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An Eigenvector Problem in Walsh-Fourier Analysis
Certain Mathematical Constants Associated with Harmonic Numbers and Higher-Dimensional Harmonic Sums
Finite-Series Solutions of Hybrid PDE Systems for Conditional Moments of Regime-Switching Extended CEV Processes with Applications in Finance

Journal Description

Mathematics

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

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), RePEc, and other databases.
Journal Rank: JCR - Q1 (Mathematics) / CiteScore - Q1 (General Mathematics )
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 17.3 days after submission; acceptance to publication is undertaken in 2.8 days (median values for papers published in this journal in the second half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Sections: published in 16 topical sections.
Companion journals for Mathematics include: Foundations, Analytics, International Journal of Topology, Geometry and Logics.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 2.3 (2025); 5-Year Impact Factor: 2.1 (2025)

Imprint Information Journal Flyer Open Access ISSN: 2227-7390

Latest Articles

26 pages, 6705 KB

Open AccessArticle

Intelligent Analysis of the Geomechanical State of Rock Masses During Underground Mining

by Dmytro Babets, Amirbek Yerkinbekov, Serik Moldabayev, Samal Assylkhanova, Volodymyr Hnatushenko and Olena Sdvyzhkova

Mathematics 2026, 14(12), 2222; https://doi.org/10.3390/math14122222 (registering DOI) - 20 Jun 2026

This study presents an intelligent framework for the analysis of multidimensional geomechanical states in underground mining systems based on numerical simulation and machine learning methods. A three-dimensional geomechanical model of the Zholymbet deposit was developed in the RS3 environment using the generalized Hoek–Brown [...] Read more.

This study presents an intelligent framework for the analysis of multidimensional geomechanical states in underground mining systems based on numerical simulation and machine learning methods. A three-dimensional geomechanical model of the Zholymbet deposit was developed in the RS3 environment using the generalized Hoek–Brown failure criterion. Numerical simulations were performed for representative mining scenarios characterized by complex excavation interaction and stress redistribution. The modelling results were transformed into a multidimensional geomechanical dataset containing stress, deformation, displacement, and yielding parameters. Principal component analysis (PCA) was applied to investigate the internal structure of the geomechanical state space and identify dominant patterns controlling the rock mass behavior. Clustering analysis revealed several geomechanical regimes corresponding to stable, transitional, and instability-prone conditions. Isolation Forest anomaly detection demonstrated that atypical geomechanical states are not randomly distributed but spatially localized near excavation systems and mining horizons. The obtained results indicate that hazardous geomechanical conditions are governed by complex interactions between stress concentration, deformation intensity, yielding processes, and excavation geometry. The proposed approach provides a basis for intelligent interpretation of large-scale numerical modelling results and may support geomechanical risk assessment in underground mining operations. Full article

(This article belongs to the Special Issue Advances in Numerical Computation and Mathematical Modelling for Mechanics and Dynamics in Geotechnical Engineering)

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22 pages, 2034 KB

Open AccessFeature PaperArticle

Fixed-Point Analysis of Supra-Contractions with Applications to Nonlinear Economic Systems

by G. Sudhaamsh Mohan Reddy, Lateef Ahmad Wani, Mudasir Younis and Saiful R. Mondal

Mathematics 2026, 14(12), 2221; https://doi.org/10.3390/math14122221 (registering DOI) - 20 Jun 2026

In this article, we construct a framework for analyzing the equilibrium and stability of networked multi-sector economic systems via fixed-point analysis. We represent directional intersectoral dependencies, nonlinear feedback effects, and heterogeneous adjustment dynamics in the model by the coupled and tripled fixed-point theory [...] Read more.

In this article, we construct a framework for analyzing the equilibrium and stability of networked multi-sector economic systems via fixed-point analysis. We represent directional intersectoral dependencies, nonlinear feedback effects, and heterogeneous adjustment dynamics in the model by the coupled and tripled fixed-point theory in the graphically extended suprametric spaces. The graphical structure encodes supply-chain and influence networks, whereas asymmetric and nonuniform interaction strengths are encoded in the suprametric setting. Furthermore, we prove the existence, uniqueness, and convergence of equilibrium solutions under new generalized contraction conditions. We apply the theoretical findings in nonlinear state systems in which prices in interdependent markets are adjusted using integral equations. The results of numerical simulations show consistent convergence, and the sensitivity parameter of the network structure significantly influences the determination of economic stability and speed of adjustment. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Applications)

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20 pages, 800 KB

Open AccessArticle

Multi-Objective Just-in-Time Permutation Flow Shop: Tools for Analysis of Different Conflict Scenarios

by Nícolas Samuel Assis, Socorro Rangel and Helio Yochihiro Fuchigami

Mathematics 2026, 14(12), 2220; https://doi.org/10.3390/math14122220 (registering DOI) - 20 Jun 2026

Permutation flow shop scheduling is an important production planning problem handled in different contexts. Just-in-time measures have been significant in the optimization of real problems and one is specifically addressed here: the total earliness and tardiness of jobs. The most used approach in [...] Read more.

Permutation flow shop scheduling is an important production planning problem handled in different contexts. Just-in-time measures have been significant in the optimization of real problems and one is specifically addressed here: the total earliness and tardiness of jobs. The most used approach in the literature to mathematically express this measure is to sum them up using unit weights thus obtainning a mono-objective function. In this paper it is shown that this is a simplification of a problem that is inherently multi-objective, highlighting how a more comprehensive approach can better support decision-making. A bi-objective mathematical optimization model and tools capable of analyzing the mono-objective solution within the multi-objective perspective are proposed. A computational study to analyze the benefits and difficulties of the solution using the bi-objective approach is presented. The results show that for large-scale instances in which the tardiness factor is small, the conflict between the objectives of minimizing the total earliness and minimizing the total tardiness of jobs increases significantly. Specifically, the mono-objective solution is unbalanced in 50.00% of the analyzed instance structures. However, in 48.12% of the instances, alternative Pareto-optimal trade-offs can be achieved with zero increase to the mono-objective optimal value. Therefore, the multi-objective approach has a greater potential to support decision-makers. Furthermore, we show that the choice of the solution method must be carefully considered, since the Pareto frontier associated with most instances has many non-supported points, representing up to 66.71% of the non-dominated set. Full article

(This article belongs to the Special Issue Innovations in Optimization and Operations Research, 2nd Edition)

14 pages, 1103 KB

Open AccessArticle

Statistical Properties of Rosenthal’s Fail-Safe Number in Meta-Analysis

by Vanusa Rocha, Miguel Felgueiras and Vera Afreixo

Mathematics 2026, 14(12), 2219; https://doi.org/10.3390/math14122219 (registering DOI) - 20 Jun 2026

Rosenthal’s fail-safe number is widely used to assess the robustness of meta-analysis results against publication bias; however, its statistical properties remain insufficiently understood. This paper re-evaluates the coverage performance of confidence intervals for the Rosenthal’s fail-safe number using an updated simulation framework that [...] Read more.

Rosenthal’s fail-safe number is widely used to assess the robustness of meta-analysis results against publication bias; however, its statistical properties remain insufficiently understood. This paper re-evaluates the coverage performance of confidence intervals for the Rosenthal’s fail-safe number using an updated simulation framework that incorporates zero truncation, an epsilon correction to the expected value, and a restriction to statistically significant meta-analyses. In addition to the standard normal bootstrap approximation, bias-corrected and accelerated bootstrap confidence intervals are considered. Simulation results show that standard bootstrap intervals tend to be conservative under symmetric settings and exhibit substantial deviations under asymmetric distributions. The bias-corrected and accelerated bootstrap method improves coverage accuracy, particularly under asymmetry and moderate sample sizes, although both methods exhibit conservative behavior in several scenarios. Overall, reliable inference for the fail-safe number depends on both appropriate parameter specification and bootstrap procedures that account for bias and asymmetry. Full article

(This article belongs to the Special Issue Advances in Statistics, Biostatistics and Medical Statistics)

43 pages, 6818 KB

Open AccessFeature PaperArticle

The Geometry of Quantum Walks on Graphs—Theory and Applications

by Ernesto Estrada

Mathematics 2026, 14(12), 2218; https://doi.org/10.3390/math14122218 (registering DOI) - 20 Jun 2026

We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex [...] Read more.

We introduce a geometric framework for continuous-time quantum walks on graphs by embedding each vertex into a Euclidean space through its time-dependent quantum probability distribution. This construction induces a rich geometry in which quantum transport is characterized by distances, radii, angles, and simplex volumes, allowing interference, localization, and spreading to be analyzed within a unified metric-angular formalism. We prove that, in contrast to classical diffusion, which collapses to a spherical geometry, quantum dynamics generate a generically non-spherical affine geometry with persistent anisotropy. Applying this theory to real-world networks—including transportation systems, semantic graphs, and neuronal connectomes—we show that quantum geometry reveals dynamically meaningful backbones, interference-based “communities”, and vulnerability structures that are invisible to classical random-walk and spectral methods. In particular, angular and radial quantum descriptors isolate functional hubs, control cores, and coherence classes without any topological or dimensionality assumptions. Together, these results demonstrate that quantum-walk-induced geometry provides a powerful new lens for understanding structure and function in complex networks. Full article

(This article belongs to the Special Issue Information-Induced Geometries in Statistics, Data Analysis, and Applied Research)

17 pages, 338 KB

Open AccessArticle

Multi-Criteria Financial Screening Under Data Uncertainty: An LLM-Extraction and Min–Max TOPSIS Approach for SMEs

by Vinicius Minatogawa, Mitsuyoshi Fukushi, Jose Garcia, Jorge Rojas, Jose Gornall, Alfredo Angulo and Jefferson Pinto

Mathematics 2026, 14(12), 2217; https://doi.org/10.3390/math14122217 (registering DOI) - 20 Jun 2026

Small and medium enterprises routinely face a paradox in financial monitoring: their accounting documents exist, but the cost of converting heterogeneous PDFs into timely financial signals is prohibitive without dedicated analytical staff or specialized software. This paper presents a two-layer artifact, designed under [...] Read more.

Small and medium enterprises routinely face a paradox in financial monitoring: their accounting documents exist, but the cost of converting heterogeneous PDFs into timely financial signals is prohibitive without dedicated analytical staff or specialized software. This paper presents a two-layer artifact, designed under Design Science Research, that bridges this gap using only public-web large language models (LLMs) and a parsimonious multi-criteria decision routine. Layer 1 implements a structured LLM-driven workflow that extracts account–value pairs from annual tax balance sheets without code, APIs, or fine-tuning. Layer 2 reconstructs auditable accounting aggregates and ranks yearly financial condition through TOPSIS with min–max normalization—a deliberate replacement for classical vector normalization, which fails when profitability indicators are negative, as routinely occurs in distress years. To avoid size effects and algebraic redundancy, the decision matrix uses only three criteria spanning liquidity, profitability, and solvency. The artifact is demonstrated in a four-year case study of an anonymized construction SME (2021–2024), with accountant-verified document-level match rates of 0.810, 0.998, 0.950, and 0.909. Equal weighting is the only weighting configuration used; a supplementary entropy-based dispersion diagnostic yields the same ordinal ranking—2024 > 2023 > 2021 > 2022—and 10,000 Monte Carlo replications, with uncertainty injected at the reconstructed-aggregate level, confirm that the extreme ranks are invariant across all runs. The contribution is methodological and practical: a transparent, low-infrastructure pipeline that brings first-pass financial screening within reach of SMEs operating under severe data and budget constraints. Full article

(This article belongs to the Special Issue Applications of Mathematics Analysis in Financial Marketing)

29 pages, 35250 KB

Open AccessArticle

Optimal Sensor Placement Based on Fisher Information Matrix and Improved Particle Swarm Optimization Algorithm for Typical Tensile Membrane Structures

by Qiu Yu, Xin Zhang, Zhiyang Jia and Chen Peng

Mathematics 2026, 14(12), 2216; https://doi.org/10.3390/math14122216 (registering DOI) - 20 Jun 2026

Large-amplitude and long-term vibration deformation under external environmental loads often occurs on tensile membrane structures. Proper sensor placement plays a vital role in effectively achieving the objectives of a structural health monitoring system. In order to optimize the sensor placement to identify the [...] Read more.

Large-amplitude and long-term vibration deformation under external environmental loads often occurs on tensile membrane structures. Proper sensor placement plays a vital role in effectively achieving the objectives of a structural health monitoring system. In order to optimize the sensor placement to identify the modal vibration parameters for tensile membrane structures, this paper proposes an optimal sensor placement method based on the Fisher information matrix (FIM) and improved random strategy discrete particle swarm optimization algorithm (IRSDPSO). Firstly, the structural modal order is selected by using the two-norm difference and trace change rate of FIM, and the number of sensors is determined based on the QR decomposition and MAC criterion. Secondly, an improved particle swarm optimization algorithm named IRSDPSO, which has the discrete characteristic, is proposed to arrange the placement of sensors. Finally, the convergence, stability and sensitivity are used to evaluate the effectiveness of optimal sensor placement results using a numerical modal test example of the plane bidirectional tensile membrane structure. The results show that the first nineteen modal frequencies can be accurately identified. This indicates that the proposed optimal sensor placement method can determine the number of sensors and arrange the placement of the sensors. The work is reasonable and feasible in the optimal sensor placement for the tensile membrane structure. Full article

(This article belongs to the Special Issue Advances in Optimization and Soft Computing Methodologies with Applications)

21 pages, 333 KB

Open AccessArticle

Quillen–Suslin Theorem for Connected Cochain DG Algebras

by Xuefeng Mao and Biyan Zhu

Mathematics 2026, 14(12), 2215; https://doi.org/10.3390/math14122215 (registering DOI) - 20 Jun 2026

Let

A

be a connected cochain differential graded algebra and P a finitely generated differential graded

A

-module. We show that P is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considered as [...] Read more.

Let

A

be a connected cochain differential graded algebra and P a finitely generated differential graded

A

-module. We show that P is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considered as a generalization of the well-known Quillen–Suslin Theorem in differential graded context. As an application, we show that the ghost length and the cone length of a compact differential graded module coincide. Full article

19 pages, 6096 KB

Open AccessArticle

A Novel Hybrid Modeling Framework Integrating Feature Engineering for Battery Remaining Useful Life Prediction

by Ru Xiao, Jiyang Xu and Jiabo Li

Mathematics 2026, 14(12), 2214; https://doi.org/10.3390/math14122214 (registering DOI) - 20 Jun 2026

Accurate remaining useful life (RUL) prediction is critical for the reliable operation of lithium-ion batteries. Traditional data-driven methods often suffer from parameter redundancy and error accumulation in state prediction. This paper proposes a hybrid data-driven RUL prediction framework based on Gaussian process regression [...] Read more.

Accurate remaining useful life (RUL) prediction is critical for the reliable operation of lithium-ion batteries. Traditional data-driven methods often suffer from parameter redundancy and error accumulation in state prediction. This paper proposes a hybrid data-driven RUL prediction framework based on Gaussian process regression (GPR) optimized by the lightning search algorithm (LSA). First, both local and global indirect health features (HFs) are extracted from the external characteristic parameter curves and the incremental capacity curves during battery charging/discharging. Second, the Pearson correlation coefficient is applied to select highly relevant features, forming a compact feature set. Third, a GPR model is developed, and the LSA is introduced to optimize its hyperparameters, overcoming the tendency of the conjugate gradient method to fall into local optima or fail to converge. Experimental results under identical conditions show that the proposed LSA–GPR model achieves a prediction error of 3% or less. Full article

(This article belongs to the Special Issue AI-Enabled Prognostics and Health Management: Mathematical Aspect and Engineering Applications)

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23 pages, 1466 KB

Open AccessArticle

A Spreadsheet Environment for Force, Torque and Strength of Materials Modeling: Bridging Analytical Mathematics and Engineering Practice

by Elisa Munich, Jérémie Schutz, Christophe Sauvey and Yves Gillet

Mathematics 2026, 14(12), 2213; https://doi.org/10.3390/math14122213 (registering DOI) - 19 Jun 2026

This paper presents and validates a unified spreadsheet-based framework for engineering mechanics education and preliminary design. Three modules are integrated within a single openly available workbook: multi-point resultant force and moment computation; axial normal stress with stress concentration effects for three geometric configurations [...] Read more.

This paper presents and validates a unified spreadsheet-based framework for engineering mechanics education and preliminary design. Three modules are integrated within a single openly available workbook: multi-point resultant force and moment computation; axial normal stress with stress concentration effects for three geometric configurations (plate with hole, shoulder plate, stepped shaft); and beam deflection for simply supported and cantilever configurations under point loads. All governing equations are implemented as explicit closed-form expressions validated against analytical reference solutions for six independent cases; relative errors fall below

10^{- 10}

in all cases. Three worked exercises demonstrate the practical scope of the framework. A biomechanical multi-point force system yields joint moments of

- 6880

, −33,421, and −58,241 N·mm at the wrist, elbow, and shoulder, respectively. A tensile shoulder plate with

K_{t} \approx 1.85

produces

σ_{max} = 232

MPa against

σ_{y} = 200

MPa, identifying a design failure; a parametric redesign with fillet radius

r = 10

mm reduces

K_{t}

to approximately 1.59 and

σ_{max}

to approximately 198.7 MPa, restoring structural safety. A cantilever beam subjected to a 20,000 N tip load yields a maximum deflection of 13,133 μm. The framework constitutes a validated intermediate layer between manual analytical derivations and high-fidelity numerical simulations, applicable to preliminary design, parametric sensitivity studies, and engineering education at the linear elastic level. Full article

(This article belongs to the Special Issue Modeling and Simulation in Engineering, 4th Edition)

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12 pages, 287 KB

Open AccessArticle

Geometric Structures and Inclusion Properties of Multivalent Mittag-Leffler-Type Poisson Subfamilies

by Feras Yousef, Tariq Al-Hawary and Ibtisam Aldawish

Mathematics 2026, 14(12), 2212; https://doi.org/10.3390/math14122212 (registering DOI) - 19 Jun 2026

The interplay between special functions and geometric function theory continues to inspire significant advances in the study of analytic and multivalent functions. In this work, we introduce and investigate several new subfamilies of multivalent functions associated with the generalized Mittag-Leffler-type Poisson distribution in [...] Read more.

The interplay between special functions and geometric function theory continues to inspire significant advances in the study of analytic and multivalent functions. In this work, we introduce and investigate several new subfamilies of multivalent functions associated with the generalized Mittag-Leffler-type Poisson distribution in the open unit disk. We establish necessary and sufficient conditions characterizing membership in these classes and derive meaningful inclusion relationships among them. Furthermore, we define a novel integral operator linked to the generalized Mittag-Leffler-type Poisson distribution and examine its mapping properties and structural connections with the proposed function classes. The results presented herein not only unify and extend a variety of earlier contributions but also demonstrate the effectiveness of distribution-theoretic methods in the analysis of multivalent functions. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions, 2nd Edition)

17 pages, 1454 KB

Open AccessArticle

A Unified Constant-Time Switch Rule for Constructing Edge-Disjoint Hamiltonian Cycles in Gaussian Networks

by Bader Albader

Mathematics 2026, 14(12), 2211; https://doi.org/10.3390/math14122211 (registering DOI) - 19 Jun 2026

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that, when the generator

α = a + b i

satisfies

gcd (a, b) = 1

, the real and imaginary dimensions directly [...] Read more.

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that, when the generator

α = a + b i

satisfies

gcd (a, b) = 1

, the real and imaginary dimensions directly form two edge-disjoint Hamiltonian cycles. A later construction extended the result to the non-coprime case

gcd (a, b) = d > 1

, but its proof relied on long node-sequence tables and separate odd/even cases for d. This paper presents a unified closed-form construction that covers both

d = 1

and

d > 1

, and both odd and even d, without separate case tables. In the rectangular representation with d rows and

r = (a^{2} + b^{2}) / d

columns, the construction uses a constant-time local switch rule, meaning constant time per individual switch, for each

q = 1, 2, \dots, d - 1

at column

a_{q} = q - 1

. Each switch removes two horizontal edges and inserts two vertical edges. The switched horizontal structure forms the first Hamiltonian cycle, while its edge-complement in the Gaussian network forms the second Hamiltonian cycle. Thus, the full edge set is partitioned into two edge-disjoint Hamiltonian cycles. The construction requires

O (d)

switch-generation time and

O (N)

time to list the two cycles, where

N = a^{2} + b^{2}

. Exhaustive validation for all

1 \leq a \leq b \leq 100

, excluding only the degenerate

N = 2

network, and large-scale validation up to

N = 3, 250, 000

confirm implementation correctness and demonstrate practical scalability. Full article

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

32 pages, 3092 KB

Open AccessReview

A Review on Deep State Space Models for Sequential Healthcare Data Prediction

by Wenjie Li, Yongming Xie and Yinglong Dai

Mathematics 2026, 14(12), 2210; https://doi.org/10.3390/math14122210 (registering DOI) - 19 Jun 2026

Sequential data prediction is a crucial area in healthcare. Healthcare data have the characteristics of non-stationarity, long-range dependence (LRD), and irregular sampling. Modeling these complex temporal features is highly challenging. Recurrent Neural Networks (RNNs) and their variants are limited in learning long-range dependencies [...] Read more.

Sequential data prediction is a crucial area in healthcare. Healthcare data have the characteristics of non-stationarity, long-range dependence (LRD), and irregular sampling. Modeling these complex temporal features is highly challenging. Recurrent Neural Networks (RNNs) and their variants are limited in learning long-range dependencies (LRDs) due to the inherent issues of vanishing and exploding gradients. Transformers alleviate this limitation by using the self-attention mechanism. Its quadratic computational complexity and memory bottleneck limit its scalability in long-range healthcare data. In this context, Structured State Space Models (SSMs) have emerged as a promising alternative. Compared with conventional RNNs, they can alleviate the difficulty of modeling LRDs more efficiently, and many modern SSM variants achieve linear time sequence modeling while reducing the computational burden associated with Transformers. In this review, we provide a formal definition of Healthcare Process Modeling, compare the core theoretical frameworks of RNNs, Transformers, and SSMs, trace the architectural evolution of SSM architectures, and provide a comprehensive review of healthcare applications and open challenges, including LSSL, S4, S5, Mamba, and their related variants. Existing studies suggest that structured SSMs are promising for selected long-sequence healthcare prediction tasks, particularly when computational efficiency and long-context retention are important. With these advantages, they may help alleviate the computational burden in certain healthcare tasks and provide a basis for further exploring the practical application of data-driven healthcare systems in clinical practice. Full article

(This article belongs to the Topic The Future of Artificial Intelligence: Trends, Challenges, and Developments)

31 pages, 452 KB

Open AccessArticle

A Banach-Space Framework for Proposed (v,w)–s–Convex Response-Curve Certification in Machine Learning

by Ahad Hamoud Alotaibi, Muhammad Saeed Ahmad, Muhammad Waseem Asghar and Mujahid Abbas

Mathematics 2026, 14(12), 2209; https://doi.org/10.3390/math14122209 (registering DOI) - 19 Jun 2026

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed

(v, w)

–s [...] Read more.

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed

(v, w)

–s–convex response-curve certification. The response curve is treated as an element of the Banach space of continuous functions under the supremum norm, while derivative-based certificates are handled in a Lipschitz and Sobolev-type norm when required. Generalized convexity is represented through a bounded structural operator, whose order condition defines a closed convex acceptance set. The violation score is measured by the positive part of the operator residual, and the Hermite–Hadamard, Fejér, and Ostrowski quantities are interpreted as bounded certificate functionals. The auxiliary profiles are constructed from validation-curve residuals through a split-calibrated procedure and then tested on held-out triples. The framework certifies only scalar response-curve summaries under explicit structural and empirical assumptions; it does not certify a full learning system, guarantee generalization, or replace dense sampling when the structural gate fails. Full article

38 pages, 2010 KB

Open AccessReview

Beyond Neural Solvers: A Critical Review of Machine Learning for Combinatorial Optimization

by Mostafa E. A. Ibrahim, Alaa E. S. Ahmed and Yassine Daadaa

Mathematics 2026, 14(12), 2208; https://doi.org/10.3390/math14122208 (registering DOI) - 19 Jun 2026

Combinatorial optimization is a key component in critical decision problems such as routing, scheduling, network design, and graph optimization. Although combinatorial optimization methods, including exact algorithms, approximation methods, constraint programming, mixed integer programming, and metaheuristics, are widely available, they often face obstacles, such [...] Read more.

Combinatorial optimization is a key component in critical decision problems such as routing, scheduling, network design, and graph optimization. Although combinatorial optimization methods, including exact algorithms, approximation methods, constraint programming, mixed integer programming, and metaheuristics, are widely available, they often face obstacles, such as limited scalability and adaptability in various applications. In this study, a systematic critical review of machine learning for combinatorial optimization is provided to characterize the usage and evaluation of learning-based approaches. A detailed analysis is used to infer and determine findings and limitations. The paper emphasizes how machine learning for computational optimization has changed over time, moving from end-to-end neural solvers to hybrid systems. Learning components are essential for directing, speeding up, or enhancing traditional solver backbones such as constraint programming and metaheuristics in hybrid systems. The review also critically examines current limits that impact performance in general, including scalability, deployment readiness, generalization, and benchmark consistency. Even though using large language models for problem formulation and heuristic synthesis has potential, more work needs to be done to ensure reliable validation. As a conclusion, this article examines recent studies’ findings, emphasizes the growing trend toward hybrid learning-driven optimization frameworks, and underlines important methodological limits and unresolved issues. Full article

(This article belongs to the Special Issue Mathematical Methods and Applications in Signal Analysis, Machine Learning, and Artificial Intelligence)

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26 pages, 1991 KB

Open AccessArticle

The Maximal Almost Sure Lyapunov Exponent of Three-Dimensional Linear Stratonovich Stochastic Differential Equations

by Jianyue Su and Ziying He

Mathematics 2026, 14(12), 2207; https://doi.org/10.3390/math14122207 (registering DOI) - 19 Jun 2026

The sign of the maximal almost sure Lyapunov exponent determines the stability of stochastic systems, while its numerical computation for three-dimensional linear Stratonovich stochastic differential equations remains challenging due to the failure of classical two-dimensional strategies. The spherical angular motion of 3D systems [...] Read more.

The sign of the maximal almost sure Lyapunov exponent determines the stability of stochastic systems, while its numerical computation for three-dimensional linear Stratonovich stochastic differential equations remains challenging due to the failure of classical two-dimensional strategies. The spherical angular motion of 3D systems produces a Fokker–Planck equation with intractable mixed partial derivatives, preventing conventional analytical solutions. This paper develops a unified computational framework for three-dimensional linear Stratonovich stochastic systems using analytical derivation for degenerate cases and physics-informed neural network (PINN) approximation for general non-degenerate scenarios. For degenerate systems, we reduce the coefficient matrix to a lower triangular form via orthogonal transformation and establish tight upper bounds based on the logarithmic growth property of the Wiener process, yielding closed-form expressions for the maximal almost sure Lyapunov exponent under all parameter sign configurations. For non-degenerate systems, we reformulate the Fokker–Planck equation in spherical coordinates and construct a customized PINN with trigonometric encoding to enforce periodic boundary conditions. The network is trained by joint loss functions of equation residuals, boundary constraints and normalization consistency, and the converged stationary density is substituted into the Furstenberg–Khasminskii formula to calculate the exponent via Gauss–Legendre quadrature. Monte Carlo simulations confirm the accuracy and robustness of the proposed method, which reliably identifies the sign of the maximal almost sure Lyapunov exponent even in near-critical regimes. Numerical experiments on a 3D stochastic Hopf bifurcation model show that noise negatively shifts the bifurcation point, with the offset linearly proportional to the squared noise intensity. This work extends Lyapunov stability analysis from two-dimensional to three-dimensional linear Stratonovich stochastic systems, offering an effective tool for stability evaluation of general three-dimensional stochastic dynamical models. Full article

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23 pages, 573 KB

Open AccessFeature PaperArticle

Data-Driven Inventory Policy Assignment in ETO Environments Using Fuzzy K-Prototypes Clustering

by Mario J. Seni Molina and David Peidro Payá

Mathematics 2026, 14(12), 2206; https://doi.org/10.3390/math14122206 - 19 Jun 2026

In engineer-to-order (ETO) manufacturing environments, the high variability of final product configurations makes it difficult to consistently estimate material consumption and, consequently, to define appropriate inventory control policies. This paper proposes a data-driven framework based on unsupervised learning to identify product typologies from [...] Read more.

In engineer-to-order (ETO) manufacturing environments, the high variability of final product configurations makes it difficult to consistently estimate material consumption and, consequently, to define appropriate inventory control policies. This paper proposes a data-driven framework based on unsupervised learning to identify product typologies from historical manufacturing orders in a real industrial context. The approach employs a fuzzy k-prototypes algorithm to cluster mixed-type data, allowing the simultaneous treatment of numerical and categorical variables. In the case study, the proposed crisp-BOM-based scenario achieved a 28.67% reduction in line-side WIP and a 10.79% reduction in linear storage space, corresponding to the release of approximately two to three assembly stations. From the resulting fuzzy memberships, probabilistic bill of materials (BOM) structures are constructed, capturing the inherent variability of material consumption across different product configurations. A defuzzification procedure is then applied to obtain a crisp BOM representation suitable for operational decision-making. Additionally, a material versatility indicator based on entropy is introduced to quantify the dispersion of each material across product typologies. This indicator, together with the estimated consumption per cluster, is used as input for an analytical inventory model that supports the classification of materials into kanban or kitting policies. The methodology is validated using real data from a high- and medium-voltage switchgear manufacturing plant, comprising over 60,000 order–material observations. The results show that the proposed framework enables a more structured characterization of material behavior, reducing reliance on planner experience and improving the consistency of inventory policy decisions. From an industrial perspective, the approach provides a practical and scalable tool for aligning inventory strategies with the actual consumption patterns of ETO systems. Full article

(This article belongs to the Special Issue Mathematical Techniques and New ITs for Smart Manufacturing Systems)

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45 pages, 566 KB

Open AccessReview

Topological Data Analysis: Foundations, Algorithms, and Emerging Applications

by Dimitrios Georgiou, Sotiris Kotsiantis and Fotini Sereti

Mathematics 2026, 14(12), 2205; https://doi.org/10.3390/math14122205 - 19 Jun 2026

Topological data analysis (TDA) has evolved into a flexible and robust paradigm for obtaining qualitative, geometry-inspired insights from high-dimensional, noisy, and complex data. Grounded in algebraic topology, geometry, statistics, and machine learning (ML), TDA provides multiscale descriptions through persistent homology, Mapper (a graph-based [...] Read more.

Topological data analysis (TDA) has evolved into a flexible and robust paradigm for obtaining qualitative, geometry-inspired insights from high-dimensional, noisy, and complex data. Grounded in algebraic topology, geometry, statistics, and machine learning (ML), TDA provides multiscale descriptions through persistent homology, Mapper (a graph-based method that summarizes the shape of high-dimensional data), and related topological signatures that are often inaccessible to standard linear and metric methods. In recent years, and especially during 2024–2025, TDA has expanded rapidly across science, engineering, biomedical research, and socio-economic studies, while also being integrated with modern learning paradigms such as deep learning (DL) and graph learning. This survey summarizes recent developments in TDA using a carefully selected set of articles, with emphasis on 2024–2025. We first present the mathematical and computational foundations of TDA, covering simplicial complexes, filtrations, persistent homology, the Mapper algorithm, and computational advances such as data simplification, stability, and efficiency. We then review applications in time series and dynamical systems, biomedical imaging and precision medicine, engineering and physical sciences, finance and risk analysis, DL and interpretability, and security and critical infrastructure systems. Throughout, we highlight how TDA can extract informative features, function as a model component, and provide a conceptual lens for studying complex systems. However, the survey also emphasizes recurrent failure patterns: TDA performance is highly sensitive to filtration, embedding, and vectorization choices; aggressive simplification can dilute or remove informative topological signals; and integration into standard ML workflows still lacks uniform validation and reporting protocols. We conclude by outlining key challenges—including scalability, statistical foundations, interpretability, and compatibility with rapidly evolving artificial intelligence (AI) paradigms—and by identifying directions for future research. The survey also provides a unifying design perspective for TDA systems, highlighting methodological trade-offs and emerging research directions for integrating topology with modern ML. Full article

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12 pages, 317 KB

Open AccessArticle

A Novel Asymptotic Technique for Integrals Involving the Hankel Contour and the Bleistein Asymptotic Formula

by Athanassios S. Fokas and Jonatan Lenells

Mathematics 2026, 14(12), 2204; https://doi.org/10.3390/math14122204 - 19 Jun 2026

Several important functions, including the gamma function, as well as several infinite sums, admit integral representations involving the Hankel contour. In addition, the large t asymptotic analysis of several recently derived identities satisfied by the Riemann zeta function requires the computation of the [...] Read more.

Several important functions, including the gamma function, as well as several infinite sums, admit integral representations involving the Hankel contour. In addition, the large t asymptotic analysis of several recently derived identities satisfied by the Riemann zeta function requires the computation of the asymptotic form of certain integrals which also involve the Hankel contour; these integrals depend on a real parameter,

α

. A rigorous asymptotic technique is presented here for computing such integrals to all orders. For certain values of

α

, the relevant formula, in addition to an asymptotic series of explicit terms, also contains a specific integral. It is shown that, remarkably, the leading behavior of this integral can be written in the form of the leading order of the Bleistein integral. The latter integral arises in the implementation of the classical steepest descent method in the case that the stationary point coincides with one of the boundary points of the integral under consideration. Full article

(This article belongs to the Section C: Mathematical Analysis)

8 pages, 224 KB

Open AccessArticle

On the Unboundedness of the Number of Natural Solutions for a Parameter-Dependent System of Equations

by Dostonjon Numonjonovich Barotov

Mathematics 2026, 14(12), 2203; https://doi.org/10.3390/math14122203 - 19 Jun 2026

In this paper, we consider a system of 10 equations from the standpoint of the number of its natural solutions, containing a non-negative integer parameter n and describing the magic state of the corresponding special table of numbers. As a result of the [...] Read more.

In this paper, we consider a system of 10 equations from the standpoint of the number of its natural solutions, containing a non-negative integer parameter n and describing the magic state of the corresponding special table of numbers. As a result of the study, it is constructively proven that, for each natural number m, there exist natural numbers

n_{m}

and

s_{m}

such that, for a non-negative integer parameter n equal to

n_{m}

, this system has at least

2^{m}

solutions, and all ten coordinates of each of these solutions are

s_{m}

-digit natural numbers, with the first, ninth, and tenth coordinates in decimal notation being represented only by the digits 0, 8, and 9, and the d-th coordinate,

d \in {2, 3, \dots, 8}

, being represented only by a single digit, equal to

(d - 1)

. This result, which constructively confirms the unboundedness of the number of solutions of this system depending on a non-negative integer parameter n, strengthens some recently published results. Full article

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