Mathematics

22 pages, 2993 KB

Open AccessArticle

Construction and Application of a Dynamic Model Integrating Technological Progress, Carbon Emissions, Economic Growth, and Energy Structure

by Xiongfei Wang, Hua Xu, Yuanyuan Song, Zhirong Sheng and Minggang Wang

Mathematics 2026, 14(9), 1575; https://doi.org/10.3390/math14091575 - 6 May 2026

Cited by 1 | Viewed by 456

Abstract

Technological progress reduces carbon emissions by promoting energy structure optimization while fostering new industries and improving efficiency, thus achieving a win–win situation for economic growth and low-carbon development. From the perspective of mechanism analysis, this paper constructs a new dynamic system model of [...] Read more.

Technological progress reduces carbon emissions by promoting energy structure optimization while fostering new industries and improving efficiency, thus achieving a win–win situation for economic growth and low-carbon development. From the perspective of mechanism analysis, this paper constructs a new dynamic system model of technological progress–carbon emissions–economic growth–energy structure based on the interdependent and mutually restrictive causal relationships among technological progress, carbon emissions, economic growth and energy structure within an economic period. The dynamical behaviors of the system and its subsystems are analyzed using Lyapunov exponents, bifurcation diagrams, equilibrium point stability theory and other methods. Numerical simulations show that the system parameter

a_{2}

(the driving coefficient of economic growth on carbon emissions) determines the threshold of state transition. With the increase in

a_{2}

, the system exhibits a clear evolutionary path from stable equilibrium to periodic state and then to chaotic state. The system enters chaos when

a_{2}

falls within the interval [0.741, 0.79]. Model parameters are estimated based on real data, the evolutionary relationships of technological progress, carbon emissions, energy structure and economic growth over time are presented, and the impacts of different regulation strategies on carbon emission reduction and economic growth are analyzed. Full article

(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)

► Show Figures

Figure 1

24 pages, 1075 KB

Open AccessArticle

The Spatio-Temporal Differentiation and Convergence Characteristics of the Coordinated Development of Digitalization and Greening in China

by Peipei Zhang and Yusen Luo

Mathematics 2026, 14(9), 1574; https://doi.org/10.3390/math14091574 - 6 May 2026

Viewed by 433

Abstract

The synergistic development of digitalization and greening is an important lever for China to accelerate the formation of new quality productive forces. This study adopted the global entropy method and coupling coordination degree model to measure the level of coordinated development between digitalization [...] Read more.

The synergistic development of digitalization and greening is an important lever for China to accelerate the formation of new quality productive forces. This study adopted the global entropy method and coupling coordination degree model to measure the level of coordinated development between digitalization and greening with the panel data of Chinese cities from 2011 to 2022. Spatio-temporal evolution characteristics were explored through kernel density estimation, the Dagum Gini coefficient, and spatial autocorrelation methods. This study further tested the convergence characteristics of coordinated development through a two-way fixed effect model and spatial econometric model. The results show the following: (1) The overall level of coordinated development of digitalization and greening in China is on the rise, with the development level in the eastern region being significantly higher than that in the central and western regions. The degree of differentiation in coordinated development shows a trend of decreasing first and then increasing, mainly due to regional differences. (2) The level of coordinated development between digitalization and greening in China shows a significant positive spatial autocorrelation feature, with a clustering pattern dominated by “low–low” clustering. (3) It is found that the coordinated development of digitalization and greening in China has significant characteristics of σ convergence, spatial β convergence and club convergence. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

► Show Figures

Figure 1

23 pages, 6086 KB

Open AccessArticle

CSA-Optimized Adaptive Weighted Centroid Algorithm for Spacecraft Structural Impact Localization Using FBG Sensors

by Jinsong Yang, Jie Luo, Xiaozhen Zhang and Chengguang Fan

Mathematics 2026, 14(9), 1573; https://doi.org/10.3390/math14091573 - 6 May 2026

Viewed by 477

Abstract

Accurate impact localization on spacecraft structural panels subjected to contact loading by on-orbit servicing robots is critical for real-time structural health monitoring (SHM), yet remains challenging due to heterogeneous elastic wave propagation in complex aluminum structures with stiffener ribs and bonded joints. Conventional [...] Read more.

Accurate impact localization on spacecraft structural panels subjected to contact loading by on-orbit servicing robots is critical for real-time structural health monitoring (SHM), yet remains challenging due to heterogeneous elastic wave propagation in complex aluminum structures with stiffener ribs and bonded joints. Conventional Received Signal Strength Indicator (RSSI)-based weighted centroid methods rely on fixed path-loss exponents that cannot accommodate spatially varying wave attenuation, resulting in position-dependent localization errors that worsen significantly near structural discontinuities. This paper proposes a Crow Search Algorithm (CSA)-optimized adaptive weighted centroid algorithm using distributed Fiber Bragg Grating (FBG) sensors, featuring three principal innovations: (i) a novel FBG wavelength-shift-to-RSSI amplitude mapping derived from elastic wave attenuation theory, bridging optical fiber sensing with centroid localization; (ii) per-event online weight optimization via CSA that adapts sensor contributions to each individual impact’s strain-wave signature; and (iii) a multi-objective fitness function simultaneously optimizing localization accuracy, noise robustness, and temporal consistency. The proposed method is validated across 200 impact events distributed over five representative positions on a 1 m³ Al6061 satellite-like structure with 64 FBG sensors (8 × 8 grid, 125 mm pitch), under three Gaussian noise levels (σ = 1%, 3%, 5% of signal RMS), and benchmarked against classical weighted centroid (WC), PSO-WC, GA-WC, DE-WC, and GWO-WC using paired t-tests (p < 0.01). CSA-WC achieves a mean localization error of 4.63 mm—an 83.29% improvement over classical WC and the lowest error among all five compared algorithms—with an average computation time of 0.14 s per event, satisfying real-time monitoring requirements. Full article

(This article belongs to the Special Issue Mathematical Models for Fault Detection and Diagnosis)

► Show Figures

Figure 1

22 pages, 3059 KB

Open AccessArticle

Multiplicative Gradient Decomposition: A Hybrid Additive–Multiplicative Optimization Framework

by Sajedeh Norozpour

Mathematics 2026, 14(9), 1572; https://doi.org/10.3390/math14091572 - 6 May 2026

Viewed by 488

Abstract

We propose a unified additive–multiplicative optimization framework, termed hybrid multiplicative gradient decomposition (HMGD), for training machine learning models. Unlike conventional gradient-based methods that rely solely on additive parameter updates, the proposed approach decomposes the gradient into complementary additive and multiplicative components, where the [...] Read more.

We propose a unified additive–multiplicative optimization framework, termed hybrid multiplicative gradient decomposition (HMGD), for training machine learning models. Unlike conventional gradient-based methods that rely solely on additive parameter updates, the proposed approach decomposes the gradient into complementary additive and multiplicative components, where the multiplicative term is defined through logarithmic derivative transformations to capture geometric scaling effects. This formulation enables the simultaneous modeling of linear and exponential parameter dynamics, which is particularly relevant in non-convex optimization settings and in models involving multiplicative interactions. The HMGD framework introduces separate momentum mechanisms for additive and multiplicative components, along with norm-based regularization to improve stability and promote structured sparsity in gradient updates. The method can be integrated into standard backpropagation by extending the chain rule to incorporate geometric derivatives. Empirical evaluations on multiple benchmark datasets demonstrate that HMGD achieves consistently faster convergence, improved robustness under multiplicative noise, and competitive or slightly improved performance compared to widely used optimizers such as Adam and RMSProp. Additional analysis shows that the proposed framework induces higher gradient sparsity and maintains stable optimization behavior across training. These results suggest that HMGD provides a flexible and theoretically grounded alternative for optimization in complex learning systems, particularly in scenarios involving nonlinear and multiplicative dynamics. Full article

► Show Figures

Figure 1

17 pages, 314 KB

Open AccessArticle

New Constructions of Complete Permutation Polynomials over Finite Fields of Even Characteristic

by Jian Li, Zhengbang Zha and Ziran Tu

Mathematics 2026, 14(9), 1571; https://doi.org/10.3390/math14091571 - 6 May 2026

Viewed by 534

Abstract

Complete permutation polynomials have many applications in mathematics and cryptography. In this paper, we study the complete permutation property of polynomial

x h {(x^{q - 1})}^{q + 1}

over

F_{q^{2}}

, where [...] Read more.

Complete permutation polynomials have many applications in mathematics and cryptography. In this paper, we study the complete permutation property of polynomial

x h {(x^{q - 1})}^{q + 1}

over

F_{q^{2}}

, where

h (x) = h_{1} (x + x^{q}) + h_{2} (x + x^{q}) x^{k}

. Based on the trace functions and Dickson polynomials, we present several new constructions of such complete permutations by choosing suitable

h_{1} (x)

,

h_{2} (x)

, and k. Full article

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

23 pages, 2996 KB

Open AccessArticle

Voxelization-Based Variable Neighborhood Tabu Search Strategy for Three-Dimensional Irregular Strip Packing

by Yue He, Shishun Cheng, Zhuo Xie, Shaowen Yao and Lijun Wei

Mathematics 2026, 14(9), 1570; https://doi.org/10.3390/math14091570 - 6 May 2026

Viewed by 478

Abstract

This paper proposes an efficient algorithm that integrates a variable neighborhood search (VNS) framework with an adaptive voxel discretization for the three-dimensional irregular packing problem. The problem arises in additive manufacturing, logistics loading, and other fields, especially in strip packing scenarios where the [...] Read more.

This paper proposes an efficient algorithm that integrates a variable neighborhood search (VNS) framework with an adaptive voxel discretization for the three-dimensional irregular packing problem. The problem arises in additive manufacturing, logistics loading, and other fields, especially in strip packing scenarios where the filling length in a virtual container with a fixed cross-section and infinite length is to be minimized. The algorithm first discretizes continuous three-dimensional geometric models into Boolean voxel matrices, thereby transforming complex geometric interference detection into efficient logical operations. An initial solution is generated using a greedy “largest-volume-first” strategy. An innovative adaptive voxel precision adjustment mechanism is introduced to dynamically modify the discretization granularity according to the current filling rate, realizing a hierarchical solution strategy of “coarse-grained fast search + fine-grained precise optimization”. On this basis, a variable-neighborhood iterative framework based on tabu search (TS-VNS) is constructed. Three complementary neighborhood operators are designed: single-item reinsertion, block exchange, and rotation perturbation, together with an adaptive operator selection mechanism driven by historical contributions. Experiments on multiple standard instances of varying scales and complexities (e.g., miniature chess pieces and engine components) show that the proposed algorithm outperforms comparative methods in both packing height and average height, achieving a favorable balance between solution efficiency and stability. Thus, it provides a reliable and efficient approach for the practical engineering application of three-dimensional irregular packing. Full article

(This article belongs to the Special Issue Computational Geometry: Theory, Algorithms and Applications)

► Show Figures

Figure 1

17 pages, 1012 KB

Open AccessArticle

Dynamic Analysis of Sugarcane Pokkah Boeng Model with Time Delay in Removal or Chemical Control

by Fengbing Li, Jiaxiang Tao, Haitao Huang and Qinlong Wang

Mathematics 2026, 14(9), 1569; https://doi.org/10.3390/math14091569 - 6 May 2026

Viewed by 409

Abstract

In this paper, a sugarcane pokkah boeng SEIR model with time delay due to removal or chemical treatment of diseased sugarcane plants is investigated. By analyzing the characteristic equations, the stability of each feasible equilibrium of the system is discussed, and the existence [...] Read more.

In this paper, a sugarcane pokkah boeng SEIR model with time delay due to removal or chemical treatment of diseased sugarcane plants is investigated. By analyzing the characteristic equations, the stability of each feasible equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the positive equilibrium is established. Furthermore, by choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as

τ

crosses some critical values. Meanwhile, we adopt a hierarchical Bayesian model to conduct statistical inference on time delay and combine it with the former to carry out an empirical analysis on the prevention and control of sugarcane pokkah boeng. These provide a more comprehensive and effective theoretical basis for decision-making in the prevention and control of sugarcane pokkah boeng. Numerical simulations are carried out to illustrate the main theoretical results. Full article

(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)

► Show Figures

Figure 1

19 pages, 854 KB

Open AccessArticle

STAGE: LLM-Driven Semantic and Topological Augmented Graph Embedding for Text-Attributed Graphs

by Shiwei Huang, Shunxin Xiao, Xu-Yao Zhang, Shunzhi Zhu, Luoqi Liu and Da-Han Wang

Mathematics 2026, 14(9), 1568; https://doi.org/10.3390/math14091568 - 6 May 2026

Viewed by 643

Abstract

Text-attributed graphs (TAGs) require models to jointly exploit node text and graph structure, yet doing so effectively remains difficult when node text is sparse and the structural context is large. Here, we propose STAGE (Semantic and Topological Augmented G [...] Read more.

Text-attributed graphs (TAGs) require models to jointly exploit node text and graph structure, yet doing so effectively remains difficult when node text is sparse and the structural context is large. Here, we propose STAGE (Semantic and Topological Augmented Graph Embedding), a two-stage framework for representation learning on TAGs. In Stage I, a frozen large language model is used offline to generate explanatory text that enriches compressed node attributes without introducing online LLM training cost. In Stage II, STAGE performs structure-aware representation learning under a fixed global token budget by combining random-walk-based structural context with graph-conditioned token reduction before PLM encoding. This design preserves informative semantic content while preventing unconstrained sequence expansion. Experiments on seven benchmark datasets show that STAGE consistently outperforms strong baselines under the same evaluation setting and maintains favorable efficiency under bounded input-length constraints. Full article

(This article belongs to the Special Issue Causal Inference and Machine Learning: Mathematical Modeling, Analysis and Applications)

► Show Figures

Figure 1

19 pages, 2947 KB

Open AccessArticle

Light-Aware Modality Balancing Network for Multimodal Pedestrian Detection

by Yu Fu, Fan Zhang and Zhou Li

Mathematics 2026, 14(9), 1567; https://doi.org/10.3390/math14091567 - 6 May 2026

Viewed by 393

Abstract

The visible light and infrared thermal multimodal images in autonomous driving provide a wealth of information for pedestrian detection, and its challenge lies in utilizing the complementary information across modalities to obtain an optimal joint representation. This study proposes a light-aware modality balancing [...] Read more.

The visible light and infrared thermal multimodal images in autonomous driving provide a wealth of information for pedestrian detection, and its challenge lies in utilizing the complementary information across modalities to obtain an optimal joint representation. This study proposes a light-aware modality balancing network (LMB-Net) for pedestrian detection by fusing visible light and infrared thermal images. We designed an alignment complementary fusion module across modalities to exchange target information. Deformable convolutions are employed to automatically perform spatial deformation on features, thereby eliminating perception biases caused by misalignment. Furthermore, as the contribution of different modalities to pedestrian detection varies under different lighting conditions, we designed a light-aware module to utilize the distinct advantages of visible light and infrared thermal images. Extensive experiments on the KAIST and LLVIP datasets demonstrate that our method achieves the best detection performance compared to some other methods. Full article

► Show Figures

Figure 1

35 pages, 11787 KB

Open AccessArticle

A New One-Parameter Model Supports an Upside-Down Bathtub Failure Rate: Theory, Inference, and Real-World Applications

by Ohud A. Alqasem and Ahmed Elshahhat

Mathematics 2026, 14(9), 1566; https://doi.org/10.3390/math14091566 - 6 May 2026

Viewed by 519

Abstract

Researchers often develop ordinal hazard distributions, whether increasing or decreasing, into multi-parameter distributions to derive various forms of the hazard function. This process necessitates the formulation of a multi-parameter hazard function, which involves a more complex mathematical expression. In contrast, this study introduces [...] Read more.

Researchers often develop ordinal hazard distributions, whether increasing or decreasing, into multi-parameter distributions to derive various forms of the hazard function. This process necessitates the formulation of a multi-parameter hazard function, which involves a more complex mathematical expression. In contrast, this study introduces a new one-parameter lifetime model, termed the Inverted Z–Lindley (IZL) distribution, which is capable of capturing an upside-down bathtub-shaped failure rate without sacrificing analytical simplicity. Fundamental distributional properties of the IZL model are rigorously established, including closed-form expressions for the probability density, cumulative distribution, reliability, and hazard rate functions. Theoretical analysis shows that the density is strictly positive, unimodal, positively skewed, and heavy-tailed, while the hazard rate is unimodal with vanishing limits at both extremes. Fractional moments are obtained, and the non-existence of classical moments is formally justified, motivating the use of quantile-based and inactivity-time reliability measures. Besides the quantile function, several key reliability measures, including the mean inactivity time and strong mean inactivity time functions, and order statistics, are also developed. Inferential procedures are constructed under Type-II censoring using both likelihood-based and Bayesian frameworks. The existence and uniqueness of the frequentist estimator are established, while Bayesian estimation is implemented via Markov chain Monte Carlo methods under informative gamma priors. Several interval estimation techniques—including asymptotic, bootstrap, Bayesian credible, and highest posterior density intervals—are developed and compared through extensive Monte Carlo simulations. The practical relevance of the proposed model is demonstrated using real datasets from environmental health and communication engineering, where the IZL distribution consistently outperforms fifteen well-established inverted lifetime models according to likelihood-based criteria, information measures, and goodness-of-fit diagnostics. Overall, the IZL model offers a powerful, interpretable, and computationally efficient alternative for modeling heavy-tailed lifetime data with non-monotone failure behavior, contributing meaningfully to modern distribution theory and applied reliability analysis. Full article

(This article belongs to the Special Issue Computational Statistics: Analysis and Applications for Mathematics)

► Show Figures

Figure 1

18 pages, 1307 KB

Open AccessArticle

LTBoost: A New High-Precision Method for Academic Early Warning and Prediction

by Hailong Sun, Shenbing Fei, Mengdi Ma, Zhiqi Yan and Wei Wang

Mathematics 2026, 14(9), 1565; https://doi.org/10.3390/math14091565 - 6 May 2026

Viewed by 401

Abstract

Currently, the research on academic early warning assessment and prediction for college students under the credit system in colleges and universities is mainly based on methods such as machine learning. However, the existing prediction models often have problems such as difficulty in network [...] Read more.

Currently, the research on academic early warning assessment and prediction for college students under the credit system in colleges and universities is mainly based on methods such as machine learning. However, the existing prediction models often have problems such as difficulty in network structure, parameter selection, and extraction of context time series information. In response to these issues, this study, based on students’ historical academic performance, proposes LTBoost, a novel framework of the XGBoost classification prediction model. The proposed model framework integrates the BiLSTM module to handle the time series information in the original data. It also integrates the proposed Enhanced Transformer module to perceive global information and obtain enhanced features. Through experiments, the LTBoost prediction model was compared with four other machine learning algorithms. The accuracy of the proposed LTBoost classification prediction model increased by 0.7% to 99.4%, demonstrating a good prediction effect on whether students are at risk of prolonging their studies. It provides a new paradigm and path for the construction of talent cultivation plans in credit-based universities. Full article

► Show Figures

Figure 1

27 pages, 601 KB

Open AccessArticle

Differentially Private Probabilistic Active Disturbance Rejection Control with Uncertainty-Calibrated Extended State Observers

by Jiahui Dai and Peng Hou

Mathematics 2026, 14(9), 1564; https://doi.org/10.3390/math14091564 - 6 May 2026

Cited by 1 | Viewed by 420

Abstract

Active disturbance rejection control (ADRC) is attractive because it estimates and compensates a lumped “total disturbance” with limited plant information, but privacy-sensitive networked deployment, measurement-noise amplification, and actuator saturation remain insufficiently addressed together. This paper proposes a Differentially Private Probabilistic ADRC (DP-PADRC) framework [...] Read more.

Active disturbance rejection control (ADRC) is attractive because it estimates and compensates a lumped “total disturbance” with limited plant information, but privacy-sensitive networked deployment, measurement-noise amplification, and actuator saturation remain insufficiently addressed together. This paper proposes a Differentially Private Probabilistic ADRC (DP-PADRC) framework for nonlinear SISO systems under saturation. In contrast to adaptive ADRC schemes that schedule gains from raw residuals, and unlike model-based differentially private filters that rely on explicit stochastic plant models, the proposed method combines a linear ESO with a lightweight uncertainty surrogate computed from clipped and privatized innovations. The resulting controller is not Bayesian; rather, it is probabilistic in the sense that second-moment information from the released innovation stream is explicitly used to calibrate observer bandwidth and disturbance compensation. We further incorporate a saturation-aware gate so that scheduling remains well behaved when the commanded and applied inputs differ. An ISS-type mean-square bound is derived for the closed loop, making the dependence on the disturbance derivative, measurement-noise variance, clipping level, and privacy parameters

(ε, δ)

explicit. We also discuss the composition of privacy loss across repeated tuning windows and quantify the privacy-induced perturbation of the scheduling signal. Simulation-based nonlinear servo benchmarks show improved tracking/noise robustness over fixed-gain LADRC and a nonlinear ADRC baseline, while clarifying the privacy–performance trade-off and the scope of the method. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Control: Challenges and Innovations)

► Show Figures

Figure 1

16 pages, 3514 KB

Open AccessArticle

Gaussian–Cubic Backward Substitution Method for Fourth-Order Stream Function in Transient Two-Dimensional Incompressible Viscous Flows

by Ji Lin, Zonghui Zhang, Yuhui Zhang and Jun Lu

Mathematics 2026, 14(9), 1563; https://doi.org/10.3390/math14091563 - 6 May 2026

Viewed by 437

Abstract

This paper presents a meshless collocation technique for the fourth-order transient stream function formulation of the Navier–Stokes equations. The technique employs a hybrid kernel function and is augmented by the ghost point method and Picard iteration. The reduction in unknowns inherent in this [...] Read more.

This paper presents a meshless collocation technique for the fourth-order transient stream function formulation of the Navier–Stokes equations. The technique employs a hybrid kernel function and is augmented by the ghost point method and Picard iteration. The reduction in unknowns inherent in this stream function approach simplifies the solution process. Introducing vorticity and stream functions enables mathematical reformulation of the coupled, time-dependent Navier–Stokes system as a fourth-order partial differential equation in one variable. The Gaussian–cubic backward substitution method and time difference method are used to solve the corresponding equation, in which the nonlinear part is generally transformed into linear equations through Picard iteration methods. This paper simulates three flows to prove the feasibility of the scheme. Full article

(This article belongs to the Special Issue Advances in Meshless Methods and Their Applications)

► Show Figures

Figure 1

10 pages, 3175 KB

Open AccessArticle

The Strong Chromatic Index of Complete Halin Graphs

by Zhiwei Bi and Yunfang Tang

Mathematics 2026, 14(9), 1562; https://doi.org/10.3390/math14091562 - 6 May 2026

Viewed by 524

Abstract

The strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if they are incident to a common edge or share an endpoint. The strong chromatic index of [...] Read more.

The strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if they are incident to a common edge or share an endpoint. The strong chromatic index of a graph G, denoted by

χ_{s}^{'} (G)

, is the minimum number of colors needed for a strong edge coloring of G. In this paper, we prove the following two theorems: (1) If

G = T \cup C

is a complete Halin graph with

Δ = 4

that contains adjacent vertices of maximum degree, then

χ_{s}^{'} (G) \leq χ_{s}^{'} (T) + 1 = 2 Δ

. In particular, when T is a regular tree,

χ_{s}^{'} (G) = χ_{s}^{'} (T) + 1 = 2 Δ

. (2) If

G = T \cup C

is a complete Halin graph with

Δ \geq 5

and

G \neq W_{n}

, then

χ_{s}^{'} (G) = χ_{s}^{'} (T) = 2 Δ - 1

when T is a regular tree. We extend the strong edge coloring results for complete cubic regular Halin graphs studied by W.C. Shiu and W.K. Tam, and improve the upper bound on the strong chromatic index of general Halin graphs established by Wei Yang and Baoyindureng Wu. Full article

► Show Figures

Figure 1

29 pages, 417 KB

Open AccessArticle

The Operational Efficiency Measurement of China’s Top 100 Digital Economy Firms: An Approach Based on DEA and Kernel Density Estimation

by Linyan Zhang, Yumeng Zhang, Kun Yang and Jian Zhang

Mathematics 2026, 14(9), 1561; https://doi.org/10.3390/math14091561 - 5 May 2026

Viewed by 577

Abstract

In recent years, China’s digital economy has become a key engine for high-quality development. Assessing the operational efficiency of leading digital enterprises is crucial for optimizing resource allocation and promoting sectoral growth. However, existing research largely remains at regional or industry levels and [...] Read more.

In recent years, China’s digital economy has become a key engine for high-quality development. Assessing the operational efficiency of leading digital enterprises is crucial for optimizing resource allocation and promoting sectoral growth. However, existing research largely remains at regional or industry levels and typically reports efficiency scores without diagnosing the root sources of inefficiency. To fill this gap, this study measures the operational efficiency of 99 firms selected from China’s Top 100 Digital Economy list (2017–2022) using the BCC-DEA model, and analyzes their dynamic evolution via kernel density estimation. The findings reveal a fluctuating upward trend in overall efficiency, and that the gap in overall technical efficiency primarily originates from scale efficiency rather than pure technical efficiency. The kernel density peak exhibits a “rise–decline–rise” pattern, indicating existing but narrowing efficiency differences among firms. By decomposing efficiency, this study further classifies firms into four types, revealing that inefficiency is heterogeneous. This paper makes three main contributions. First, it identifies scale efficiency as the main source of efficiency gaps. Second, it classifies firms into four types, revealing that inefficiency is heterogeneous. Third, it uses kernel density estimation to track the dynamic evolution of efficiency, showing a narrowing efficiency gap but a persistent superstar effect. Two policy implications follow: firms with low pure technical efficiency should focus on management training and technology adoption, while firms with low scale efficiency should pursue scale expansion through mergers or partnerships. Full article

(This article belongs to the Special Issue New Advances of Optimization and Data Envelopment Analysis)

► Show Figures

Figure 1

24 pages, 312 KB

Open AccessFeature PaperArticle

Inverse-Limit Formulas and Stable-Range Rigidity for Cyclotomic Sums

by Juan D. Vélez and Carlos Cadavid

Mathematics 2026, 14(9), 1560; https://doi.org/10.3390/math14091560 - 5 May 2026

Viewed by 379

Abstract

We study truncation-compatible families

F = {(F_{m})}_{m \geq 1}

over

Q [z]

through an inverse-limit formalism, and we evaluate them at the punctured cyclotomic cosine points [...] Read more.

We study truncation-compatible families

F = {(F_{m})}_{m \geq 1}

over

Q [z]

through an inverse-limit formalism, and we evaluate them at the punctured cyclotomic cosine points

α_{k, n} = cos (2 π k / n)

with the specialization

z = n - 1

. For symmetric families of uniformly bounded total x-degree

\leq d

, we prove a stable-range rigidity theorem: for all

n \geq d + 2

, the cosine-point evaluation factors through the finitely many punctured cosine power sums

P_{1} (n), \dots, P_{d} (n)

. In the purely polynomial case, this implies eventual polynomiality in n. We then extend the framework to include fixed-product factors and package their cosine-point contribution in multiplicative invariants

M_{Q} (n)

. In the stable range, the bounded-degree symmetric part collapses as before; any remaining cyclotomic dependence occurs only through these explicit product terms. Finally, we show that coefficient extraction from such products produces further bounded-degree symmetric families, and we apply this to complete symmetric functions

h_{r}

evaluated at cosine points. Full article

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

25 pages, 432 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Dimension-Independent Approximations on Low-Dimensional Manifolds Using Transformers

by Ji Shi and Demetrio Labate

Mathematics 2026, 14(9), 1559; https://doi.org/10.3390/math14091559 - 5 May 2026

Viewed by 528

Abstract

Deep neural networks have been remarkably successful in high-dimensional learning and scientific computing, often succeeding where classical discretization methods fail due to the curse of dimensionality. This efficacy is often explained by their approximation properties combined with the manifold hypothesis: the idea that [...] Read more.

Deep neural networks have been remarkably successful in high-dimensional learning and scientific computing, often succeeding where classical discretization methods fail due to the curse of dimensionality. This efficacy is often explained by their approximation properties combined with the manifold hypothesis: the idea that although data are embedded in dimension D, the effective degrees of freedom are governed by a much smaller intrinsic dimension

d ≪ D

. Under this hypothesis, data are concentrated near a low-dimensional manifold that neural networks can approximate efficiently. While the approximation theory for fully-connected ReLU networks on manifolds is well established, a comparable theory for transformer architectures, the dominant model class in modern foundation models, is still emerging. In this paper, we prove a new non-asymptotic, uniform approximation theorem for a class of single-head ReLU-transformers acting on vector inputs, where the approximation error depends only on the intrinsic dimension d rather than on the ambient dimension D. To the best of our knowledge, this is the first transformer approximation result that combines an intrinsic-dimensional rate with an ambient-dimension-independent multiplicative constant. We include a numerical experiment using a circle embedded in ambient dimensions of various sizes, showing that the observed error remains nearly unchanged as D varies, in agreement with the predicted ambient-dimension independence. Full article

(This article belongs to the Special Issue Mathematical Foundations of Deep Learning for Imaging)

► Show Figures

Figure 1

25 pages, 3616 KB

Open AccessFeature PaperArticle

Simultaneous Decompositions of Two Sets of Five Quaternion Tensors and Applications in Color Videos Processing

by Zhuo-Heng He, Yu-Fei Jiang, Mei-Ling Deng and Shao-Wen Yu

Mathematics 2026, 14(9), 1558; https://doi.org/10.3390/math14091558 - 5 May 2026

Viewed by 511

Abstract

This paper extends the theory of equivalence canonical forms from quaternion matrices to quaternion tensors under the Einstein product. Motivated by recent results on the simultaneous decomposition of two specific configurations of five quaternion matrices, we establish a comprehensive framework for the corresponding [...] Read more.

This paper extends the theory of equivalence canonical forms from quaternion matrices to quaternion tensors under the Einstein product. Motivated by recent results on the simultaneous decomposition of two specific configurations of five quaternion matrices, we establish a comprehensive framework for the corresponding configurations of five quaternion tensors. The core approach leverages bijective transformation maps that establish isomorphisms between quaternion tensor spaces and matrix spaces, allowing us to systematically construct invertible transformation tensors that simultaneously reduce the given tensor quintuples to canonical forms consisting solely of binary entries (0 and 1). A detailed structural analysis of the resulting canonical tensor forms is provided, including explicit dimension formulas for all identity blocks derived from precise rank conditions. To demonstrate practical utility, we integrate the proposed tensor decomposition with the discrete wavelet transform to construct a color video encryption and decryption system. Experimental results confirm perfect reconstruction (PSNR exceeding 300 dB, SSIM equal to 1) and strong security performance: NPCR of 49.8%, UACI of 49.6%, information entropy of 0.9986 bits per pixel, adjacent pixel correlation below 0.03 in absolute value, and a key space exceeding 2⁵¹². The developed theory significantly extends the existing literature on quaternion tensor decompositions and provides powerful tools for multidimensional signal processing. Full article

► Show Figures

Figure 1

26 pages, 344 KB

Open AccessArticle

Acting Fibrations and Lifting Functions in the Homotopy Theory of Single Intersection Graphs over Topological Semigroups

by Fozaiyah Alhubairah, Adem Kiliçman, Maryam F. Alshammari and Altaf Alshuhail

Mathematics 2026, 14(9), 1557; https://doi.org/10.3390/math14091557 - 4 May 2026

Viewed by 408

Abstract

In this paper, we study the homotopy theory of single intersection graphs arising from acting spaces over topological semigroups. An acting space

(S, B)

is defined as a topological space

B

equipped with a continuous action of a topological semigroup [...] Read more.

In this paper, we study the homotopy theory of single intersection graphs arising from acting spaces over topological semigroups. An acting space

(S, B)

is defined as a topological space

B

equipped with a continuous action of a topological semigroup

S

, generalizing the notion of algebraic actions in a topological setting. To connect this structure with graph theory, we associate to each acting space a single intersection graph

G_{S B}

, whose vertices are proper

S B

-subacting spaces, and two vertices are adjacent if their intersection is a singleton set. This graph construction encodes both algebraic and topological interactions between subacting spaces and provides a framework to study connectivity and homotopical properties via combinatorial methods. We then work within a categorical framework, where objects are graphical acting semigroups and morphisms are

S

-acting maps, allowing us to systematically study structural properties and their invariance under morphisms. In this setting, we introduce the notion of acting fibrations and formulate the corresponding lifting problem. Our main result establishes that an

S

-acting map is an acting fibration if and only if it admits an

A

-lifting function, providing a characterization analogous to classical fibration theory. Furthermore, we introduce

A

-regular lifting functions and analyze their role in preserving homotopical structures, including a natural homotopy extension property. Full article

► Show Figures

Figure 1

43 pages, 22952 KB

Open AccessArticle

Parameters Estimation and Reliability Analysis for Burr XII Distribution Under Adaptive Progressive First-Failure Censoring: Systematic Techniques with Application

by Rashad M. EL-Sagheer, Mohamed H. El-Menshawy, Mahmoud E. Bakr, Noha A. Tashkandi, Oluwafemi Samson Balogun and Mahmoud M. Ramadan

Mathematics 2026, 14(9), 1556; https://doi.org/10.3390/math14091556 - 4 May 2026

Viewed by 521

Abstract

An adaptive progressive first-failure censoring scheme is used to enhance the efficiency of statistical analyses and minimize test time in life-testing experiments. This paper focuses on statistical inferences for the unknown parameters, survival, and hazard rate functions of the Burr XII distribution under [...] Read more.

An adaptive progressive first-failure censoring scheme is used to enhance the efficiency of statistical analyses and minimize test time in life-testing experiments. This paper focuses on statistical inferences for the unknown parameters, survival, and hazard rate functions of the Burr XII distribution under this censoring scheme. Since the maximum likelihood estimates for the model parameters and reliability characteristics cannot be obtained explicitly, the Newton–Raphson method is employed for numerical derivation. The delta method is used to determine the variances of reliability characteristics and is applied to construct confidence intervals. Bayesian estimates of the unknown parameters and reliability characteristics are derived under the squared error and linear exponential loss functions. As these estimates are not explicitly obtainable, the Lindley and Markov chain Monte Carlo methods are used as approximation techniques. Additionally, asymptotic confidence intervals and highest posterior density credible intervals are developed for the parameters and reliability characteristics. A Monte Carlo simulation is performed to evaluate the proposed estimators, and the methodology is validated through a real dataset analysis on arthritic patients. Full article

► Show Figures

Figure 1

13 pages, 253 KB

Open AccessArticle

Existence and Maximal Regularity of Solutions for a Class of Third-Order Differential Equations with an Unbounded Coefficient

by Sabit Igissinov, Gulmira Nigmetova, Talgat Akhazhanov, Dauren Matin and Manat Shomanbayeva

Mathematics 2026, 14(9), 1555; https://doi.org/10.3390/math14091555 - 4 May 2026

Viewed by 440

Abstract

This paper investigates a class of third-order partial differential equations with an unbounded lower-order coefficient in the Hilbert space

L_{2} (R^{2})

. The study is motivated by the wide use of third-order equations, particularly of Korteweg–de Vries type, in [...] Read more.

This paper investigates a class of third-order partial differential equations with an unbounded lower-order coefficient in the Hilbert space

L_{2} (R^{2})

. The study is motivated by the wide use of third-order equations, particularly of Korteweg–de Vries type, in mathematical physics and wave theory, as well as by the limited development of the corresponding theory in the presence of unbounded coefficients. The main focus is on the existence, uniqueness, and maximal regularity of solutions. Within a functional-analytic framework, the well-posedness of the problem is established in natural function spaces under minimal assumptions on the coefficients. In particular, a priori estimates ensuring maximal regularity are derived. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Waves)

32 pages, 594 KB

Open AccessArticle

Design-Aware Predictive and Causal Modeling of Cardiovascular Risk in Chronic Kidney Disease Using Penalized and Double Machine Learning Approaches

by Fernando Rojas, Axa Tapia and Hilda Espinoza

Mathematics 2026, 14(9), 1554; https://doi.org/10.3390/math14091554 - 4 May 2026

Viewed by 519

Abstract

We develop a design-aware framework that combines penalized prediction and causal inference for finite populations observed through complex survey designs. The framework integrates survey-weighted pseudo-likelihoods,

ℓ_{1}

-penalized estimation, Neyman-orthogonal moment functions, and a bootstrap procedure that resamples primary sampling units within strata. [...] Read more.

We develop a design-aware framework that combines penalized prediction and causal inference for finite populations observed through complex survey designs. The framework integrates survey-weighted pseudo-likelihoods,

ℓ_{1}

-penalized estimation, Neyman-orthogonal moment functions, and a bootstrap procedure that resamples primary sampling units within strata. Methodologically, the contribution is an explicit pipeline that supports design-based inference while separating predictive associations from structurally adjusted effects in high-dimensional, clustered data. We illustrate the framework using data from the Chilean National Health Survey (ENS) 2016–2017 to study the relationship between chronic kidney disease (CKD) and high cardiovascular (CV) risk. In the ENS adult population, the survey-weighted prevalence of CKD was 3.1% (95% CI: 2.4–3.8), and the prevalence of high CV risk was 23.9% (95% CI: 21.5–26.3). High CV risk was markedly more frequent among individuals with CKD than among those without CKD (90.9% versus 21.5%). Predictive and associational analyses combined survey-weighted penalized logistic regression (LASSO) with refitted unpenalized models. In conventional survey-weighted logistic regressions, CKD showed a strong association with high CV risk (odds ratio = 5.66; 95% CI: 2.71–11.82;

p < 0.001

), and effect sizes remained stable after LASSO-based variable selection. To assess causal relevance under confounding and potential endogeneity, we implemented two endogeneity-aware estimators: two-stage residual inclusion (2SRI) and double/debiased machine learning (DML). The DML estimator, defined as the primary causal estimand, reports an orthogonalized estimate of the average treatment effect of CKD on the probability of high CV risk. After adjustment for age and major cardiometabolic comorbidities, the DML estimate was attenuated and statistically non-significant (average treatment effect =

- 0.094

; 95% CI:

[- 0.409, 0.220]

). The 2SRI approach yielded unstable estimates with wide confidence intervals, consistent with the limited effective sample size of CKD cases (

n_{C K D} \approx 190

in a sample with n ≈ 6233) and weak identification conditions under low-prevalence settings. Simulation experiments under ENS-like complex sampling suggest that naive predictive associations may overestimate the structural contribution of CKD under confounding, whereas orthogonalized estimators yield more conservative estimates when identification holds. The causal interpretation relies on a conditional mean independence assumption given observed covariates and survey design, while control-function specifications are treated as diagnostic sensitivity analyses due to the absence of credible exclusion-based instruments. Overall, the results demonstrate a fundamental divergence between predictive relevance and causal importance in finite-population settings, underscoring the need for design-aware and endogeneity-robust methods in statistical modeling. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

► Show Figures

Figure 1

28 pages, 16866 KB

Open AccessArticle

Quantifying Terrain Effects on Turbine Wake Recovery with Field Data and Simulation of a Real Wind Farm

by Andrea Torrejón-Fontana, Luis Silva-Llanca, Sonia Montecinos and Charles Meneveau

Mathematics 2026, 14(9), 1553; https://doi.org/10.3390/math14091553 - 4 May 2026

Viewed by 719

Abstract

Global reliance on wind energy continues to grow, leading to an increasing number of wind farms implemented in complex topographies. However, there remains a significant research gap on how the terrain’s features affect the wake recovery, especially when the irregularities scale with the [...] Read more.

Global reliance on wind energy continues to grow, leading to an increasing number of wind farms implemented in complex topographies. However, there remains a significant research gap on how the terrain’s features affect the wake recovery, especially when the irregularities scale with the wind turbine’s size. This study uses field data and Reynolds-averaged simulations to quantify the influence of topographical features on a wind farm’s wake recovery and power generation. To characterize the terrain surrounding the turbines, this study introduces two parameters—the Downwind Slope and the surface complexity length

ζ

—which quantify the local average terrain unevenness. The findings demonstrate that turbines in terrains with streamwise positive slopes exhibit faster wake recovery, averaging 6.35D in length (D = turbine diameter), followed by complex-flat terrain (8.7D on average), then descending terrains with the least beneficial wake recovery (9.2D on average). A terrain with a higher surface complexity also improves wake recovery owing to the turbulent entrainment that enhances momentum transport exchange into the wake. Additionally, simulations of the same turbine distribution, but in a completely flat terrain, showed that the complex terrain may lead to lower performance compared to the idealized flat terrain: 11.5% of power generation decrease in our case. The latter highlights the importance of considering topographic effects when planning wind energy projects. Full article

(This article belongs to the Special Issue Mathematical Methods in Computational Fluid Dynamics: Theory and Applications)

► Show Figures

Graphical abstract

18 pages, 408 KB

Open AccessArticle

Biquadratic SOS Rank and Augmented Zarankiewicz Number

by Liqun Qi, Chunfeng Cui and Yi Xu

Mathematics 2026, 14(9), 1552; https://doi.org/10.3390/math14091552 - 3 May 2026

Cited by 2 | Viewed by 463

Abstract

This paper introduces the concepts of the augmented Zarankiewicz number

z_{A} (m, n)

and the limited augmented Zarankiewicz number

z_{L} (m, n)

, which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers [...] Read more.

This paper introduces the concepts of the augmented Zarankiewicz number

z_{A} (m, n)

and the limited augmented Zarankiewicz number

z_{L} (m, n)

, which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers arise from augmented bipartite graphs that may contain both standard edges (1-edges) and pairs of edges representing squares of binomials (2-edges). The main theoretical result establishes the inequality chain

BSR (m, n) \geq z_{A} (m, n) \geq z_{L} (m, n) \geq z (m, n)

, linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of

z_{L} (m, n)

for the cases

(m, 2)

,

(3, 3)

,

(4, 3)

, and

(4, 4)

and provide new lower bounds for the cases

(5, 3)

,

(5, 4)

, and

(5, 5)

. These results yield improved lower bounds for the maximum SOS rank of biquadratic forms, demonstrating that

z_{L} (m, n)

can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

► Show Figures

Figure 1

20 pages, 965 KB

Open AccessArticle

Fast Finite-Time Position Tracking Control of Electro-Hydraulic Servo Systems with Parametric Uncertainty via Dynamic Surface and Neural Adaptive Method

by Shuai Li, Yaya Yan, Yue Yu, Qishui Zhong, Lanfeng Hua and Daixi Liao

Mathematics 2026, 14(9), 1551; https://doi.org/10.3390/math14091551 - 3 May 2026

Cited by 8 | Viewed by 766

Abstract

In research on electro-hydraulic servo systems, nonlinearity deeply affects dynamic performance, such as the output of hydraulic actuators and the generation of control signals, leading to response hysteresis and control complexity. Moreover, during the control process, changes in the external environment and component [...] Read more.

In research on electro-hydraulic servo systems, nonlinearity deeply affects dynamic performance, such as the output of hydraulic actuators and the generation of control signals, leading to response hysteresis and control complexity. Moreover, during the control process, changes in the external environment and component loss lead to model parameter distort, which reduces control capability. To address these challenges, this paper conducts a structural transformation on the traditional dynamic surface controller in combination with the fast finite-time stability theorem and proposes a novel finite-time dynamic surface control strategy, which can not only overcome the differential explosion phenomenon in the recursive backstepping iterative process but also enhance the transient dynamic response speed. Furthermore, the neural network adaptive algorithm is adopted to handle the negative dynamic effect caused by parametric uncertainty. The theoretical results are verified by the Lyapunov stability method and numerical simulation. Full article

(This article belongs to the Special Issue Advanced Control of Complex Dynamical Systems and Robotics with Applications)

► Show Figures

Figure 1

36 pages, 6439 KB

Open AccessEditor’s ChoiceArticle

Modelling Workload and Injury Risk in Elite Touch Rugby with Clustering Effect: A Time-Scaled Shared Frailty Approach

by Tom Huang, Shu Su, Nuttanan Wichitaksorn and Kirsten Spencer

Mathematics 2026, 14(9), 1550; https://doi.org/10.3390/math14091550 - 3 May 2026

Viewed by 511

Abstract

In this study, we propose a general mathematical modelling framework based on the characteristics of elite athletes’ movements in the touch rugby matches to investigate the dynamic relationship between physical workload and injury risk over time. Our framework extends the Cox-based model in [...] Read more.

In this study, we propose a general mathematical modelling framework based on the characteristics of elite athletes’ movements in the touch rugby matches to investigate the dynamic relationship between physical workload and injury risk over time. Our framework extends the Cox-based model in the context of touch rugby by incorporating a time-scaling component and cluster-specific heterogeneity simultaneously. In addition, we allow for the inclusion of covariates (e.g., velocity variation) to capture their effects. We applied our model to high-frequency wearable sensor data collected from 27 elite athletes (15 men and 12 women). The empirical study results show that our model, time-scaled frailty model (TSFM), demonstrates better goodness-of-fit than traditional frailty and Andersen–Gill models. The results reveal that higher velocity variation, particularly during high-intensity phases, and longer time of continuous exposure to the workload spike state significantly increased overload risk, ultimately resulting in injury. It also highlights the importance of individual differences, even under the same exercise intensity. These insights provide coaches with an evidence-based framework for athlete monitoring, allowing for more personalized training loads, tactical deployment, and injury prevention strategies in elite touch rugby environments. Full article

(This article belongs to the Special Issue Mathematical Foundations of Reliability Theory with Applications in Engineering and Applied Statistics)

► Show Figures

Figure 1

20 pages, 398 KB

Open AccessEditor’s ChoiceArticle

Robust-Mean–Geometric-Mean and Robust Haberman Linking with Invariant Item Discriminations Under Sparse Differential Item Functioning

by Alexander Robitzsch

Mathematics 2026, 14(9), 1549; https://doi.org/10.3390/math14091549 - 2 May 2026

Viewed by 422

Abstract

Comparison of two or multiple groups based on dichotomous items is a central task in item response theory (IRT) linking. This article considers the two-parameter logistic scaling model under sparse differential item functioning (DIF) in item intercepts and DIF-free item discriminations. Robust-mean-geometric-mean (RMGM) [...] Read more.

Comparison of two or multiple groups based on dichotomous items is a central task in item response theory (IRT) linking. This article considers the two-parameter logistic scaling model under sparse differential item functioning (DIF) in item intercepts and DIF-free item discriminations. Robust-mean-geometric-mean (RMGM) and robust Haberman (RHAB) linking are compared across several loss functions and under scaling models with noninvariant or invariant item discriminations. Two simulation studies show that invariant item discriminations improve the precision of estimated group means. In addition, the

L_{0}

loss function is generally preferable to the

L_{1}

and

L_{0.5}

loss functions when DIF proportions or sample sizes are large. Several empirical examples illustrate the proposed specifications. Full article

(This article belongs to the Special Issue Computational Statistics, Data Analysis and Applications)

9 pages, 230 KB

Open AccessArticle

On a Norm Inequality for Three 2 × 2 Matrices with One Normal Factor

by Na Li and Fen Wang

Mathematics 2026, 14(9), 1548; https://doi.org/10.3390/math14091548 - 2 May 2026

Viewed by 454

Abstract

In this paper, we continue to investigate the norm inequality for three real matrices that was recently conjectured by L. László. We establish the validity of the conjecture for the case where

n = 2

and one of the matrices is normal. Full article

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

15 pages, 289 KB

Open AccessArticle

Riemann Solitons and Ricci Bi-Conformal Vector Fields on 4-Dimensional Oscillator Group

by Bang-Yen Chen, Foued Aloui, Majid Ali Choudhary and Mohammad Nazrul Islam Khan

Mathematics 2026, 14(9), 1547; https://doi.org/10.3390/math14091547 - 2 May 2026

Viewed by 472

Abstract

We consider Riemann soliton vector fields and Ricci bi-conformal vector fields on the oscillator group. We prove that the oscillator group admits Riemann solitons. Subsequently, we provide a complete classification of all Ricci bi-conformal vector fields admitted by the oscillator group and identify [...] Read more.

We consider Riemann soliton vector fields and Ricci bi-conformal vector fields on the oscillator group. We prove that the oscillator group admits Riemann solitons. Subsequently, we provide a complete classification of all Ricci bi-conformal vector fields admitted by the oscillator group and identify those that belong to specific categories, namely gradient-type vector fields, Killing vector fields, and Ricci collineation using the partial differential equations. Full article

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

64 pages, 137860 KB

Open AccessArticle

An Artistic Image Segmentation Method Using an Art-Design-Inspiration-Driven Ivy Algorithm

by Xiaoning Wang, Fan Liu, Xianmeng Zhao and Hui Zhang

Mathematics 2026, 14(9), 1546; https://doi.org/10.3390/math14091546 - 2 May 2026

Cited by 1 | Viewed by 435

Abstract

To overcome the limitations of the original Ivy Algorithm (IVYA), including insufficient population diversity, limited step-size adaptability, and premature convergence, this paper proposes a multi-strategy enhanced Ivy optimization algorithm (MEIVYA). The proposed method integrates chaotic population initialization, adaptive growth-rate regulation, and an elite-guided [...] Read more.

To overcome the limitations of the original Ivy Algorithm (IVYA), including insufficient population diversity, limited step-size adaptability, and premature convergence, this paper proposes a multi-strategy enhanced Ivy optimization algorithm (MEIVYA). The proposed method integrates chaotic population initialization, adaptive growth-rate regulation, and an elite-guided cooperative search strategy to improve global exploration, local exploitation, and convergence stability. Experimental results on the CEC2014 and CEC2017 benchmark suites show that MEIVYA achieves competitive convergence accuracy, robustness, and stability compared with several state-of-the-art metaheuristic algorithms. In addition, MEIVYA is applied to multi-threshold image segmentation based on the Otsu criterion, where it produces clearer segmentation structures and better visual quality. The results demonstrate that MEIVYA is an effective and robust approach for both numerical optimization and artistic image segmentation. Full article

(This article belongs to the Special Issue Metaheuristic Algorithms, 2nd Edition)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Mathematics, Volume 14, Issue 9 (May-1 2026) – 175 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI