Mathematics

12 pages, 446 KB

Open AccessArticle

Parity-Based Level-Set Approach to the Collatz Conjecture

by Selcuk Koyuncu, Thevasha Sathiyakumar, Praise Alayode, Christopher Ellis and Peyton Thomas

Mathematics 2026, 14(10), 1763; https://doi.org/10.3390/math14101763 - 20 May 2026

The Collatz conjecture concerns the iteration of the map

f (n) = 3 n + 1

for odd n and

f (n) = n / 2

for even n. In this paper, we study the level sets [...] Read more.

The Collatz conjecture concerns the iteration of the map

f (n) = 3 n + 1

for odd n and

f (n) = n / 2

for even n. In this paper, we study the level sets

l_{x} = {n \in N ∣ L (n) = x}

, where

L (n)

denotes the Collatz length. Using the parity representation of Collatz trajectories, we partition each

l_{x}

according to the number of odd steps and analyze the corresponding means

μ_{x, k}

. Under a natural scaling assumption, these means satisfy an approximate geometric progression, so that

log μ_{x, k}

is approximately linear in k. Computations for

n \leq 100, 000

and

10 \leq x \leq 50

show highly stable regression parameters and near-perfect linear fits. Full article

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

30 pages, 3640 KB

Open AccessArticle

Intermittent Control for Synchronization-Like Behavior of State-Dependent Impulsive Neural Networks via Interval–Impulse Differential Inequality

by Yanshou Dong, Junfang Zhao, Jinqiu Li, Tingting Dai and Yan Han

Mathematics 2026, 14(10), 1762; https://doi.org/10.3390/math14101762 - 20 May 2026

Abstract

This paper investigates the synchronization problem for a class of master–slave neural networks with state-dependent impulses. Different from fixed-time impulsive systems, the impulsive instants considered here depend on the current states of the neural networks, which makes the synchronization analysis more complicated. In [...] Read more.

This paper investigates the synchronization problem for a class of master–slave neural networks with state-dependent impulses. Different from fixed-time impulsive systems, the impulsive instants considered here depend on the current states of the neural networks, which makes the synchronization analysis more complicated. In particular, when both the master and slave systems possess their own state-dependent impulses, the corresponding impulsive instants are generally asynchronous, so the synchronization error evolves over an impulsive interval rather than undergoing only a single instantaneous jump. To address this difficulty, two easily verifiable conditions are first proposed to guarantee that each trajectory intersects every impulsive surface exactly once, thereby excluding the beating phenomenon. Then, an interval–impulse differential inequality is established to characterize the error evolution on non-impulsive subintervals and to handle the mismatch between the impulsive times of the master and slave systems. Based on this inequality, an intermittent controller activated only outside the impulsive interval is designed so that the controller does not destroy the intrinsic state-dependent impulsive rhythm of the master system. By combining Lyapunov analysis with matrix inequality techniques, verifiable criteria are derived for local exponential synchronization-like behavior of the considered neural networks. Here, synchronization-like refers to exponential decay of the synchronization error on the non-mismatched time intervals since the master and slave systems generally possess asynchronous state-dependent impulsive instants. Finally, numerical examples are presented to illustrate the effectiveness of the proposed conditions and control strategy. The simulation results show that the designed controller can effectively suppress synchronization error and that increasing the control gain can significantly accelerate the convergence process. Full article

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

29 pages, 2780 KB

Open AccessArticle

Enhanced Transmission Loss and Modal Coupling in Dual-Membrane Flexible-Shell Cylindrical Waveguides: A Rigorous Mode-Matching–Galerkin Framework

by Mohammed Alkinidri

Mathematics 2026, 14(10), 1761; https://doi.org/10.3390/math14101761 - 20 May 2026

Abstract

This paper develops an analytical treatment of vibro-acoustic wave propagation in a cylindrical waveguide containing two clamped elastic membranes and a central flexible-shell segment. The acoustic field obeys the time-harmonic Helmholtz equation, the shell motion is described by Donnell–Mushtari thin-shell theory under axisymmetric [...] Read more.

This paper develops an analytical treatment of vibro-acoustic wave propagation in a cylindrical waveguide containing two clamped elastic membranes and a central flexible-shell segment. The acoustic field obeys the time-harmonic Helmholtz equation, the shell motion is described by Donnell–Mushtari thin-shell theory under axisymmetric loading, and the membrane response is governed by classical membrane theory and incorporated through a tailored Galerkin scheme. The resulting coupled fluid–structure boundary-value problem is solved by the Mode-Matching Method: the acoustic potentials are expanded in orthogonal radial eigenfunctions within each subregion, and continuity of pressure, normal velocity, and structural displacement are enforced at every interface. The mirror symmetry of the configuration is exploited by an exact decomposition into symmetric and anti-symmetric sub-problems, each of which reduces to a truncated linear algebraic system of dimension

4 N + 4

for the unknown modal amplitudes. Acoustic power-balance identities provide a quantitative consistency check on the numerical implementation and diagnose convergence with respect to the truncation order; structural damping is accommodated through complex-modulus substitutions for the shell and the membrane tension without altering the algebraic structure of the system. The numerical results demonstrate that the dual-membrane configuration delivers transmission-loss values exceeding

25 dB

across the low-frequency band relevant to HVAC and automotive applications, with a representative plateau near

13 dB

at the reference geometry, through resonance-driven modal coupling between the acoustic field and the compliant interfaces. Parametric studies identify the excitation frequency, the inner-membrane radius, the shell radius, and the chamber length as effective design parameters for tuning the attenuation. The formulation furnishes a unified and computationally efficient analytical tool for predicting and optimising noise attenuation in flexibly coupled cylindrical duct systems. Full article

(This article belongs to the Section E4: Mathematical Physics)

41 pages, 1003 KB

Open AccessArticle

Geometric Structure of Genomes Across the Tree of Life: Toward a Geometric Theory of Sequence Structure

by Valentin E. Brimkov and Reneta P. Barneva

Mathematics 2026, 14(10), 1760; https://doi.org/10.3390/math14101760 - 20 May 2026

Abstract

This work develops a geometric and statistical framework for analyzing the structure of biological sequences and explores its implications for understanding the emergence and evolution of life. Motivated by questions concerning the transition from prebiotic chemistry to living systems, the quantification of negentropy [...] Read more.

This work develops a geometric and statistical framework for analyzing the structure of biological sequences and explores its implications for understanding the emergence and evolution of life. Motivated by questions concerning the transition from prebiotic chemistry to living systems, the quantification of negentropy in organic matter, and the distinction between random and biologically viable sequences, we introduce mathematical descriptors that measure deviation from linearity and related geometric irregularities of self-replicating macromolecules. These descriptors reveal a pronounced geometric separation between biological DNA and random sequences, underscoring the non-random structural organization characteristic of living systems. Using these descriptors, we compare a broad range of species across the Tree of Life and examine how geometric complexity varies between primitive and more advanced organisms. We further investigate whether these measures provide a natural way to compare organismal complexity, characterize the structure of viable sequence space, and identify potential constraints on evolutionary trajectories. The framework also offers an initial perspective on how natural selection and stochastic mutations may jointly influence genomic organization. Finally, we outline speculative connections between increasing geometric irregularity and the emergence of biological complexity, suggesting that such geometric transitions may offer insight into the origins of life and the theoretical limits of evolutionary development. Full article

(This article belongs to the Section E3: Mathematical Biology)

37 pages, 4969 KB

Open AccessFeature PaperArticle

Fuzzy Iterative Learning Contouring Control

by Thanh-Quan Ta and Shyh-Leh Chen

Mathematics 2026, 14(10), 1759; https://doi.org/10.3390/math14101759 - 20 May 2026

Abstract

Iterative learning contouring control (ILCC) improves contouring accuracy in multi-axis motion systems via the equivalent contour error formulation. However, its convergence strongly depends on the learning gain. Large gains may induce overly aggressive updates and local divergence, degrading performance, whereas small gains lead [...] Read more.

Iterative learning contouring control (ILCC) improves contouring accuracy in multi-axis motion systems via the equivalent contour error formulation. However, its convergence strongly depends on the learning gain. Large gains may induce overly aggressive updates and local divergence, degrading performance, whereas small gains lead to slow convergence. Moreover, contour error convergence is typically non-uniform along the trajectory, and local divergence may still occur despite global convergence, particularly near error saturation regions. To address these issues, a fuzzy inference mechanism is integrated into the online ILCC framework, yielding an online ILCC with fuzzy-regulated convergence parameters (online ILCCf), enabling adaptive regulation of the learning gain. Two regulation strategies are developed: (i) online ILCCf

_{i}

, an independent multi-parameter regulation scheme; and (ii) online ILCCf

_{u}

, a unified single-parameter regulation scheme. The fuzzy mechanism adaptively adjusts the convergence parameters online according to the instantaneous magnitude of the equivalent contour error. Experimental results on a six-axis industrial robot demonstrate fast convergence while maintaining satisfactory contouring performance. Among all comparison cases , online ILCCf

_{i}

achieves the best performance, reducing the RMS position error from

7.26 \times 10^{- 1}

mm to

5.93 \times 10^{- 2}

mm and the RMS orientation error from

6.95 \times 10^{- 4}

rad to

5.64 \times 10^{- 5}

rad, without oscillation or local divergence. Further simulations confirm robustness under model uncertainty and measurement noise. Full article

41 pages, 7512 KB

Open AccessArticle

Geometric Interpretation of Frequency Domain Robustness Constraints and Closed-Loop Pole Locations

by Vesela Karlova-Sergieva

Mathematics 2026, 14(10), 1758; https://doi.org/10.3390/math14101758 - 20 May 2026

Abstract

Requirements for robustness and performance in the frequency domain in control theory are usually formulated as constraints on the modulus of complex functions describing the open-loop system, the sensitivity function, and the complementary sensitivity function. These constraints generate circular sets that can be [...] Read more.

Requirements for robustness and performance in the frequency domain in control theory are usually formulated as constraints on the modulus of complex functions describing the open-loop system, the sensitivity function, and the complementary sensitivity function. These constraints generate circular sets that can be interpreted as admissible or forbidden regions in the complex plane, particularly in the Nyquist diagram. In engineering practice, they are often treated as method-specific constructions, without clarifying the general geometric mechanism by which they arise. This study develops a geometric interpretation in which a broad class of frequency-domain robustness constraints is represented as level sets of analytic and fractional-linear functions. The resulting circular sets in the Nyquist plane are characterized in a unified manner and mapped to admissible regions in the complex s-plane through preimage transformations. The approach is formulated entirely using complex transfer functions, remaining within the classical frequency-domain framework, without state-space representations, linear matrix inequalities, or optimization methods. Classical robustness measures, including gain margin, phase margin, and constraints on sensitivity and complementary sensitivity, are shown to be special cases of the same geometric structure. The main insight of this work is that these apparently different robustness constraints arise from the same underlying geometric mechanism. This interpretation establishes a geometric link between frequency domain robustness constraints and the location of closed-loop poles, allowing a qualitative assessment of robustness and dynamic properties of control systems without introducing new stability criteria or design procedures. The resulting admissible regions provide a geometric interpretation of frequency domain robustness specifications in terms of pole locations in the s-plane. Full article

(This article belongs to the Special Issue Advances in Robust Control Theory and Its Applications)

15 pages, 313 KB

Open AccessArticle

Relational Nonlinear Almost Contractions of Mukherjea Type and Applications to Boundary Value Problems

by Doaa Filali, Mohammed Zayed Alruwaytie, Abdulaziz Abbas Alshammari, Adel Alatawi, Fahad M. Alamrani and Faizan Ahmad Khan

Mathematics 2026, 14(10), 1757; https://doi.org/10.3390/math14101757 - 20 May 2026

Abstract

In this paper, we explore specific fixed-point outcomes under a Mukherjea-Type nonlinear description of almost contractions in a metric space with a locally ℏ-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. To demonstrate [...] Read more.

In this paper, we explore specific fixed-point outcomes under a Mukherjea-Type nonlinear description of almost contractions in a metric space with a locally ℏ-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. To demonstrate the reliability of our results, a couple of instances are supplied. Our outcomes are deployed to evaluate the accuracy of the unique solution of a boundary value problem. Full article

22 pages, 4316 KB

Open AccessArticle

Spatiotemporal Forecasting of Seismic Activity Trends Using Wiener Filtering and Artificial Neural Networks

by Pengfei Ren, Peijia Li, Xiaoyang Chen, Tingkai Gu, Xiaoyu Song, Cong Wang and Kai Yan

Mathematics 2026, 14(10), 1756; https://doi.org/10.3390/math14101756 - 20 May 2026

Abstract

Reliable forecasting of seismic activity trends is essential for regional seismic hazard analysis. Based on earthquake catalogs from 1500 to 2026, this study investigates the spatiotemporal evolution of seismic activity in the North-South Seismic Belt using a hybrid framework that integrates Wiener filtering [...] Read more.

Reliable forecasting of seismic activity trends is essential for regional seismic hazard analysis. Based on earthquake catalogs from 1500 to 2026, this study investigates the spatiotemporal evolution of seismic activity in the North-South Seismic Belt using a hybrid framework that integrates Wiener filtering and artificial neural networks. Seismic activity is modeled as a discrete-time stochastic process, and a time series of earthquakes with magnitudes ≥ 6.0 is constructed. Wiener filtering is applied to establish an optimal linear relationship between input and output under the minimum mean square error criterion, and multi-origin extrapolation is employed to predict earthquakes with magnitudes ≥ 7.0 over the next century. The results reveal several stable peaks or peak clusters that agree well with historical strong earthquakes, with prediction errors generally within approximately three years. Sensitivity analyses indicate that longer time series (∼500 years) and higher threshold magnitudes (≥6.0) enhance prediction stability, although the method shows limitations in spatial prediction. To address this issue, a 16–8–4 artificial neural network model is developed, and seismic sequence features are extracted using a sliding time window approach to perform both temporal and spatial forecasting. The artificial neural network achieves high accuracy in temporal prediction (maximum error ≈ 0.5) and outperforms Wiener filtering in spatial prediction, capturing the migration characteristics of seismic activity. The results further suggest that earthquakes with magnitudes ≥ 7.0 are more likely to occur within the latitude range of 30.5–33.0° N in the near future. Full article

(This article belongs to the Special Issue Advances of Mathematics and Artificial Intelligence in Engineering Applications)

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16 pages, 1770 KB

Open AccessArticle

A Hybrid AI Approach for Intelligent Group Buying and Digital Marketing Strategy Optimization Based on Machine Learning and Evolutionary Algorithms

by Zhansaya Abildaeva, Raissa Uskenbayeva, Zhuldyz Kalpeyeva, Aizhan Kassymova, Aigul Dauitbayeva and Adranova Asselkhan

Mathematics 2026, 14(10), 1755; https://doi.org/10.3390/math14101755 - 20 May 2026

Abstract

This study considers the digital transformation of Kazakhstan’s agro-industrial complex, which has created an urgent need for scientifically grounded methods that can optimize marketing strategies under conditions of resource limitations, production seasonality, and heterogeneous consumer behavior. This study proposes a hybrid decision-support framework [...] Read more.

This study considers the digital transformation of Kazakhstan’s agro-industrial complex, which has created an urgent need for scientifically grounded methods that can optimize marketing strategies under conditions of resource limitations, production seasonality, and heterogeneous consumer behavior. This study proposes a hybrid decision-support framework integrating a modified NSGA-III algorithm with machine learning techniques for optimizing digital marketing strategies in the agro-industrial complex of Kazakhstan. The model considers three objectives: maximizing channel efficiency and audience reach while minimizing marketing costs. Experimental results based on a dataset of N = 1200 observations demonstrate that the proposed approach improves the composite performance indicator by 12.4% compared to baseline single-objective optimization methods. Pareto front analysis reveals three distinct clusters of strategies, corresponding to (1) high-impact integrated digital TV strategies, (2) cost-efficient traditional channel strategies, and (3) high-risk high-return allocations. The clustering validity is confirmed by a silhouette score of 0.624, indicating strong separation between strategy groups. The results highlight the practical significance of adaptive budget allocation and demonstrate the effectiveness of combining evolutionary optimization with machine learning for decision support in complex marketing environments. Full article

(This article belongs to the Topic AI and Data-Driven Advancements in Industry 4.0, 2nd Edition)

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10 pages, 830 KB

Open AccessArticle

Finite Element Analysis of Shear Strength Improvements in Fibre-Reinforced Soils Using the Mohr–Coulomb Model

by Thomas Houghton and Alireza Ahangar Asr

Mathematics 2026, 14(10), 1754; https://doi.org/10.3390/math14101754 - 20 May 2026

Abstract

With increasing emphasis on sustainable construction, fibre reinforcement of soils presents a viable approach for enhancing mechanical performance while minimising environmental impact. This study investigates the influence of fibreglass fibres (0–3% by dry weight) on soil behaviour under triaxial loading using finite element [...] Read more.

With increasing emphasis on sustainable construction, fibre reinforcement of soils presents a viable approach for enhancing mechanical performance while minimising environmental impact. This study investigates the influence of fibreglass fibres (0–3% by dry weight) on soil behaviour under triaxial loading using finite element modelling. The results demonstrate that fibre inclusion improves peak deviator stress and reduces volumetric strain, with the magnitude of enhancement dependent on both fibre content and confining pressure. Optimal performance was observed at 2% fibre content, which increased performance by approximately 20% under low confining pressures (50–100 kPa). At higher pressures (200–500 kPa), optimal performance was from 3% fibre content, which increased performance by approximately 5%. Improvements are attributed to mechanisms reported in the literature, including the ability of fibres to redistribute stresses and bridge microcracks. These findings provide insights into the behaviour of fibre-reinforced soils for applications such as slope stabilisation, foundations, and retaining structures, supporting more sustainable geotechnical engineering practices. These findings are not intended for direct design application. Full article

(This article belongs to the Special Issue Numerical Analysis and Finite Element Method with Applications)

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25 pages, 1072 KB

Open AccessArticle

RBFNN-Based Secure Tracking Control for a Class of Strict-Feedback Nonlinear Systems with Asymmetric Output Constraints and Its Application to UAVs

by Lijun Zhang, Meiru Jiang, Jiahao Li, Na Liu, Jiyong Lu and Kai Cui

Mathematics 2026, 14(10), 1753; https://doi.org/10.3390/math14101753 - 20 May 2026

Abstract

This paper investigates a tracking control problem for a class of strict-feedback nonlinear systems with time delays, asymmetric output constraints, and deception attacks on the controller. First, by introducing a new error conversion technology, any nonzero and bounded initial state is converted to [...] Read more.

This paper investigates a tracking control problem for a class of strict-feedback nonlinear systems with time delays, asymmetric output constraints, and deception attacks on the controller. First, by introducing a new error conversion technology, any nonzero and bounded initial state is converted to zero, which not only solves the overshoot/oscillation problem of the output during the constraint switching phase but also unifies the control design of constrained and unconstrained systems. Second, a barrier function with asymmetric output constraints is designed, which converts the problem of satisfying the tracking control of nonlinear systems under output constraints into one of ensuring the boundedness. In addition, radial basis function neural networks (RBFNNs) are utilized to handle both unknown uncertain terms and deception attacks simultaneously. By utilizing the new asymmetric delayed barrier function error together with an RBFNN technique, the tracking error is ultimately uniformly bounded, regardless of the presence or absence of output constraints. Finally, the superiority of the proposed strategy is verified through its simulation on an unmanned aerial vehicle (UAV) system. Full article

(This article belongs to the Special Issue Computational Approaches to Control Systems: Methods and Applications)

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24 pages, 3629 KB

Open AccessArticle

Compact Orbits and Topological Sensitivity on Locally Compact Spaces

by Sanil Jose and Vinod Kumar P.B.

Mathematics 2026, 14(10), 1752; https://doi.org/10.3390/math14101752 - 19 May 2026

Abstract

We introduce a cover-based formulation of topological sensitivity and sensitivity at a point for continuous self-maps on locally compact spaces, extending the classical metric framework. Using the one-point compactification, we analyze the basin of attraction of infinity and relate it to escaping dynamics. [...] Read more.

We introduce a cover-based formulation of topological sensitivity and sensitivity at a point for continuous self-maps on locally compact spaces, extending the classical metric framework. Using the one-point compactification, we analyze the basin of attraction of infinity and relate it to escaping dynamics. We study the set

K (f)

consisting of points whose forward orbits are contained in a compact subset of the phase space, establishing its fundamental topological properties under suitable assumptions on the map f. In particular, we show that for proper maps,

K (f)

coincides with the complement of the escaping set. Under additional hypotheses on f, we prove that the boundary of the set of points with compact orbits, the boundary of the basin of attraction of infinity, and the set of sensitive points coincide. This provides a topological generalization of the classical dichotomy between Fatou and Julia sets in complex dynamics. Full article

26 pages, 3007 KB

Open AccessArticle

Rayleigh Wave Propagation on the Partially Saturated Poro-Thermo-Viscoelastic Half-Space Based on Fractional Order Viscoelasticity

by Li Li and Wei Zhuang

Mathematics 2026, 14(10), 1751; https://doi.org/10.3390/math14101751 - 19 May 2026

Abstract

This paper probes into the propagation characteristics of Rayleigh waves in a partially saturated, porous, thermo-viscoelastic half-space, with full consideration of the fractional viscoelastic effect and thermal coupling effect. A fractional Zener model is introduced to depict the thermo-viscoelastic mechanical behavior of the [...] Read more.

This paper probes into the propagation characteristics of Rayleigh waves in a partially saturated, porous, thermo-viscoelastic half-space, with full consideration of the fractional viscoelastic effect and thermal coupling effect. A fractional Zener model is introduced to depict the thermo-viscoelastic mechanical behavior of the solid skeleton by constructing a complete set of governing equations that include mass balance, generalized Darcy’s law, momentum balance, and generalized heat conduction. Field equations are derived by means of Helmholtz vector decomposition, and the dispersion equation, and the phase velocity expression of Rayleigh waves are obtained by combining the traction-free and adiabatic boundary conditions of the medium. The impacts of key material properties, such as medium saturation, intrinsic permeability, medium viscoelasticity, and thermal expansion coefficient, on the dispersion feature of Rayleigh waves are discussed in detail. Numerical analysis results show that an increase in the thermal expansion coefficient will lead to a rise in Rayleigh wave phase velocity, in which the increase in P1 compressional wave velocity plays a dominant role among the velocities of various types of waves. Meanwhile, the attenuation coefficient of Rayleigh waves presents a decreasing trend and gradually tends to be stable with the growth of the thermal expansion coefficient. Similarly, the phase velocity of Rayleigh waves also increases with the rise in fractional order index, which is jointly dominated by the velocity enhancement of P1 waves and S waves. In addition, the attenuation coefficient of Rayleigh waves increases first and then decreases with the increase in fractional order index and reaches the peak value when the fractional order index is about 0.4. The research results reveal the influence of laws of thermal expansion characteristics and viscoelasticity on Rayleigh wave propagation and provide theoretical support for the analysis of wave propagation characteristics in porous media in relevant engineering applications. Full article

(This article belongs to the Special Issue Advances in Fractional Order Models and Applications)

13 pages, 1141 KB

Open AccessArticle

Self-Adaptive Limited-Memory Quasi-Newton Method with Function Value Information for Large-Scale Unconstrained Optimization

by Jiangwen Ju, Weixin Lin and Hao Liu

Mathematics 2026, 14(10), 1750; https://doi.org/10.3390/math14101750 - 19 May 2026

Abstract

We extend the modified BFGS algorithm to a limited-memory framework, and propose a self-adaptive limited-memory quasi-Newton method, denoted as LADBFGS, for large-scale unconstrained optimization. The proposed method fully exploits function value information to improve the curvature approximation of the objective function, while enabling [...] Read more.

We extend the modified BFGS algorithm to a limited-memory framework, and propose a self-adaptive limited-memory quasi-Newton method, denoted as LADBFGS, for large-scale unconstrained optimization. The proposed method fully exploits function value information to improve the curvature approximation of the objective function, while enabling dynamic and adaptive adjustment of parameters. We establish the global R-linear convergence of the proposed algorithm for uniformly convex problems. Numerical experiments on 102 standard unconstrained test functions with dimensions of no less than 1000 show that the proposed LADBFGS method outperforms the standard limited-memory BFGS method in terms of iteration count, number of function and gradient evaluations, and computational time, and also achieves a higher success rate for solving the test problems. Full article

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

21 pages, 379 KB

Open AccessArticle

On the Structural Solvability of MLWE with Rank-Deficient Public Matrices

by Nor Siti Khadijah Arunah, Amir Hamzah Abd Ghafar, Muhammad Asyraf Asbullah and Muhammad Rezal Kamel Ariffin

Mathematics 2026, 14(10), 1749; https://doi.org/10.3390/math14101749 - 19 May 2026

Abstract

The security of Module Learning With Errors (MLWE) relies on the assumption that the public matrix is sampled uniformly and forms a full-rank operator. In this work, we examine the structural consequences of relaxing this assumption by considering public matrices that demonstrate slot-wise [...] Read more.

The security of Module Learning With Errors (MLWE) relies on the assumption that the public matrix is sampled uniformly and forms a full-rank operator. In this work, we examine the structural consequences of relaxing this assumption by considering public matrices that demonstrate slot-wise rank deficiency under the Number Theoretic Transform (NTT). Focusing on the case where each NTT slot matrix has rank 1, we show that this leads to enlarged left nullspace, which allows the elimination of the secret component

s_{1}

, reducing the original relation to a linear system consisting only of

s_{2}

. Given partial knowledge of

s_{2}

, this projected system admits a unique solution once a sufficient number of independent constraints is available. After recovering

s_{2}

, the problem of determining

s_{1}

reduces to solving a bounded linear system, which can be viewed as a structured instance of the Short Integer Solution (SIS) problem. These results provide a dimension-based characterization of solvability under slot-wise rank-deficient public matrices. Using ML-DSA as a concrete instantiation, we illustrate how such structural deviations affect the behavior of the system and discuss simple safeguards, such as rank verification during key generation, to mitigate these issues. Full article

(This article belongs to the Special Issue Advances in Mathematics Cryptography and Information Security)

19 pages, 1765 KB

Open AccessArticle

SetConv++: Point Cloud Scene Flow Estimation Constrained by Feature Self-Supervision

by Fei Zhang, Yinghui Wang, Yang Xi and Chunhao Hua

Mathematics 2026, 14(10), 1748; https://doi.org/10.3390/math14101748 - 19 May 2026

Abstract

Point cloud scene flow estimation aims to capture the three-dimensional motion of each point in a sequence of point clouds. Although progress has occurred in this field, existing methods often face significant challenges. In particular, two key issues persist: the absence of corresponding [...] Read more.

Point cloud scene flow estimation aims to capture the three-dimensional motion of each point in a sequence of point clouds. Although progress has occurred in this field, existing methods often face significant challenges. In particular, two key issues persist: the absence of corresponding local information from the source point cloud to the target, preventing correct feature matching, and the presence of highly similar adjacent structures in target regions, which leads to ambiguous correspondences due to indistinguishable point features. To address these problems, this paper introduces a novel self-supervised method for point cloud scene flow estimation. Theoretically, we establish a new framework that integrates discriminative feature learning with probabilistic flow refinement. A new network architecture, SetConv++, is designed to learn more discriminative point feature representations, enhancing differentiation in similar structures. Additionally, a refinement module uses the random walk algorithm to adjust initial flow estimates. This approach reconstructs low-confidence flows with high-confidence surrounding ones, reducing missing correspondence issues. Crucially, a new flow smoothing loss term ensures local consistency while suppressing error propagation—a fundamental limitation in existing methods. Through comprehensive experiments on the KITTI Scene Flow dataset, our method demonstrates superior performance. It significantly outperforms existing self-supervised approaches across multiple standard evaluation metrics. Specifically, on the KITTI Scene Flow dataset, our method reduces the Endpoint Error (EPE) by 13.6% (from 0.0411 to 0.0355) and improves Accuracy Strict (AS) by 2.43 percentage points (from 92.68% to 95.11%) compared to baseline self-supervised approaches, while also reducing the outlier rate (Out) by 1.5 percentage points. This advancement not only provides a robust theoretical framework for handling ambiguous correspondences but also enables more reliable and efficient downstream applications—such as autonomous driving perception systems requiring real-time motion accuracy in complex scenes. Full article

27 pages, 1461 KB

Open AccessArticle

A Legendre Spectral Operational Matrix Method with Convergence Analysis for Two-Dimensional Integro-Differential Equations

by Ishtiaq Ali

Mathematics 2026, 14(10), 1747; https://doi.org/10.3390/math14101747 - 19 May 2026

Abstract

In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the physical domain onto a reference square and approximates the unknown solution using a [...] Read more.

In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the physical domain onto a reference square and approximates the unknown solution using a tensor-product Legendre polynomial expansion. Exact operational matrices for differentiation and lower-limit integration are constructed, allowing the original integro-differential problem to be reduced systematically to a finite-dimensional algebraic system for the spectral coefficients. The formulation provides a unified treatment of differential, Volterra, and Fredholm operators within a single spectral framework and avoids complicated discretizations of multidimensional integral terms. For a specialized linear form of the problem, rigorous convergence estimates are established in both

L^{2}

and

L^{\infty}

norms under suitable regularity assumptions on the coefficients and kernels. The analysis shows that the dominant convergence behavior is governed by the differential operator, while the integral terms contribute only higher-order consistency effects. Several benchmark examples involving both linear and nonlinear two-dimensional integro-differential equations are presented to demonstrate the performance of the proposed method. Numerical results exhibit rapid spectral-type error decay as the polynomial degree increases, with the numerical errors approaching machine precision for moderate truncation orders. These results confirm the accuracy, efficiency, and reliability of the proposed Legendre spectral operational matrix framework for solving a broad class of multidimensional integro-differential equations with nonlocal operators. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis and Approximation)

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45 pages, 28274 KB

Open AccessArticle

Efficiency and Stability of a New Hybrid Unconstrained Optimization Algorithm with Quasi-Newton Updates and Higher-Order Methods

by Alicia Cordero, Javier G. Maimó, Juan R. Torregrosa and Natanael Ureña Castillo

Mathematics 2026, 14(10), 1746; https://doi.org/10.3390/math14101746 - 19 May 2026

Abstract

We propose the higher-order quasi-Newton (HOQN) method, a hybrid algorithm for unconstrained optimization that combines Newtonian predictors with higher-order correctors derived from vector extensions of the Traub, Chun, and Ostrowski methods, along with quasi-Newton updates of the inverse Hessian using Broyden–Fletcher–Goldfarb–Shanno (BFGS) or [...] Read more.

We propose the higher-order quasi-Newton (HOQN) method, a hybrid algorithm for unconstrained optimization that combines Newtonian predictors with higher-order correctors derived from vector extensions of the Traub, Chun, and Ostrowski methods, along with quasi-Newton updates of the inverse Hessian using Broyden–Fletcher–Goldfarb–Shanno (BFGS) or Davidon–Fletcher–Powell (DFP) formulas. We demonstrate that the resulting scheme achieves cubic local convergence order, representing a substantial improvement over the superlinear convergence typical of classical quasi-Newton methods, while maintaining a cost of

O (n^{2})

per iteration. We also analyze variants that incorporate two successive quasi-Newton updates, and show that they retain the same cubic order. Numerical experiments with the benchmark functions of Himmelblau and Freudenstein–Roth confirm the theoretical convergence order and show that the hybrid variants consistently require fewer iterations than BFGS, DFP, and Symmetric Rank-One (SR1). In the case of the Booth function, given its strictly convex quadratic structure, the proposed hybrid methods reach the global minimum in just two iterations and exhibit numerical accuracy superior to that of classical quasi-Newton methods. In addition, limited-memory variants (L-HOQN) are introduced; these are evaluated during the training of a convolutional neural network on the MNIST dataset, where they achieve test accuracies exceeding

99 %

and outperform L-BFGS and standard stochastic gradient descent (SGD) at all tested learning rates. Full article

25 pages, 528 KB

Open AccessArticle

A Break-Regime Score-Driven Model for Tail-Risk Forecasting in China’s Carbon Market Under Policy Shifts

by Xinshu Gong and Bin Zheng

Mathematics 2026, 14(10), 1745; https://doi.org/10.3390/math14101745 - 19 May 2026

Abstract

Accurate tail-risk measurement in carbon markets is challenging because carbon allowance prices are shaped not only by heavy-tailed return dynamics, but also by policy changes that can alter the underlying risk dynamics. Models that ignore such structural shifts may perform reasonably well in [...] Read more.

Accurate tail-risk measurement in carbon markets is challenging because carbon allowance prices are shaped not only by heavy-tailed return dynamics, but also by policy changes that can alter the underlying risk dynamics. Models that ignore such structural shifts may perform reasonably well in normal periods while still understating downside risk when market conditions change. To address this issue, this paper proposes a break-regime generalized autoregressive score model with Student-t innovations, denoted as BR-GAS-t, for one-step-ahead forecasting of Value-at-Risk and Expected Shortfall. Using daily spot data from China’s carbon market, we compare BR-GAS-t with historical simulation, GARCH-N, GARCH-t, and regime-free GAS-t benchmarks. The results show that carbon returns are strongly heavy-tailed and that the post-break regime is characterized by stronger shock sensitivity, lower persistence, and a higher long-run conditional scale. Out-of-sample evidence further indicates that BR-GAS-t delivers the strongest overall VaR backtesting performance and the lowest average Fissler–Ziegel (FZ) loss in joint VaR–ES evaluation. Its advantage is most pronounced at the 2.5% and 1% tails, where downside risk is hardest to forecast. Robustness checks based on alternative break dates, window lengths, recursive schemes, and distributional assumptions confirm that the main conclusion is stable. The findings suggest that explicitly incorporating observed policy breaks improves tail-risk forecasting in policy-driven carbon markets. Full article

(This article belongs to the Special Issue Mathematical Modelling in Financial Economics)

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