waves, breathers, dark/bright solitons, bright–dark interaction solitons, and rogue-like solutions. To better illustrate the complex nonlinear dynamics of the high-order Korteweg–de Vries wave equation, representative solutions are selected, and their moduli are visualized using Maple software through three-dimensional, two-dimensional, and contour plots. Full article

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26 pages, 1181 KB

Open AccessArticle

An Integrated Hybrid Model for Evaluating Performance and Allocating Incentives to Order Pickers in E-Commerce Fulfillment

by Milan Andrejić and Vukašin Pajić

Mathematics 2026, 14(5), 885; https://doi.org/10.3390/math14050885 - 5 Mar 2026

Abstract

E-commerce has been a rapidly growing sales channel in recent years, with a strong trend toward further expansion. However, logistics companies face significant challenges in the preparation and sorting of orders when delivering shipments purchased through e-commerce platforms. In this process, order pickers [...] Read more.

E-commerce has been a rapidly growing sales channel in recent years, with a strong trend toward further expansion. However, logistics companies face significant challenges in the preparation and sorting of orders when delivering shipments purchased through e-commerce platforms. In this process, order pickers play a pivotal role, as their efficiency directly impacts both the operational performance of logistics companies and the quality of service provided to customers. During peak periods of high order volumes, it is common for order pickers to exceed the prescribed work norm, making them eligible for performance-based bonuses. This study aims to develop a model for evaluating order picker efficiency, ranking them, and determining the optimal allocation of bonuses. It addresses a critical gap in the existing literature, as only a handful of studies have explored this issue in depth. To assess the efficiency of 56 order pickers, the DEA method was applied, incorporating three input and five output variables. The analysis identified 18 order pickers as fully efficient. These individuals were then ranked using the IMF SWARA and COPRAS methods, where IMF SWARA was employed to determine the weights of nine evaluation criteria, while COPRAS was used for the final ranking process. Based on the ranking results, a structured bonus allocation model was developed, encompassing four distinct scenarios. Furthermore, a sensitivity analysis and model validation were conducted to ensure the robustness and reliability of the proposed approach. Full article

45 pages, 2817 KB

Open AccessArticle

Assessment of Various Three-Phase PLLs Based on SOGI-QSG for Grid Synchronization Under Unbalanced Grid Conditions

by Atif Ali Alqarni, Abdullah Ali Alhussainy, Fahd Hariri, Sultan Alghamdi and Yusuf A. Alturki

Mathematics 2026, 14(5), 884; https://doi.org/10.3390/math14050884 - 5 Mar 2026

Abstract

In grid-connected inverter systems, the Phase-Locked Loop (PLL) is fundamental for achieving and maintaining precise synchronization between the inverter and the electrical grid. Developing an efficient and robust PLL is essential to ensure reliable operation, particularly in the presence of abnormal grid conditions. [...] Read more.

In grid-connected inverter systems, the Phase-Locked Loop (PLL) is fundamental for achieving and maintaining precise synchronization between the inverter and the electrical grid. Developing an efficient and robust PLL is essential to ensure reliable operation, particularly in the presence of abnormal grid conditions. Among the existing synchronization methods, the Synchronous Reference Frame-based PLL (SRF-PLL) is widely adopted due to its robust performance; however, it suffers from degraded accuracy under unbalanced voltage conditions. To address this limitation, the Second-Order Generalized Integrator-Quadrature Signal Generator (SOGI-QSG) was proposed in previous studies as an alternative approach. Despite its advantages, the SOGI-PLL exhibits weak filtering capability for lower-order harmonics and remains sensitive to DC offset, both of which can affect synchronization quality. As a result, numerous advanced PLLs based on SOGI-QSG have been proposed in the literature to address SOGI-QSG limitations by enhancing DC offset rejection, filtering capability, and dynamic response. This article provides a comprehensive assessment of various three-phase PLLs based on SOGI-QSG under unbalanced grid conditions, focusing on peak-to-peak frequency error, filtering performance, and DC offset rejection. The operational principles and mathematical models of each technique are discussed, and their performances are validated using MATLAB/Simulink (R2025b). The results show that the SRF-PLL exhibits oscillatory behavior under unbalanced conditions, whereas the PLLs based on SOGI-QSG demonstrate stable synchronization with different trade-offs between filtering strength and dynamic response. Therefore, the selection of the appropriate PLLs based on SOGI-QSG depends on the priorities of the specific application. Full article

17 pages, 330 KB

Open AccessFeature PaperArticle

Boundary Value Problems and Propagation of Singularities for Several Partial Differential Equations of Mathematical Physics

by Angela Slavova and Petar Popivanov

Mathematics 2026, 14(5), 883; https://doi.org/10.3390/math14050883 - 5 Mar 2026

Abstract

This paper deals with several equations of mathematical physics written in explicit form with their solutions. In Theorem 1, an oblique derivative problem for the string equation is studied. More precisely, the initial-boundary value problem for the string equation is investigated. The corresponding [...] Read more.

This paper deals with several equations of mathematical physics written in explicit form with their solutions. In Theorem 1, an oblique derivative problem for the string equation is studied. More precisely, the initial-boundary value problem for the string equation is investigated. The corresponding vector field on the boundary is non-vanishing and does not have a characteristic direction, but can be tangential to some part of the boundary, and it is allowed to change sign. A classical solution exists with suitable compatibility conditions at the corner points. The picture changes significantly in the case of the wave equation with several (say two: 2D) space variables in a circular cylinder. The initial-boundary value problem turns out to be underdetermined with an infinite-dimensional kernel if the boundary vector field is orthogonal to the time axis. By prescribing extra conditions on the generatrices of the cylinder where the vector field is tangential to the cylinder, we obtain a unique classical solution. In Theorem 2, we consider the Cauchy problem in the interior of the parabola of the Lorentzian-type eikonal equation and find its unique classical solution in

{0 \leq x_{2} \leq 1 / 2} \cap {x_{2} \geq \frac{x_{1}^{2}}{2}}

. Propagation of singularities for the D and 3 D hyperbolic (Klein–Gordon) equations in

R^{4}

,

R^{8}

is studied in Theorem 3. In the double characteristic points, the wave front propagates either along the surface of the characteristic cone, or in the solid cone starting from

(t_{0}, x_{0})

. Full article

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

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21 pages, 2711 KB

Open AccessArticle

Refined Conditions for the Inclusion Properties of Special Functions in Lemniscate and Nephroid Domains

by Saiful R. Mondal and Lateef Ahmad Wani

Mathematics 2026, 14(5), 882; https://doi.org/10.3390/math14050882 - 5 Mar 2026

Abstract

For

D : = {z \in C : | z | < 1}

, this paper derives refined conditions for the inclusion of special functions in lemniscate and nephroid domains focusing on solutions to the differential equations of the form [...] Read more.

For

D : = {z \in C : | z | < 1}

, this paper derives refined conditions for the inclusion of special functions in lemniscate and nephroid domains focusing on solutions to the differential equations of the form

z^{n} F^{″} (z) + a (z) z^{n - 1} F^{'} (z) + b (z) F (z) + d (z) = 0, n \in {1, 2}, z \in D,

with normalization

F (0) = 1

, where

a (z)

,

b (z)

and

d (z)

are analytic in

D

. Using advanced techniques from geometric function theory, we generalize and improve existing results, particularly for classes of functions defined by differential equations. Specific applications include generalized Bessel functions, regular Coulomb wave functions, and associated Laguerre polynomials, where we derive improved bounds for their inclusion in lemniscate domains. Additionally, we present open problems, supported by numerical experiments, to guide future research in this direction. Full article

(This article belongs to the Special Issue Special Functions with Applications)

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34 pages, 2002 KB

Open AccessArticle

A Topological Framework for Atmospheric River Interaction Using Framed Braids

by Ioannis Diamantis

Mathematics 2026, 14(5), 881; https://doi.org/10.3390/math14050881 - 5 Mar 2026

Abstract

Atmospheric Rivers (ARs) are filamentary moisture pathways responsible for a large fraction of extreme precipitation and often occur as interacting filament bundles within the same synoptic regime. Existing diagnostics typically analyze ARs in isolation, despite the frequent coexistence and interaction of multiple filaments. [...] Read more.

Atmospheric Rivers (ARs) are filamentary moisture pathways responsible for a large fraction of extreme precipitation and often occur as interacting filament bundles within the same synoptic regime. Existing diagnostics typically analyze ARs in isolation, despite the frequent coexistence and interaction of multiple filaments. We introduce a topological framework for AR analysis based on framed braids and framed braidoids, which encodes both the geometric interaction of AR centroids and the internal evolution of moisture transport. In this approach, AR filaments are represented as strands whose time-ordered crossings form braid words, while moisture-based framing captures internal intensification or weakening along each filament. Applying this framework to reanalysis-derived Atmospheric River track data, we construct braid and framed braid representations over sliding time windows and analyze a strongly interacting multi-filament AR episode in the North Pacific. The results show that braid-based indicators capture structural reorganizations and moisture intensification episodes that are not apparent from centroid geometry or IVT magnitude alone, offering a complementary structural perspective on atmospheric moisture transport. Full article

(This article belongs to the Section B: Geometry and Topology)

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14 pages, 1320 KB

Open AccessFeature PaperArticle

An Adaptive Damped Double-Inertial Parallel Algorithm for Common Fixed-Point Problems with Applications to Image Restoration

by Supalin Tiammee, Suthep Suantai and Jukrapong Tiammee

Mathematics 2026, 14(5), 880; https://doi.org/10.3390/math14050880 - 5 Mar 2026

Abstract

Inertial methods are widely used to accelerate the convergence of iterative algorithms for solving fixed-point problems. However, standard inertial terms often introduce undesirable oscillations, particularly in high-dimensional settings. In this paper, we propose a novel parallel double inertial algorithm with adaptive damping control [...] Read more.

Inertial methods are widely used to accelerate the convergence of iterative algorithms for solving fixed-point problems. However, standard inertial terms often introduce undesirable oscillations, particularly in high-dimensional settings. In this paper, we propose a novel parallel double inertial algorithm with adaptive damping control (D-DIMPMHA) for finding a common fixed point of a finite family of nonexpansive mappings in real Hilbert spaces. By integrating a double inertial step with a self-adaptive damping parameter, the proposed method effectively balances momentum and stability, thereby mitigating numerical oscillations without requiring vanishing inertial conditions. We establish the weak convergence theorem of the generated sequence under suitable control conditions. Furthermore, the practical efficiency of the algorithm is demonstrated through numerical experiments on large-scale convex feasibility problems and image restoration problems. Comparative results indicate that the proposed algorithm achieves superior convergence speed and higher restoration quality compared to existing single inertial methods and FISTA. Full article

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

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34 pages, 1511 KB

Open AccessArticle

Finite-Time Contractivity Profiling of a Two-Parameter Parallel Root-Finding Scheme via a kNN–LLE Proxy

by Mudassir Shams, Andrei Velichko and Bruno Carpentieri

Mathematics 2026, 14(5), 879; https://doi.org/10.3390/math14050879 - 5 Mar 2026

Abstract

Parallel iterative schemes are widely used for the simultaneous computation of all distinct roots of nonlinear equations in scientific computing and engineering. While high-order parallel methods can provide substantial acceleration, their practical performance is often dominated by the choice of internal real-valued parameters [...] Read more.

Parallel iterative schemes are widely used for the simultaneous computation of all distinct roots of nonlinear equations in scientific computing and engineering. While high-order parallel methods can provide substantial acceleration, their practical performance is often dominated by the choice of internal real-valued parameters introduced by correction/acceleration mechanisms, which may strongly affect convergence speed and numerical robustness. Classical parameter-selection strategies—based on analytical sufficient conditions, trial-and-error experimentation, or qualitative dynamical diagnostics (basins of attraction, bifurcation-style inspection, and parameter planes)—are typically problem-dependent, expensive to scale, and difficult to automate reproducibly. In this work, we propose a data-driven framework for systematic parameter optimization based on finite-time contractivity profiling. The approach uses k-nearest neighbors (kNN) micro-series analysis to estimate a proxy profile of the largest Lyapunov exponent (LLE) along the iteration index, summarizing the transient contraction/expansion behavior of the solver trajectories. Two profile-based scores, the minimum score

S_{min}

and the moment score

S_{mom}

, are introduced to rank candidate parameter pairs and to construct stability landscapes over

(α, β)

grids. As a testbed, we apply the framework to a bi-parametric two-step parallel Weierstrass-type scheme and demonstrate that the learned parameter regions yield faster and more reliable convergence than generic or manually tuned choices. Extensive numerical experiments show that the proposed profiling-based optimization consistently improves convergence rate and robustness across the considered nonlinear test problems, providing a scalable and reproducible alternative to heuristic and dynamical-system-based tuning. Full article

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

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21 pages, 575 KB

Open AccessArticle

An Adaptive Online Knowledge Distillation Algorithm for Edge Computing Models Enhanced by Elite-Students

by Jincheng Xia, Yan Zhou, Xu Yang and Chengyan Zhao

Mathematics 2026, 14(5), 878; https://doi.org/10.3390/math14050878 - 5 Mar 2026

Abstract

In recent years, deep learning models have exhibited exceptional performance across several tasks. However, the substantial computational and storage demands impede implementation on edge devices with constrained resources. Online Knowledge Distillation (OKD) has arisen as an effective model compression strategy that removes the [...] Read more.

In recent years, deep learning models have exhibited exceptional performance across several tasks. However, the substantial computational and storage demands impede implementation on edge devices with constrained resources. Online Knowledge Distillation (OKD) has arisen as an effective model compression strategy that removes the reliance on pre-trained teachers characteristic of conventional distillation approaches. Nonetheless, OKD persists in facing challenges, including substantial performance variances within student networks, insufficient learning capacity for difficult data, and network homogeneity. To address those issues, this paper proposes an Elite-Student-Enhanced Adaptive Online Knowledge Distillation (ESAKD) algorithm. ESAKD introduces a patience factor-based adaptive temperature scheduling mechanism to dynamically balance knowledge clarity and richness during knowledge transfer. This mechanism utilizes the performance benefits of elite-students, particularly during initial training phases, to offer superior supervision that successfully transcends the learning capacity limitations of current OKD approaches. This method promotes swift convergence and substantially enhances the ultimate precision of the standard-student models. Furthermore, a confidence-weighted ensemble student model is designed to improve collective decision-making. Experimental assessments indicate that ESAKD provides substantial performance improvements over supervised learning without distillation and other leading online distillation techniques. On the CIFAR-100 dataset, ESAKD improves the test accuracy of various models by 1.49–6% over the undistilled baselines, and by 0.27–2.18% compared to advanced online distillation algorithms. Moreover, it exhibits enhanced performance on the Tiny-ImageNet-200 dataset as well. Full article

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26 pages, 4337 KB

Open AccessFeature PaperArticle

Data-Driven Multi-Objective Optimization of Conformal Cooling Channels for Energy-Efficient Injection Molding

by Carlos Pereira, António J. Pontes and António Gaspar-Cunha

Mathematics 2026, 14(5), 877; https://doi.org/10.3390/math14050877 - 5 Mar 2026

Abstract

Injection molding is widely used for plastic parts, but its performance is limited by the cooling stage, which dominates cycle time and affects dimensional stability and energy consumption. Conformal cooling channels, which can be manufactured using additive technologies, improve thermal efficiency but introduce [...] Read more.

Injection molding is widely used for plastic parts, but its performance is limited by the cooling stage, which dominates cycle time and affects dimensional stability and energy consumption. Conformal cooling channels, which can be manufactured using additive technologies, improve thermal efficiency but introduce a high-dimensional design problem. This work proposes an integrated methodology for optimizing injection molds with conformal cooling channels that combines parametric CAD (Computer-Aided Drawing), simulation, non-linear principal component analysis, artificial neural network, and multi-objective evolutionary optimization. The workflow is applied to a case study with five cooling layouts. An initial set of 36 metrics related to temperature gradients, warpage, shrinkage, and energy is reduced to a small number of latent objectives, simplifying the search space while preserving the main physical trends. Artificial neural networks surrogates accurately reproduce numerical results, enabling exploration of the design space at a fraction of the computational cost. The optimization yields diverse Pareto-optimal solutions that balance cycle time, dimensional stability, and energy consumption, assisting the design of more sustainable injection molds. Sensitivity analysis identifies mold temperature and channel position/diameter as key design levers. The proposed methodology reduces dependence on expensive simulations and is readily transferable to industrial mold design. Full article

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

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18 pages, 334 KB

Open AccessArticle

Three-Dimensional Second-Order Rational Difference Equations: Explicit Formulas and Simulations

by Ahmed Ghezal, Hassan J. Al Salman and Ahmed A. Al Ghafli

Mathematics 2026, 14(5), 876; https://doi.org/10.3390/math14050876 - 5 Mar 2026

Abstract

This paper studies a three-dimensional second-order rational difference system, which generalizes earlier scalar and bidimensional models. We derive explicit closed-form solutions for general initial conditions and identify special cases that simplify the system’s structure. These explicit solutions are particularly significant, as they not [...] Read more.

This paper studies a three-dimensional second-order rational difference system, which generalizes earlier scalar and bidimensional models. We derive explicit closed-form solutions for general initial conditions and identify special cases that simplify the system’s structure. These explicit solutions are particularly significant, as they not only enable rigorous stability analysis but also provide a precise analytical characterization of the system’s long-term behavior, offer deeper insight into its underlying periodic structure, and establish a solid theoretical foundation for potential future applications in the control, prediction, and optimization of multivariable dynamical systems. The analysis carefully addresses conditions ensuring well-defined solutions, avoiding singularities. Numerical simulations illustrate various dynamic behaviors, including oscillations, convergence, and sensitivity to parameters, confirming the richness and robustness of the system’s temporal evolution. Full article

(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)

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15 pages, 2136 KB

Open AccessArticle

Efficient Time-Domain Dimension Reduction Methods for Simulating Stationary Stochastic Processes

by Guoyu Liu, Shiwei Yin, Xiaojiao Fu and Zixin Liu

Mathematics 2026, 14(5), 875; https://doi.org/10.3390/math14050875 - 5 Mar 2026

Abstract

The high-dimensional stochastic space caused by a large number of random variables remains a significant challenge hindering the practical application of stochastic process simulation in engineering. Although various dimension reduction techniques have been developed, their direct integration into time-domain simulation frameworks remains limited. [...] Read more.

The high-dimensional stochastic space caused by a large number of random variables remains a significant challenge hindering the practical application of stochastic process simulation in engineering. Although various dimension reduction techniques have been developed, their direct integration into time-domain simulation frameworks remains limited. To address this issue, this paper proposes two efficient time-domain dimension reduction methods for simulating stationary stochastic processes. The methods reduce the number of input random variables required for simulation to a single variable, while the randomness of the output stochastic process remains unchanged. The proposed methods are theoretically motivated by spectral decomposition of processes using two distinct strategies and explicitly incorporate the decay characteristics of the impulse response function associated with the stochastic process. Based on this, the random orthogonal functions can be naturally introduced to simulate the stationary stochastic process, which effectively resolves the high-dimensional random variables encountered in conventional time-domain simulations. Furthermore, the incorporation of a number-theoretic method enables uncertainty quantification of stochastic process samples. Numerical simulations demonstrate that the proposed methods reduce the random variable dimension from 2400 to 1 (99.95% reduction). Relative error of the simulated power spectral density remains below 2%, while computational time is reduced by approximately 4% compared with the conventional time-domain methods. These results demonstrate the effectiveness and practical applicability of the proposed approach in engineering stochastic process simulation. Full article

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28 pages, 747 KB

Open AccessArticle

Optimization Analysis of a Multi-Server Queueing System with Two-Phase Heterogeneous Services and Synchronous Working Vacation Interruptions

by Wei Xu, Linhong Li and Wentao Fan

Mathematics 2026, 14(5), 874; https://doi.org/10.3390/math14050874 - 5 Mar 2026

Abstract

Despite extensive research on queueing models in flexible manufacturing systems (FMS), few studies have simultaneously considered the heterogeneity of servers and the impact of synchronous working vacations on system performance. To fill this research gap, this study proposes a novel multi-server queueing model [...] Read more.

Despite extensive research on queueing models in flexible manufacturing systems (FMS), few studies have simultaneously considered the heterogeneity of servers and the impact of synchronous working vacations on system performance. To fill this research gap, this study proposes a novel multi-server queueing model that uniquely integrates two-phase heterogeneous services with synchronous working vacation interruptions. The innovation lies in capturing the complex task processing mechanism where the two-phase service is provided by different servers, and they conduct working vacations synchronously when the system is empty and terminate vacations once the system population reaches a specified threshold. Based on the matrix-analytic approach, this research investigates the stability condition and the stationary distribution of the system. A key numerical finding is the diminishing marginal returns of server numbers, where exceeding an optimal count increases congestion, thereby degrading overall performance. The vacation interruption threshold l is also shown to significantly influence server state allocation. A comparative analysis of three different system configurations, demonstrates that dynamic server number or a vacation interruption threshold adjustment effectively mitigates congestion; and in order to enhance the system clearing capability, when vacations are longer, opting for multiple servers is preferable, whereas a single server is more suitable during shorter vacation periods. For cost optimization, three algorithms (CPSO, JAYA, MLS-JAYA) consistently converge to the same robust optimal solution for server count and vacation rate. Furthermore, to simultaneously minimize waiting time, we apply the MOEA/D algorithm to study the bi-objective optimization problem. Full article

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