Mathematics

22 pages, 1392 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Resilient Lyapunov-Based Model Predictive Control for Wind Power System Under False Data Injection Attacks

by Ningchen Luo and Langwen Zhang

Mathematics 2026, 14(13), 2291; https://doi.org/10.3390/math14132291 - 28 Jun 2026

Viewed by 240

Wind power systems operating in networked environments are vulnerable to stochastic disturbances, measurement noise, model mismatch and false data injection (FDI) attacks. These uncertainties may corrupt feedback information and degrade closed-loop control performance. This paper proposes an integrated extended Kalman filter (EKF)-based resilient [...] Read more.

Wind power systems operating in networked environments are vulnerable to stochastic disturbances, measurement noise, model mismatch and false data injection (FDI) attacks. These uncertainties may corrupt feedback information and degrade closed-loop control performance. This paper proposes an integrated extended Kalman filter (EKF)-based resilient Lyapunov model predictive control (RLMPC) framework for the secure control of wind power systems under bounded stochastic FDI attacks. A residual-based chi-square (

χ^{2}

) detector is embedded into the EKF update to evaluate the credibility of received measurements, and the resulting attack-aware state estimate is applied to the RLMPC controller at each sampling instant, constructing an EKF-RLMPC strategy. The proposed EKF-RLMPC scheme therefore links attack detection, state estimation, and predictive control within a unified secure-control framework for wind power systems. It is proved that the posterior estimation error remains bounded and that the closed-loop state is ultimately bounded under the proposed EKF-RLMPC scheme. Simulation studies under different FDI attack probabilities show that the proposed method improves state-estimation accuracy and control performance. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling, System Identification, and Advanced Control for Complex Engineering Systems)

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11 pages, 276 KB

Open AccessEditor’s ChoiceArticle

On the Supremum of Singleton Ratios in Submodular Functions

by Laszlo Csirmaz

Mathematics 2026, 14(12), 2223; https://doi.org/10.3390/math14122223 - 21 Jun 2026

Viewed by 178

Abstract

Let N be a finite set of cardinality n, and let

a \in N

. A submodular function f on N with

f (a) = 1

is defined to be a-reduced if, for any decomposition [...] Read more.

Let N be a finite set of cardinality n, and let

a \in N

. A submodular function f on N with

f (a) = 1

is defined to be a-reduced if, for any decomposition

f = g + h

into submodular functions, where h does not depend on a, it follows that h is identically zero. The maximal possible value of f on the remaining singletons defines a quantity

λ

that characterizes the degree to which one variable can constrain the value of another; geometrically, it also limits the possible elongation of the associated submodular base polytope. The parameter has concrete relevance: it caps the share-size lower bounds provable for secret-sharing schemes via the basic Shannon inequalities, and it controls the geometry of the base polytopes on which greedy submodular-optimization algorithms operate. We construct an example demonstrating that

λ

can be as large as

Ω (n / log n)

. Furthermore, we establish a doubly exponential upper bound on

λ

. The problem of narrowing the gap between these bounds remains open. Full article

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

12 pages, 328 KB

Open AccessEditor’s ChoiceArticle

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

Viewed by 195

Abstract

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)

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11 pages, 994 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Simulation of Signed Probability Distributions

by Igor Podlubny

Mathematics 2026, 14(12), 2189; https://doi.org/10.3390/math14122189 - 18 Jun 2026

Viewed by 283

Abstract

The notion of negative probability is almost one hundred years old, and so far, some results have been obtained in the direction of the theoretical development of the notion of extended probability. However, there is still a strong need for computational methods and [...] Read more.

The notion of negative probability is almost one hundred years old, and so far, some results have been obtained in the direction of the theoretical development of the notion of extended probability. However, there is still a strong need for computational methods and tools, and the presented article is aimed at filling this gap. Several examples, including Feynman’s problem, of the numerical simulation of signed probability distributions are provided for the first time, along with the results of the simulation using the developed software toolbox. The presented methods and results might open wide possibilities for using signed probabilities in the fields of quantum computing, decision making, finance, insurance, large language models and artificial intelligence, and other fields where the use of signed probability distributions can extend the current level of mathematical modeling. Full article

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14 pages, 19433 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Neighborhood Width Transform: A Structural Stability Framework for Peak Selection in Noisy 1D Coherence Curves

by Sicheng Li and Zhaohui Ye

Mathematics 2026, 14(12), 2156; https://doi.org/10.3390/math14122156 - 16 Jun 2026

Viewed by 242

Abstract

Slowness extraction via Slowness Time Coherence (STC) serves as a fundamental technique for formation evaluation in oil and gas geophysics. Conventional amplitude-dependent peak selection methods often exhibit limitations in complex logging scenarios, including weak wave arrivals, high noise floors, and spurious local maxima. [...] Read more.

Slowness extraction via Slowness Time Coherence (STC) serves as a fundamental technique for formation evaluation in oil and gas geophysics. Conventional amplitude-dependent peak selection methods often exhibit limitations in complex logging scenarios, including weak wave arrivals, high noise floors, and spurious local maxima. To address these challenges, this paper proposes Neighborhood Width Transform (NWT), an unsupervised data-driven mathematical framework that distinguishes genuine peaks from noise by quantifying local neighborhood structural stability rather than relying on amplitude magnitude. The core of NWT lies in a bilateral neighborhood width metric and a minimum-pooling fusion strategy, which suppresses narrow pseudo-peaks effectively. Experimental validation demonstrates that the proposed method outperforms seven representative peak-selection baseline methods (CWT Ridge Analysis, Gaussian Mixture Fitting, AMPD, SG Derivative Crossing, NMS, Random Forest, and 1D-Unet) in terms of detection reliability and accuracy on the tested challenging logging datasets. The proposed method provides an interpretable, high-throughput mathematical solution for automated geophysical signal processing. Full article

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

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10 pages, 273 KB

Open AccessEditor’s ChoiceArticle

On the Resonance Varieties of Vector Bundles on Curves

by Edoardo Ballico

Mathematics 2026, 14(12), 2129; https://doi.org/10.3390/math14122129 - 14 Jun 2026

Viewed by 229

Abstract

We use the resonance (Papadima, Sociu, Aprodu and others) to study vector bundles E on a smooth curve X of genus g. Our key observation is that if we know it for a specific integer c, then we get strong information [...] Read more.

We use the resonance (Papadima, Sociu, Aprodu and others) to study vector bundles E on a smooth curve X of genus g. Our key observation is that if we know it for a specific integer c, then we get strong information on X and E. Fix an integer

c \geq g + 1

. Take genus g curves X and Y and vector bundles E on X and F on Y with the same ranks and degrees. Our main result is that if E and F are sufficiently positive and they have birational resonance in degree c, then X and Y have isomorphic Jacobians. We also study the resonance for a singular curve with arithmetic genus 1. Full article

30 pages, 1949 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Use of Algebra in Genetics: From Phenotype to Genotype

by Ioannis G. Diamataris, Ioanna Maroulakou and Georgios C. Boulougouris

Mathematics 2026, 14(11), 1987; https://doi.org/10.3390/math14111987 - 4 Jun 2026

Cited by 1 | Viewed by 391

Abstract

Understanding how observed phenotype frequencies relate to underlying genetic variation remains a central challenge in population genetics. Traditional approaches are primarily statistical or based on machine learning and often lack a unified analytical framework that explicitly characterizes the space of genotype distributions compatible [...] Read more.

Understanding how observed phenotype frequencies relate to underlying genetic variation remains a central challenge in population genetics. Traditional approaches are primarily statistical or based on machine learning and often lack a unified analytical framework that explicitly characterizes the space of genotype distributions compatible with observed phenotypes. Here, we present an algebraic framework based on linear algebra that analytically relates phenotypic frequencies to compatible genotypic and allelic frequencies. In cases of complete penetrance, the proposed relations between phenotypic and genotypic frequencies are analytical for any possible sample realization whereas in the case of partial penetrance the same relations hold for the average frequency values and become exact as the size of the sample tends to infinity. Using the Moore–Penrose pseudoinverse and a constrained inference strategy, we express all genotypic frequency distributions consistent with observed phenotype data and a given genotype–phenotype mapping. We further introduce a method: Constrained Observation and Null Space-based Inference (CONSPIN), for reconstructing genotype–phenotype relationships from samples that share identical phenotype distributions. Implemented in Python 3.8.18, this approach enables systematic analysis of allele frequencies, genotype–phenotype mappings, and dominance relations, providing a powerful tool for interpreting genetic datasets, including high-throughput sequencing data and complex trait analyses. By explicitly characterizing the constraints imposed by phenotype frequencies on genotype space, this framework offers a new perspective on genetic variation and has potential applications in population genetics, complex trait analysis, and data-driven modeling of biological systems. Full article

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

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33 pages, 517 KB

Open AccessFeature PaperEditor’s ChoiceArticle

From Kernel Matrices to Kernel Functions: An Eigenfunction-Based Approach

by Alberto Muñoz, Aida Torres and Elvira Muñoz García

Mathematics 2026, 14(11), 1971; https://doi.org/10.3390/math14111971 - 3 Jun 2026

Viewed by 476

Abstract

Kernel-combination procedures used in classification often return only a combined kernel matrix on the training sample, rather than a kernel function that can be evaluated consistently at new points. This limitation is especially important for supervised or label-aware combinations, whose entries may depend [...] Read more.

Kernel-combination procedures used in classification often return only a combined kernel matrix on the training sample, rather than a kernel function that can be evaluated consistently at new points. This limitation is especially important for supervised or label-aware combinations, whose entries may depend on training labels and therefore have no immediate out-of-sample meaning. We study the problem of constructing an inductive, finite-rank kernel extension from such empirical matrices. The proposed framework makes the non-uniqueness of this extension explicit: it is determined by empirical coordinates, a positive-semidefinite coefficient matrix, and a continuation model for the coordinates. Experiments on vector, tabular, and relational classification problems give a deliberately diagnostic picture. Smooth direct combinations are stable: on Synthetic, the direct mean gives error

0.0793 \pm 0.0227

, essentially matching the best individual RBF kernel (

0.0809 \pm 0.0231

), and on Telco it remains close to the best individual polynomial kernel (

0.2061 \pm 0.0154

versus

0.2045 \pm 0.0154

). In the controlled Synthetic oracle diagnostic, reconstructing a smooth sum/mean gives relative Frobenius error

4.13 \times 10^{- 6} \pm 9.41 \times 10^{- 6}

and functional MSE at numerical scale. By contrast, abrupt label-aware matrix-only rules are less robust: the Synthetic percentile_inout_auto rule has error

0.1404 \pm 0.1198

, Telco matrix-only supervised rules are around

0.307

–

0.326

error, and the Chickenpieces pickout_auto rule fails under strict out-of-sample reconstruction (

0.3545 \pm 0.2666

error), whereas direct relational combinations match the best individual relational kernel within

10^{- 3}

. Overall, the empirical evidence supports the method as a bridge from finite matrix-level information fusion to deployable kernels, while also identifying abrupt label-aware geometries as the main limitation for stable generalization. Full article

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33 pages, 4302 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Development of a Low-Cost Open-Architecture 2-DOF Shake Table: Design, Modeling, and Control

by Diego Armando Ramírez-Zúñiga, Antonio Concha-Sánchez, Suresh Kumar Gadi, Suresh Thenozhi, Juan Luis Mata-Machuca and Yajaira Concha-Sánchez

Mathematics 2026, 14(11), 1918; https://doi.org/10.3390/math14111918 - 1 Jun 2026

Viewed by 1248

Abstract

This paper presents the mechatronic design, mathematical modeling, parameter identification, and nonlinear position control of an open-architecture biaxial shake table capable of generating base acceleration along two orthogonal horizontal directions. The shake table is tailored for engineering research and education. Addressing the limitations [...] Read more.

This paper presents the mechatronic design, mathematical modeling, parameter identification, and nonlinear position control of an open-architecture biaxial shake table capable of generating base acceleration along two orthogonal horizontal directions. The shake table is tailored for engineering research and education. Addressing the limitations of proprietary “black-box” systems, the platform is constructed using standard industrial components (HLTNC-CNC modules and NEMA 23 BLDC motors) to ensure reproducibility. A core contribution is the characterization of the system’s nonlinear dynamics to enhance tracking fidelity. The mathematical model, derived via the Euler–Lagrange formulation, incorporates viscous and Coulomb friction phenomena, which are critical for accurately reproducing zero-velocity crossings in seismic signals. System parameters are identified using the Recursive Least Squares (RLS) algorithm combined with State Variable Filters (SVFs) to process the regression vector. To enable precise closed-loop performance, a nonlinear state observer incorporating the identified friction dynamics is designed for velocity estimation. Furthermore, a Computed Torque Control (CTC) strategy is synthesized and compared against a conventional Proportional-Velocity (PV) controller. Experimental validations using historical ground motions, including the 1986 Colima earthquake, confirm that the CTC strategy reduces the maximum absolute tracking error by more than 75% compared to the PV approach, bounding the peak error to

0.36 mm

across both axes. Furthermore, in high-amplitude scenarios, the proposed model-based approach achieved an RMS tracking error reduction of more than 83%. These results validate the proposed platform as a reliable and accessible tool for structural dynamics testing. Full article

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98 pages, 1262 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Asymptotic Learning Theory for Conditional U–Statistics Based on Delta Sequences Under Missing at Random Mechanisms

by Salim Bouzebda

Mathematics 2026, 14(11), 1899; https://doi.org/10.3390/math14111899 - 29 May 2026

Cited by 2 | Viewed by 525

Abstract

This article develops a unified asymptotic theory for conditional U-statistics based on delta-sequence smoothing, thereby extending, in a substantial and conceptually coherent manner, the classical kernel-based framework for localized nonlinear conditional functionals. The proposed methodology is formulated in a highly general nonparametric [...] Read more.

This article develops a unified asymptotic theory for conditional U-statistics based on delta-sequence smoothing, thereby extending, in a substantial and conceptually coherent manner, the classical kernel-based framework for localized nonlinear conditional functionals. The proposed methodology is formulated in a highly general nonparametric setting and includes, as particular cases, the estimator of Stute, histogram-type procedures, orthogonal series methods, and a broad family of approximation schemes generated by positive delta sequences. In contrast with the existing literature, the present work explicitly incorporates response missingness under a Missing-at-Random mechanism, a setting of considerable methodological importance in modern statistical inference. Within this incomplete-data framework, we introduce a complete-case conditional U-statistic estimator and establish its asymptotic properties under general smoothness, integrability, and positivity conditions. Our first main contribution is the derivation of non-asymptotic exponential concentration inequalities for the proposed estimator, both in the bounded-kernel case and in the more delicate unbounded regime, with the latter being handled through a conditional Bernstein-type moment assumption. These inequalities provide a sharp probabilistic control of the stochastic fluctuations and constitute a fundamental technical device for the subsequent asymptotic analysis. Our second contribution is the establishment of strong consistency with explicit convergence rates, together with asymptotic normality of the localized estimator. In particular, the analysis makes precise the manner in which smoothing, dimensionality, interaction order, and missingness jointly determine the asymptotic bias and variance structure. The missing-data mechanism enters the limiting theory in a nontrivial yet fully quantifiable way through the observation probabilities, thereby yielding a refined description of the effective loss of information induced by incomplete responses. The scope of the theory is sufficiently broad to cover a wide class of nonlinear statistical functionals arising in discrimination, metric learning, multipartite ranking, conditional dependence analysis, generalized multi-sample U-statistics, and set-indexed conditional inference. To complement the theoretical developments, we conduct an extensive simulation study under several data-generating schemes, smoothing configurations, and missingness intensities. The numerical results corroborate the asymptotic theory, illustrate the finite-sample bias–variance trade-off inherent in delta-sequence localization, and demonstrate the stability and practical accuracy of the proposed estimator over a wide range of relevant regimes. Taken together, these results show that delta-sequence conditional U-statistics provide a flexible, mathematically rigorous, and broadly applicable framework for higher-order nonparametric inference with incomplete data. Full article

(This article belongs to the Section D1: Probability and Statistics)

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10 pages, 248 KB

Open AccessFeature PaperEditor’s ChoiceArticle

The Monogenity of Pure Quintic Fields: The Power of Sieving

by István Gaál

Mathematics 2026, 14(11), 1810; https://doi.org/10.3390/math14111810 - 23 May 2026

Viewed by 217

Abstract

We provide a simple algorithm for calculating all generators of power integral bases in pure quintic fields. This procedure involves the usual standard elements like Baker’s method and LLL reduction. The main purpose of this paper is to introduce a new idea to [...] Read more.

We provide a simple algorithm for calculating all generators of power integral bases in pure quintic fields. This procedure involves the usual standard elements like Baker’s method and LLL reduction. The main purpose of this paper is to introduce a new idea to considerably diminish the number of small exponents to be considered after the reduction step. This new idea allows us to test all remaining small exponents within a few minutes, using an appropriate sieve method, which turns out to be surprisingly fast. This idea will be applicable in many similar cases. Full article

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

22 pages, 796 KB

Open AccessEditor’s ChoiceArticle

Multi-View Clustering via Projection-Enhanced Bipartite Graph Learning and Consensus Fusion

by Xun Liu, Qing-Wen Wang and Jiang-Feng Chen

Mathematics 2026, 14(10), 1767; https://doi.org/10.3390/math14101767 - 21 May 2026

Viewed by 304

Abstract

Anchor-based bipartite graph methods provide scalable solutions for multi-view clustering, but most of them construct graphs in the original feature space, where high dimensionality distorts the proximity between samples and anchors and degrades graph quality. In addition, the K-means step commonly used to [...] Read more.

Anchor-based bipartite graph methods provide scalable solutions for multi-view clustering, but most of them construct graphs in the original feature space, where high dimensionality distorts the proximity between samples and anchors and degrades graph quality. In addition, the K-means step commonly used to discretize spectral embeddings may produce different cluster assignments across random seeds. To address these limitations, this paper proposes projection-enhanced bipartite graph learning (PEBGL), which first projects each view onto a compact PCA subspace and then jointly performs bipartite graph construction, consensus graph fusion with adaptive view weighting, spectral embedding, and discrete label assignment within an alternating optimization framework. Most subproblems admit closed-form or efficient projection-based updates, and the final labels are obtained by connected-component detection on the learned consensus graph, reducing the dependence on K-means post-processing. Experiments on six benchmark datasets demonstrate that PEBGL achieves competitive clustering performance against recent graph-based and bipartite graph-based methods. These results validate the effectiveness of the proposed framework. Full article

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12 pages, 4863 KB

Open AccessEditor’s ChoiceArticle

Quantitative Analysis of the Reciprocity Gap Dichotomy for Inclusions with Variable Conductivity

by Michele Di Cristo

Mathematics 2026, 14(10), 1717; https://doi.org/10.3390/math14101717 - 16 May 2026

Viewed by 295

Abstract

We study the quantitative structure of the reciprocity gap method for inclusions with spatially varying conductivity. Motivated by the variable-coefficient reciprocity gap identity, we investigate the discrete approximation mechanism underlying the bounded/blow-up dichotomy for sampling points inside and outside the inclusion. The reciprocity [...] Read more.

We study the quantitative structure of the reciprocity gap method for inclusions with spatially varying conductivity. Motivated by the variable-coefficient reciprocity gap identity, we investigate the discrete approximation mechanism underlying the bounded/blow-up dichotomy for sampling points inside and outside the inclusion. The reciprocity gap functional is discretized by harmonic test functions, and the resulting ill-conditioned linear system is regularized by a Tikhonov term consistent with the

L^{2} (\partial D)

trace norm appearing in the weighted formulation. The regularization parameter is selected by the L-curve criterion. For constant, radially varying, and angularly oscillating contrasts, the numerical results show that exterior sampling points exhibit an approximately exponential growth of

∥ v_{N} {(z) ∥}_{L^{2} (\partial D)}

with respect to the harmonic order N, whereas interior points remain bounded. This behavior is quantified through fitted growth rates and contrast indicators, and its dependence on geometry and model parameters is examined. The results provide a quantitative description of the reciprocity gap approximation mechanism in heterogeneous media and show that the bounded/blow-up dichotomy remains numerically detectable beyond the constant-coefficient setting. Full article

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

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35 pages, 38226 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Analysis of a Serial Supply Network Operating Under VMI Policy with Stochastic Replenishment Times and External Demand

by Georgios Varlas, Stelios Koukoumialos, Alexandros Diamantidis, Michael Vidalis and Evangelos Ioannidis

Mathematics 2026, 14(10), 1696; https://doi.org/10.3390/math14101696 - 15 May 2026

Viewed by 320

Abstract

An algorithm based on matrix analytic methods for the exact numerical performance evaluation of a two-echelon vendor-managed inventory system with lost sales is presented in this paper. Both supply and demand uncertainties are taken into consideration. Lead times are modeled using a phase-type [...] Read more.

An algorithm based on matrix analytic methods for the exact numerical performance evaluation of a two-echelon vendor-managed inventory system with lost sales is presented in this paper. Both supply and demand uncertainties are taken into consideration. Lead times are modeled using a phase-type (Coxian) distribution with two phases, while the stochastic nature of external demand is captured with a compound Poisson distribution comprised of a pure Poisson arrival process and a discrete empirical distribution for the demand of individual customers. A computer program based on the algorithm is developed and then it is used for an extensive numerical investigation with a view to obtain insights of possible managerial importance. Full article

(This article belongs to the Special Issue Modeling and Optimization in Supply Chain Management)

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36 pages, 1256 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Goal-Induced Pareto Fronts for a Bi-Criterion Truck–Multiple-Drone Routing Problem

by Pedro Luis González Rodríguez, David Sánchez-Wells, José Miguel León-Blanco, Marcos Calle Suárez and José Luis Andrade Pineda

Mathematics 2026, 14(10), 1635; https://doi.org/10.3390/math14101635 - 12 May 2026

Viewed by 407

Abstract

Truck–multiple-drone routing problems involve conflicting operational criteria and are therefore naturally suited to multiobjective analysis. In practical settings, however, decision makers may also specify aspiration levels for the considered criteria, which call for a target-oriented perspective. This paper studies a bi-criterion truck–multiple-drone routing [...] Read more.

Truck–multiple-drone routing problems involve conflicting operational criteria and are therefore naturally suited to multiobjective analysis. In practical settings, however, decision makers may also specify aspiration levels for the considered criteria, which call for a target-oriented perspective. This paper studies a bi-criterion truck–multiple-drone routing problem through a goal-induced deviation framework in which the original objectives are transformed to normalized positive deviations with respect to prescribed targets. First, a general mathematical framework is introduced, and several structural properties are established, including dominance preservation, invariance under positive weighting, equivalence with the original Pareto structure when all the targets are violated, and the loss of discrimination when the targets are attainable. To address this latter effect, an enhanced goal-programming scalarization is proposed and shown to preserve consistency with the Pareto efficiency. The framework is then specialized to a truck–multiple-drone routing problem with truck time and makespan as criteria and evaluated on representative benchmark instances together with a broader attainable-target benchmark battery, using a common agent-based metaheuristic search framework adapted from literature. This search framework is employed both to estimate a reference Pareto frontier and to solve the GP and EGP scalarizations under the same computational scheme. The computational results illustrate two target regimes: When the targets are unattainable, both formulations are mainly driven by the minimization of positive deviations; when they are attainable, classical goal programming may return satisfactory but dominated solutions, whereas the enhanced formulation preserves discrimination and selects Pareto-efficient alternatives. Full article

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

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24 pages, 1540 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Branch-And-Price Approach to the Platform Supply Vessel Routing and Scheduling Problem with Uncertain Demand

by Bin Ji, Jing Liu and Samson S. Yu

Mathematics 2026, 14(10), 1630; https://doi.org/10.3390/math14101630 - 11 May 2026

Viewed by 317

Abstract

With the expansion of offshore oil and gas exploration into deep-water regions, the efficient scheduling of platform supply vessels (PSVs) is critical to offshore operations. The platform supply vessel routing and scheduling problem (PSVRSP) is an NP-hard combinatorial optimization problem, which is further [...] Read more.

With the expansion of offshore oil and gas exploration into deep-water regions, the efficient scheduling of platform supply vessels (PSVs) is critical to offshore operations. The platform supply vessel routing and scheduling problem (PSVRSP) is an NP-hard combinatorial optimization problem, which is further complicated by uncertainty in offshore demand. Existing studies reveal a methodological gap: exact optimization algorithms have rarely been applied to this problem, as most prior research relies on heuristic methods that cannot guarantee optimality. To address this gap, this study proposes a novel enhanced branch-and-price (B&P) algorithm for the platform supply vessel routing and scheduling problem with uncertain demand (PSVRSP-UD). The proposed approach integrates NG-route labeling, a group-representative label mechanism, and a two-level branching strategy to efficiently obtain globally optimal solutions under demand uncertainty. A scenario-based mixed-integer linear programming (MILP) model is formulated, in which demand uncertainty is captured using Latin hypercube sampling (LHS) combined with Cholesky decomposition and sample-based reduction (SBR). Based on Dantzig–Wolfe decomposition, the proposed B&P algorithm integrates NG-route labeling and a two-level branching strategy to achieve global optimization. Computational experiments show that the B&P algorithm outperforms CPLEX in both computational efficiency and solution quality. Sensitivity analyses examine the impacts of scenario number, demand fluctuation, time window tightness, and weight coefficients on the results. The new results in this study can provide a practical decision-support tool for offshore logistics operations. Full article

(This article belongs to the Special Issue Advances in Combinatorial Optimization: Theory, Algorithms, and Applications)

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24 pages, 3169 KB

Open AccessEditor’s ChoiceArticle

Non-Singular Fast Terminal Sliding Mode Control Based Trajectory Tracking Control of Cable-Driven Manipulators Subject to Lumped Mismatched Uncertainties

by Tran Buu Thach Nguyen, Hoai Vu Anh Truong and Kyoung Kwan Ahn

Mathematics 2026, 14(10), 1602; https://doi.org/10.3390/math14101602 - 8 May 2026

Cited by 1 | Viewed by 366

Abstract

Cable-driven manipulators have emerged as a compelling alternative to traditional manipulators (driven by either electrical, hydraulic, or pneumatic motors), especially for operations in constrained and complex environments. Despite offering many advantages, they still pose significant control challenges. Therefore, this paper presents a novel [...] Read more.

Cable-driven manipulators have emerged as a compelling alternative to traditional manipulators (driven by either electrical, hydraulic, or pneumatic motors), especially for operations in constrained and complex environments. Despite offering many advantages, they still pose significant control challenges. Therefore, this paper presents a novel position tracking control framework for an n-DOF cable-driven manipulator subject to lumped mismatched uncertainties arising from unknown dynamic errors and external disturbances. The proposed approach is built upon non-singular fast terminal sliding mode control (NFTSMC), which provides robustness, high-precision tracking, fast finite-time convergence, chattering-free torque input, and complete elimination of singularity issues. To further enhance control performance, an extended state observer (ESO) is incorporated to accurately estimate unmeasured states and suppress lumped uncertainties. The stability of the closed-loop system under the proposed method is rigorously proven by the Lyapunov theorem. Finally, the superiority of the proposed method over existing controllers is demonstrated by comparative simulations to highlight its potential for practical implementation in complex robotic environments. Full article

(This article belongs to the Special Issue Mathematics Methods of Robotics and Intelligent Systems)

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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 416

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)

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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 407

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)

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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 346

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)

37 pages, 921 KB

Open AccessEditor’s ChoiceArticle

One-Dimensional Solitary-Wave Solutions in Scalar–Tensor Gravity Coupled to Aharonov–Bohm Electrodynamics

by Rosario Pullano, Fernando Minotti and Giovanni Modanese

Mathematics 2026, 14(9), 1517; https://doi.org/10.3390/math14091517 - 30 Apr 2026

Viewed by 338

Abstract

A recently proposed tensor–scalar extension of gravity coupled to extended Aharonov–Bohm electrodynamics admits one-variable traveling reductions in which a longitudinal electromagnetic scalar mode

S = \partial_{μ} A^{μ}

couples nonlinearly to gravitational scalars. In the weak-field regime outside sources, a one-dimensional traveling [...] Read more.

A recently proposed tensor–scalar extension of gravity coupled to extended Aharonov–Bohm electrodynamics admits one-variable traveling reductions in which a longitudinal electromagnetic scalar mode

S = \partial_{μ} A^{μ}

couples nonlinearly to gravitational scalars. In the weak-field regime outside sources, a one-dimensional traveling ansatz depending on

ξ = x - v t

reduces the field equations to a coupled autonomous ODE system. The mathematical core of the reduction is a singular Newton-type equation whose classical mechanics counterpart is known; the novelty here lies in its derivation from the scalar–tensor/Aharonov–Bohm field system, in the physically motivated normalization of the traveling-wave families, and in the resulting phase–space interpretation for source-generated pulse selection. We provide a systematic classification of all admissible initial data and of the corresponding maximal solutions, distinguishing repulsive/attractive regimes and subcritical/critical/supercritical behaviors through a normalized parameter map. In particular, attractive branches may reach the singularity in finite time with a universal collision exponent

2 / 3

, while escaping branches exhibit asymptotically uniform motion with a computable logarithmic correction. Finally, we construct a numerical atlas of representative trajectories and validate the computations by cross-checking direct time integration against numerical inversion of the implicit quadrature, together with energy-defect diagnostics. Full article

(This article belongs to the Special Issue Numerical Solution of Differential Equations and Their Applications)

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25 pages, 4104 KB

Open AccessEditor’s ChoiceArticle

Kalman Filter Method with Iterative Sparse Regularization and Its Application to the Retrieval of the Initial Field for the Convection–Diffusion Equation

by Xuan Deng and Yuepeng Wang

Mathematics 2026, 14(9), 1483; https://doi.org/10.3390/math14091483 - 28 Apr 2026

Viewed by 380

Abstract

Sparse regularization methods play an important role in inverse problems for extracting key features of underlying parameters and have attracted increasing attention in meteorological data assimilation. However, when the condition number of the background error covariance matrix is extremely large (e.g., 10¹² [...] Read more.

Sparse regularization methods play an important role in inverse problems for extracting key features of underlying parameters and have attracted increasing attention in meteorological data assimilation. However, when the condition number of the background error covariance matrix is extremely large (e.g., 10¹²), the instability of the inverse problem makes accurate reconstruction difficult. To address this issue, a gradient operator is incorporated into the sparse regularization term of the cost function, and a Kalman filter (KF) algorithm is developed within a majorization–minimization (MM) framework to solve the resulting optimization problem. The problem is reformulated as a weighted least-squares problem via the MM strategy and further decomposed into two subproblems in the null space and its oblique complementary space through oblique projection, which are then solved using the KF method. This approach avoids the use of an adjoint model typically required in four-dimensional variational data assimilation (4D-Var). In addition, a modified f-slope strategy with a constrained search interval is introduced to adaptively select the regularization parameter during computation. Numerical experiments on the initial condition inversion of the Convection–Diffusion equation demonstrate that the proposed method achieves more accurate reconstruction of key features than the l₁-norm regularized 4D-Var method, particularly in capturing sharp gradients and sparse structures. The adaptive regularization strategy automatically balances sparsity and smoothness without manual tuning. The inversion errors remain low even when the condition number ranges from 10⁸ to 10¹⁴, with relative MSE and MAE below 0.01 and relative bias below 0.005, indicating improved robustness and reconstruction accuracy under severely ill-conditioned settings. Full article

(This article belongs to the Special Issue Computational Approaches in Computer Science: Methods, Algorithms, and Applications)

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28 pages, 7429 KB

Open AccessEditor’s ChoiceArticle

Nash Bargaining-Based Cooperative Dispatch of Electric–Thermal–Hydrogen Multi-Microgrids Under Wind–Solar Uncertainty

by Wenyuan Yang, Tongwei Wu, Xiaojuan Wu and Jiangping Hu

Mathematics 2026, 14(9), 1465; https://doi.org/10.3390/math14091465 - 27 Apr 2026

Viewed by 674

Abstract

This paper proposes a collaborative optimal scheduling strategy based on asymmetric Nash bargaining for the integrated electricity–heat–hydrogen multi-microgrid system, which can minimize the overall system operation cost while guaranteeing the dynamic fairness of multi-microgrids energy transactions with full consideration of wind–solar uncertainty. First, [...] Read more.

This paper proposes a collaborative optimal scheduling strategy based on asymmetric Nash bargaining for the integrated electricity–heat–hydrogen multi-microgrid system, which can minimize the overall system operation cost while guaranteeing the dynamic fairness of multi-microgrids energy transactions with full consideration of wind–solar uncertainty. First, a scenario generation method based on temporally correlated Latin hypercube sampling and Wasserstein probability distance-based scenario reduction is adopted to construct representative wind–solar uncertainty scenarios, which effectively mitigates the operational risks arising from wind and solar power output fluctuations in the coordinated dispatch of multi-microgrids. Then, an asymmetric Nash bargaining-based cooperative game model for energy trading is established, with each microgrid’s optimal independent operation cost as the negotiation breakdown point. The alternating direction method of multipliers is used for a distributed solution to obtain the optimal scheme that balances total system cost and trading fairness. Simulation results verify that the proposed strategy can effectively suppress operation risks from renewable uncertainty, significantly cut total system cost by 36.85%, and fully ensure trading fairness among multi-microgrid entities, with favorable engineering application value. Full article

(This article belongs to the Special Issue Mathematical Methods Applied in Artificial Intelligence and Multi-Agent Systems, 2nd Edition)

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11 pages, 646 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Interpolating Missing Spatial Data Using Graph Laplacian Eigenbasis

by Zihan Jin and Hiroshi Yamada

Mathematics 2026, 14(9), 1435; https://doi.org/10.3390/math14091435 - 24 Apr 2026

Viewed by 347

Abstract

This paper addresses the problem of interpolating missing spatial data at the vertices of a connected undirected simple graph. We show that, by exploiting the eigenbasis of the graph Laplacian, all missing values can be reconstructed even from a single observation. This work [...] Read more.

This paper addresses the problem of interpolating missing spatial data at the vertices of a connected undirected simple graph. We show that, by exploiting the eigenbasis of the graph Laplacian, all missing values can be reconstructed even from a single observation. This work establishes a novel connection between spatial statistics and spectral graph theory. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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19 pages, 4090 KB

Open AccessFeature PaperEditor’s ChoiceArticle

PDGM-PINN: Partial Derivative Guided Multi-Branch Physics-Informed Neural Network

by Shangpeng Lei, Chenghan Yang, Roberts Grants, Uldis Grunde and Nadezhda Kunicina

Mathematics 2026, 14(8), 1349; https://doi.org/10.3390/math14081349 - 17 Apr 2026

Viewed by 513

Abstract

With the development of scientific machine learning (SciML), the proposal of physics-informed neural networks (PINNs) has provided a powerful paradigm for solving partial differential equations (PDEs). While PINNs perform well in solving high-dimensional PDEs, they perform worse than traditional numerical methods for low-dimensional [...] Read more.

With the development of scientific machine learning (SciML), the proposal of physics-informed neural networks (PINNs) has provided a powerful paradigm for solving partial differential equations (PDEs). While PINNs perform well in solving high-dimensional PDEs, they perform worse than traditional numerical methods for low-dimensional problems. This discrepancy arose from potential convergence conflicts induced by distinct physical magnitude of loss terms. To decouple the convergence conflicts, we propose a partial derivative guided multi-branch physics-informed neural network (PDGM-PINN). Inspired by SciML, we treat both the solution and partial derivatives as dependent variables to be predicted. The partial derivatives are directly predicted by sub-branches, while the main branch approximates the PDE solution, and all branches share error backpropagation information. Furthermore, we redesign the loss function. The loss of the governing equation is computed with the solution and partial derivatives predicted by the main and sub-branches. Schwarz’s theorem and Kullback–Leibler divergence are incorporated into the loss terms as soft constraints of partial derivatives continuity and residual distributions consistency for the governing equations. We conducted comprehensive experimental evaluations on seven PDEs, and ablation experiments, sensitivity analyses, and complexity analyses were carried out to investigate the rationality of PDGM-PINN. The results demonstrate that PDGM-PINN achieves the best performance among PINN variants with the fewest trainable parameters, effectively avoiding architectural redundancy. Full article

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

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19 pages, 2284 KB

Open AccessEditor’s ChoiceArticle

Multiscale Kinetic Model for Immune Reaction in Coeliac Disease

by Diletta Burini, Damian A. Knopoff and Lidia Serrano

Mathematics 2026, 14(8), 1333; https://doi.org/10.3390/math14081333 - 16 Apr 2026

Cited by 1 | Viewed by 467

Abstract

This paper presents a multiscale model that captures the complex dynamics of coeliac disease, a chronic T-cell-mediated autoimmune disorder triggered by gluten ingestion in genetically susceptible individuals. Grounded in the mathematical kinetic theory of active particles, the model captures the intricate dynamics of [...] Read more.

This paper presents a multiscale model that captures the complex dynamics of coeliac disease, a chronic T-cell-mediated autoimmune disorder triggered by gluten ingestion in genetically susceptible individuals. Grounded in the mathematical kinetic theory of active particles, the model captures the intricate dynamics of the immunological cascade that leads to small intestinal damage, involving complex interactions across molecular, cellular, and tissue scales. This approach explicitly models the different functional subsystems involved and examines the role of molecular messengers in shaping these dynamics, focusing on gluten recognition and immune system activation. Simulations demonstrate that the model can reproduce significant clinical and biological observations of coeliac disease, suggesting its potential to inform therapeutic strategies. Full article

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22 pages, 778 KB

Open AccessEditor’s ChoiceArticle

Decentralization Under Energy Growth: Geographic Reallocation and Convergence in Bitcoin Mining

by Angeliki Papana and Konstantinos Katrakilidis

Mathematics 2026, 14(8), 1309; https://doi.org/10.3390/math14081309 - 14 Apr 2026

Viewed by 774

Abstract

Understanding how Bitcoin mining is distributed across countries is important for evaluating both the sustainability and resilience of the network. In this study, we examine the evolution of total Bitcoin electricity consumption alongside the geographic distribution of Bitcoin mining. Data are provided by [...] Read more.

Understanding how Bitcoin mining is distributed across countries is important for evaluating both the sustainability and resilience of the network. In this study, we examine the evolution of total Bitcoin electricity consumption alongside the geographic distribution of Bitcoin mining. Data are provided by the Cambridge Centre for Alternative Finance (Licensed under CC BY–NC–SA 4.0): Annual data from the Cambridge Bitcoin Electricity Consumption Index (2010–2025) and a monthly panel of country-level Bitcoin hashrate shares for 105 countries (September 2019–January 2022). To assess the degree of decentralization in the global mining network, we employ entropy-based measures, inequality indices, and panel convergence tests. The results indicate that total electricity consumption grew exponentially during the early years of Bitcoin, but later transitioned to a more stable and approximately linear path. Country-level permutation entropy reveals highly volatile and dynamic mining trajectories. The Theil index shows that cross-sectional inequality declines over time, while increasing symbolic entropy reflects a progressively more even cross-country distribution of mining activity. Further evidence from

σ

-convergence supports a statistically significant reduction in cross-country dispersion of mining shares. Dynamic panel fixed-effects estimates reveal mean-reverting behavior in relative country shares, consistent with stochastic convergence. Finally, Phillips–Sul analysis points to heterogeneous early transition paths but ultimately supports convergence toward a single global club. The gradual geographical decentralization occurs alongside persistent core–periphery asymmetries in long-run mining shares. Overall, our findings suggest that Bitcoin mining behaves as a globally integrated industry in which computational capacity reallocates rapidly across countries in response to economic and regulatory conditions. Full article

(This article belongs to the Special Issue Innovative Methods in Financial Modeling: From Econometrics to Entropy)

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16 pages, 1295 KB

Open AccessEditor’s ChoiceArticle

Approximation of Offset Surfaces Using Generalized Wendland Radial Basis Functions

by Loubna Omri, Abdelouahed Kouibia, Hossain Oulad Yakhlef and Hamza El Bazi

Mathematics 2026, 14(8), 1300; https://doi.org/10.3390/math14081300 - 13 Apr 2026

Viewed by 478

Abstract

We introduce a new methodological approach for the approximation of generalized offset surfaces using smoothing radial basis functions (RBFs). Existing offset surface generation methods often exhibit limitations such as self-intersections and singularities, which affect their robustness and accuracy. To address these issues, we [...] Read more.

We introduce a new methodological approach for the approximation of generalized offset surfaces using smoothing radial basis functions (RBFs). Existing offset surface generation methods often exhibit limitations such as self-intersections and singularities, which affect their robustness and accuracy. To address these issues, we employ generalized Wendland radial basis functions, which are compactly supported and provide enhanced stability. We establish the existence and uniqueness of the solution of the proposed method, analyze its computational aspects, and prove convergence results. Finally, numerical experiments are presented to demonstrate its effectiveness as well as to compare it with an existing method from the literature. Full article

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

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15 pages, 311 KB

Open AccessEditor’s ChoiceReview

Some Remarks on Fourth-Order Tensor Fields on Space-Times

by Graham Hall

Mathematics 2026, 14(8), 1238; https://doi.org/10.3390/math14081238 - 8 Apr 2026

Viewed by 451

Abstract

This paper is a contribution to Einstein’s general relativity theory and is mostly a review of known work. It concentrates attention on four fourth-order tensors which arise on the space-time manifold describing this theory and which are very useful. These are the (Riemann) [...] Read more.

This paper is a contribution to Einstein’s general relativity theory and is mostly a review of known work. It concentrates attention on four fourth-order tensors which arise on the space-time manifold describing this theory and which are very useful. These are the (Riemann) curvature tensor, the Weyl conformal tensor, the “E” tensor and the Weyl projective tensor. The first of these, the curvature tensor, plays an important role in the formulation and interpretation of Einstein’s theory. Next, the Weyl conformal tensor is introduced and its conformal properties described and with it, the Petrov classification of gravitational fields which arises from this tensor. This, in turn, gives rise to the Bel criteria for distinguishing Petrov types at a point by an alignment of certain null directions at that point. The third of these tensors, the “E” tensor, is an important tensor in calculations due to its close connection to the Ricci tensor. The fourth tensor, the Weyl projective tensor, is then described together with its properties relating to the geodesic structure of space-time. As examples of the combined usefulness of these tensors, pp-waves and generalised pp-waves are discussed and related, and a review of the geodesic structure of vacuum metrics is given. Full article

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

22 pages, 389 KB

Open AccessEditor’s ChoiceArticle

Adaptive Multipath Proofs for Privacy Protection and Security in Payment Channel Networks

by Wenqi Li, Zijie Pan and Yunqing Yang

Mathematics 2026, 14(7), 1199; https://doi.org/10.3390/math14071199 - 3 Apr 2026

Viewed by 435

Abstract

Payment channel networks enable scalable off-chain payments, but their practical deployment remains constrained by a persistent tension among routing efficiency, liquidity visibility, transaction privacy, and settlement security. Existing multipath routing mechanisms can improve payment success under fragmented liquidity, yet they often expose sensitive [...] Read more.

Payment channel networks enable scalable off-chain payments, but their practical deployment remains constrained by a persistent tension among routing efficiency, liquidity visibility, transaction privacy, and settlement security. Existing multipath routing mechanisms can improve payment success under fragmented liquidity, yet they often expose sensitive balance information, leak structural features of payment routes, and enlarge the attack surface for probing, channel exhaustion, and selective forwarding. This paper presents a novel framework, Adaptive Multipath Proofs (AMPs), for privacy protection and security in payment channel networks. The core idea is to bind multipath routing decisions with lightweight zero-knowledge verifiability, allowing intermediate nodes to validate path feasibility, fragment consistency, and settlement constraints without learning exact channel balances, the complete payment amount, or the global route structure. AMP integrates three mechanisms: a hidden-liquidity feasibility proof that supports privacy-preserving route selection, an adaptive payment-splitting strategy that dynamically determines fragment allocation according to network congestion and balance uncertainty, and a proof-coupled settlement guard that enforces atomicity and timeout consistency across all payment fragments. Together, these mechanisms reduce information leakage while preserving robust payment execution under dynamic network conditions. Experimental evaluation on real Lightning Network topologies and synthetic stress scenarios demonstrates that AMP significantly lowers balance disclosure and endpoint inference risk, improves payment completion under skewed liquidity distributions, and introduces only moderate computational and communication overhead. The results indicate that adaptive proof-carrying multipath routing offers a practical and effective direction for building secure, privacy-preserving, and high-success payment channel networks. Full article

(This article belongs to the Special Issue Advances in Computational Intelligence and Their Applications in Cybersecurity)

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25 pages, 8081 KB

Open AccessEditor’s ChoiceArticle

Generalized Traub Family for Solving Nonlinear Systems: Fourth-Order Optimal Method and Dynamical Analysis

by Alicia Cordero, Miguel A. Leonardo Sepúlveda, Juan R. Torregrosa, Antmel Rodríguez Cabral and María P. Vassileva

Mathematics 2026, 14(7), 1161; https://doi.org/10.3390/math14071161 - 31 Mar 2026

Cited by 1 | Viewed by 472

Abstract

A novel two-stage procedure for approximating solutions of nonlinear systems is introduced. The scheme employs two evaluations of the vector function F together with a single Jacobian computation, followed by the resolution of two linear subproblems that share an identical coefficient matrix. This [...] Read more.

A novel two-stage procedure for approximating solutions of nonlinear systems is introduced. The scheme employs two evaluations of the vector function F together with a single Jacobian computation, followed by the resolution of two linear subproblems that share an identical coefficient matrix. This structure reduces the computational cost and enhances the adaptability of the method with respect to existing alternatives. The design of the algorithm is motivated by criteria relating efficiency to the total number of functional evaluations, ensuring that the resulting strategy achieves the optimal convergence order permitted within this framework. A proof of the local convergence order is provided, and its accuracy is supported by a series of experiments on distinct nonlinear models, including problems arising from differential equations. The numerical evidence confirms that the developed technique reaches the theoretical convergence rate and performs favorably when compared with other methods of equal order. Moreover, we examine the dynamical features of the related parametric variant, offering additional understanding of its stability properties and iterative behavior. Full article

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

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16 pages, 740 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Mathematically Exact Non-Square-Integrable Solutions in Schrödinger-Equivalent Diffusion Dynamics

by László Mátyás and Imre Ferenc Barna

Mathematics 2026, 14(7), 1162; https://doi.org/10.3390/math14071162 - 31 Mar 2026

Viewed by 562

Abstract

We analyze the spherically symmetric complex diffusion and special type of the complex reaction–diffusion equations. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we [...] Read more.

We analyze the spherically symmetric complex diffusion and special type of the complex reaction–diffusion equations. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we found a new realm of solutions which lie between the solutions of the classical regular diffusion equation and the usual quantum mechanical solutions of the Schrödinger equation. As the solution method, we applied the the self-similar Ansatz, which reduces the original partial differential equation (PDE) to an ordinary differential equation (ODE) which can be solved analytically. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation which has to be used in ordinary quantum mechanics for time-independent Hamiltonian. For the complex diffusion equation—without any additional source term—the solutions are the Kummer’s M and Kummer’s U functions. For some parameter values we found

L^{2}

integrability, as in the Cartesian case. We interpret that this property can be a “quantum mechanical heritage” and can be a far relation to ordinary quantum mechanics. Therefore, in this sense, our solutions might have quantum mechanical interest in the future. For the complex reaction–diffusion-type equation we derived the Whittaker M and Whittaker W functions as solutions. These solutions have no

L^{2}

integrability at all. All derived solutions have complex quadratic arguments. These kind of analytic solutions are new and cannot be found in the existing scientific literature. Finally, the role of the complex angular momentum was investigated as well. Full article

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

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26 pages, 2035 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Stability Dependence on Inertia in the Driven Damped Pendulum: A Master Control Parameter Analysis

by Alexander N. Pisarchik

Mathematics 2026, 14(6), 1060; https://doi.org/10.3390/math14061060 - 20 Mar 2026

Viewed by 830

Abstract

The driven damped pendulum is a foundational model in nonlinear dynamics, with applications ranging from Josephson junctions to MEMS oscillators. Conventional dimensionless treatments obscure the common physical origin of damping and driving in the inertia coefficient. Here we restore this dependence and establish [...] Read more.

The driven damped pendulum is a foundational model in nonlinear dynamics, with applications ranging from Josephson junctions to MEMS oscillators. Conventional dimensionless treatments obscure the common physical origin of damping and driving in the inertia coefficient. Here we restore this dependence and establish inertia as a master control parameter governing stability, resonance, and bifurcations. Through linear analysis and perturbation theory, we derive universal scaling laws revealing a fundamental dichotomy: quantities at resonance—peak amplitude and nonlinear frequency shift—are independent of inertia due to exact algebraic cancellation between the inertia dependence of the effective driving amplitude and effective damping coefficient. Off resonance, however, amplitude scales inversely with inertia, bandwidth narrows proportionally, and the bistability threshold exhibits an even steeper dependence. A critical inertia separates underdamped from overdamped regimes, yielding non-monotonic relaxation times that maximize attractor memory at extreme inertia values. These scaling laws provide design guidelines: low inertia promotes broadband response for energy harvesting; high inertia suppresses off-resonant vibrations for precision timing and quantum applications. By establishing inertia as a physically realizable path through parameter space, this work unifies disparate phenomena and provides a framework for understanding stability in inertial-driven systems. Full article

(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)

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12 pages, 267 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Approximate Bi-Affine Mappings

by Jae-Hyeong Bae and Won-Gil Park

Mathematics 2026, 14(6), 1056; https://doi.org/10.3390/math14061056 - 20 Mar 2026

Viewed by 300

Abstract

In this paper, we introduce a multi-variable bi-affine functional equation of the form [...] Read more.

In this paper, we introduce a multi-variable bi-affine functional equation of the form

f (\sum_{i = 1}^{m} α_{i} x_{i}, \sum_{j = 1}^{n} β_{j} y_{j}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} α_{i} β_{j} f (x_{i}, y_{j})

, where m and n are integers and

m, n \geq 2

and

α_{i}, β_{j}

are nonzero scalars. We investigate the Hyers–Ulam stability of this functional equation in Banach spaces using the direct method. The results obtained in this paper can be regarded as a generalization of stability results for the classical bi-Jensen functional equation and its multi-variable mean-type variants. Full article

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

26 pages, 666 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Quantum Heuristic Approach to Vehicle Routing Problem

by Jun Suk Kim, Donghyeon Lee and Chang Wook Ahn

Mathematics 2026, 14(6), 1026; https://doi.org/10.3390/math14061026 - 18 Mar 2026

Viewed by 1038

Abstract

Quantum optimization has recently drawn considerable attention as one of the possible applications of noisy intermediate-scale quantum computation, yet the problem of qubit requirement remains a major bottleneck when combinatorial optimization problems are converted into quantum circuits. This issue becomes especially critical in [...] Read more.

Quantum optimization has recently drawn considerable attention as one of the possible applications of noisy intermediate-scale quantum computation, yet the problem of qubit requirement remains a major bottleneck when combinatorial optimization problems are converted into quantum circuits. This issue becomes especially critical in solving the capacitated vehicle routing problem (CVRP) with the quantum approximate optimization algorithm (QAOA), since the number of required qubits increases polynomially with respect to the number of nodes. This study investigates whether a heuristic divide-and-conquer strategy can be adapted to the quantum setting so as to improve qubit efficiency while preserving the optimization capability to a reasonable extent. The proposed method decomposes a single CVRP into multiple traveling salesman problems (TSPs) by the sweeping-based clustering method, searches for the sector configuration with the smallest angle sum by Grover’s search algorithm, and then solves each sector-wise TSP with the QAOA aided by the gravitational search algorithm. Experiments on five benchmark datasets show that the proposed approach attains feasible solutions within 3.4 to 12.7% of the reinforcement-learning baseline on the main test set. These results suggest that the proposed approach serves as a plausible quantum heuristic framework for constrained routing optimization, with the advantage of reducing the qubit burden by decomposing the original problem into smaller subproblems. Full article

(This article belongs to the Special Issue Theoretical and Applied Mathematics in Supply Chain Management, 2nd Edition)

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25 pages, 389 KB

Open AccessEditor’s ChoiceArticle

FedQuAD: Fast-Converging Curvature-Aware Federated Learning for Credit Default Prediction from Private Accounting Data

by Dingwen Bai, MuGa WaEr and Qichun Wu

Mathematics 2026, 14(6), 1012; https://doi.org/10.3390/math14061012 - 17 Mar 2026

Cited by 1 | Viewed by 698

Abstract

Credit default prediction from firm-level accounting statements is central to risk management, yet the underlying financial data are highly sensitive and often siloed across banks, auditors, and platforms. Federated learning (FL) offers a practical route to collaborative modeling without centralizing raw records, but [...] Read more.

Credit default prediction from firm-level accounting statements is central to risk management, yet the underlying financial data are highly sensitive and often siloed across banks, auditors, and platforms. Federated learning (FL) offers a practical route to collaborative modeling without centralizing raw records, but standard FL optimization can converge slowly under severe client heterogeneity, heavy-tailed accounting features, and label imbalance typical of default events. This paper proposes FedQuAD, a novel fast-converging FL algorithm that couples (i) quasi-Newton curvature aggregation on the server with a lightweight limited-memory update to accelerate global progress, (ii) a proximal variance-reduced local solver that stabilizes client drift under non-IID accounting distributions, and (iii) federated robust standardization of tabular financial ratios via secure aggregated quantile statistics to mitigate scale instability and outliers. FedQuAD is communication-efficient by design: It transmits compact gradient and curvature sketches and adapts local computation to each client’s stochasticity and drift. We provide convergence guarantees for strongly convex default-risk objectives (logistic and calibrated GLM losses) under bounded heterogeneity, and extend the analysis to nonconvex deep tabular models via expected stationarity bounds. Experiments on public credit-risk benchmarks with simulated cross-silo (institutional) partitions demonstrate that FedQuAD reaches target AUC and calibration error with substantially fewer communication rounds than representative baselines while maintaining privacy constraints compatible with secure aggregation and optional client-level differential privacy accounting. Full article

(This article belongs to the Special Issue Applied Mathematics, Computing, and Machine Learning)

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12 pages, 260 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Certain Mathematical Constants Associated with Harmonic Numbers and Higher-Dimensional Harmonic Sums

by Junesang Choi

Mathematics 2026, 14(6), 962; https://doi.org/10.3390/math14060962 - 12 Mar 2026

Viewed by 957

Abstract

The Euler–Mascheroni constant

γ

, defined as the limiting difference between the harmonic numbers

H_{n}

and

log n

, has long been studied and appears in diverse areas of number theory, analysis, and special functions. In this paper, we establish a unified [...] Read more.

The Euler–Mascheroni constant

γ

, defined as the limiting difference between the harmonic numbers

H_{n}

and

log n

, has long been studied and appears in diverse areas of number theory, analysis, and special functions. In this paper, we establish a unified formula for

(k + 1)

-fold harmonic sums expressed in terms of harmonic numbers. Several particular cases are examined in detail, and their asymptotic expansions are derived, leading to the identification of both classical and additional limiting constants. These results place higher-order harmonic sums within a common analytic framework and clarify the structure of their normalized limits. The broader mathematical significance of the additional constants arising from this approach remains to be determined and may warrant further investigation. Full article

(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics, 2nd Edition)

28 pages, 854 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Stability and Bifurcations in a Discrete-Time Eco-Evolutionary Logistic Model

by Rafael Luís

Mathematics 2026, 14(6), 928; https://doi.org/10.3390/math14060928 - 10 Mar 2026

Viewed by 535

Abstract

In this paper I study a two-dimensional discrete-time evolutionary logistic-type model describing the coupled dynamics of population density and a continuously evolving trait. I provide a local bifurcation analysis of the equilibria, deriving explicit conditions for their existence and local stability. In particular, [...] Read more.

In this paper I study a two-dimensional discrete-time evolutionary logistic-type model describing the coupled dynamics of population density and a continuously evolving trait. I provide a local bifurcation analysis of the equilibria, deriving explicit conditions for their existence and local stability. In particular, I show that the boundary and interior equilibria exchange stability through a transcritical bifurcation, and I characterize analytically the subsequent loss of stability of the interior equilibrium via period-doubling and Neimark–Sacker bifurcations. Transversality is established in all cases, and the criticality of the bifurcations is determined through normal form and Lyapunov coefficient computations. I show that the period-doubling bifurcation can be supercritical or subcritical, while the Neimark–Sacker bifurcation is generically nondegenerate and may be either supercritical or subcritical, depending on parameter values. Full article

(This article belongs to the Section C2: Dynamical Systems)

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39 pages, 507 KB

Open AccessEditor’s ChoiceArticle

An LM-Type Unit Root Test for Functional Time Series

by Yichao Chen and Chi Seng Pun

Mathematics 2026, 14(5), 916; https://doi.org/10.3390/math14050916 - 8 Mar 2026

Cited by 1 | Viewed by 516

Abstract

In this paper, we propose a Lagrange multiplier (LM)-type unit root test for functional time series. The key novelty lies not in introducing a new LM principle but in establishing the asymptotic validity of such a test under the functional random walk null [...] Read more.

In this paper, we propose a Lagrange multiplier (LM)-type unit root test for functional time series. The key novelty lies not in introducing a new LM principle but in establishing the asymptotic validity of such a test under the functional random walk null hypothesis without relying on functional principal component analysis (FPCA) or finite-dimensional unit root subspace assumptions. We derive the limit distribution of our proposed test statistics under the null hypothesis of a random walk and its asymptotic behavior of alternative hypotheses of trend stationary, weakly dependent stationary, and autoregressive stationary models. Specifically, we establish the theoretical consistency of the test under all aforementioned alternative hypotheses. Simulation studies corroborate these theoretical findings and demonstrate the desirable finite-sample performance of the proposed functional unit root test. The proposed test is also applied to real data of intraday stock price curves, and the test results are plausible. Full article

(This article belongs to the Special Issue New Challenges in Statistical Analysis and Multivariate Data Analysis)

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21 pages, 394 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Geometric Properties of Infinite Direct Sums

by Paweł Kolwicz

Mathematics 2026, 14(5), 906; https://doi.org/10.3390/math14050906 - 7 Mar 2026

Viewed by 535

Abstract

We show exactly when the topology of convergence in measure in Banach ideal spaces is linear (equivalently, coarser than the norm topology). Next, we present the relationship between the Kadets–Klee and suitable monotonicity properties with respect to global convergence in measure. Applying these [...] Read more.

We show exactly when the topology of convergence in measure in Banach ideal spaces is linear (equivalently, coarser than the norm topology). Next, we present the relationship between the Kadets–Klee and suitable monotonicity properties with respect to global convergence in measure. Applying these results, we characterize the Kadets–Klee property with respect to the global convergence in measure in infinite direct sums. We also prove the criteria of some related monotonicity properties in infinite direct sums. Furthermore, we solve the fundamental lifting (inheritance) problem completely for all these properties. We finish the paper with concrete examples showing how our general results can be applied. Full article

(This article belongs to the Special Issue New Advances in Complex Analysis and Functional Analysis)

17 pages, 330 KB

Open AccessFeature PaperEditor’s ChoiceArticle

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

by Angela Slavova and Petar Popivanov

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

Viewed by 634

Abstract

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

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

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

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

R^{4}

,

R^{8}

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

(t_{0}, x_{0})

. Full article

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

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20 pages, 1623 KB

Open AccessEditor’s ChoiceArticle

Deep Contextual Bandits with Multivariate Outcomes: Empirical Copula Normalization, Temporal Feature Learning, and Doubly Robust Policy Evaluation

by Jong-Min Kim

Mathematics 2026, 14(5), 846; https://doi.org/10.3390/math14050846 - 2 Mar 2026

Viewed by 687

Abstract

We develop and evaluate a deep contextual bandit framework for multivariate off-policy evaluation within a controlled simulation-based validation setting. Using real covariate distributions from the Adult, Boston Housing, and Wine Quality datasets, we construct synthetic treatment assignments and multivariate potential outcomes to enable [...] Read more.

We develop and evaluate a deep contextual bandit framework for multivariate off-policy evaluation within a controlled simulation-based validation setting. Using real covariate distributions from the Adult, Boston Housing, and Wine Quality datasets, we construct synthetic treatment assignments and multivariate potential outcomes to enable rigorous benchmarking under known data-generating processes. We compare CNN-LSTM, LSTM, and Feed-forward Neural Network (FNN) architectures as nonlinear action-value estimators. To examine representation learning under structured dependence, an

A R (1)

feature augmentation scheme is employed, while multivariate outcomes are standardized using empirical copula transformations to preserve cross-dimensional dependence. Policy values are estimated using Stabilized Importance Sampling (SIPS) and doubly robust (DR) estimators with bootstrap inference. Although the decision problem is strictly one-step, empirical results indicate that CNN-LSTM architectures provide competitive action-value calibration under temporal augmentation. Across all datasets, the DR estimator demonstrates substantially lower variance and greater stability than SIPS, consistent with its theoretical variance-reduction properties. Diagnostic analyses—including propensity overlap assessment, cumulative oracle regret (with oracle values known by construction), calibration evaluation, and sensitivity analysis—support the reliability of the proposed evaluation framework. Overall, the results demonstrate that combining copula-normalized multivariate outcomes with doubly robust off-policy evaluation yields a statistically principled and variance-efficient approach for offline policy learning in high-dimensional simulated environments. Full article

(This article belongs to the Special Issue Advances in Statistical AI and Causal Inference)

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30 pages, 1351 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Recursive Least-Squares Algorithm Based on a Fourth-Order Tensor Decomposition for Acoustic Echo Cancellation

by Radu-Andrei Otopeleanu, Laura-Maria Dogariu, Constantin Paleologu, Jacob Benesty, Cristian-Lucian Stanciu and Ruxandra-Liana Costea

Mathematics 2026, 14(5), 812; https://doi.org/10.3390/math14050812 - 27 Feb 2026

Cited by 2 | Viewed by 486

Abstract

Adaptive filtering algorithms based on tensor decomposition represent appealing choices for system identification problems, especially when dealing with the estimation of long-length impulse responses, like in acoustic echo cancellation. The topic has recently been addressed in the literature, showing that the gain (compared [...] Read more.

Adaptive filtering algorithms based on tensor decomposition represent appealing choices for system identification problems, especially when dealing with the estimation of long-length impulse responses, like in acoustic echo cancellation. The topic has recently been addressed in the literature, showing that the gain (compared to the conventional approach) is twofold in terms of both better performance and lower complexity. The main idea is that a system identification problem with a large parameter space (i.e., a long-length filter) is reformulated based on a group of shorter filters, while their coefficients are combined using the Kronecker product. Nevertheless, one of the main challenges is related to handling the tensor rank, which is particularly addressed for each specific decomposition order. Previous solutions have been designed for second-order (matrix case) and third-order tensorial decompositions. In this paper, we develop a recursive least-squares adaptive filtering algorithm that exploits a fourth-order tensor decomposition, aiming for further performance improvements compared to the existing solutions. In this framework, the influence of the decomposition setup is investigated, which is also related to the main parameters of the algorithm, i.e., the forgetting factors. Simulations performed in the context of acoustic echo cancellation support the theoretical findings and indicate the good performance of the proposed algorithm. Full article

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11 pages, 723 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Conceptual Model for Growth by Capital–Education Investments

by Ferdinand Verhulst

Mathematics 2026, 14(5), 747; https://doi.org/10.3390/math14050747 - 24 Feb 2026

Viewed by 509

Abstract

In a first approximation, economic growth depends on capital investments and on investments in education and innovation. The macro-economic model introduced here will specifiy aggregate output as determined by aggregate supply of capital and education investment. We will consider the effectiveness of education [...] Read more.

In a first approximation, economic growth depends on capital investments and on investments in education and innovation. The macro-economic model introduced here will specifiy aggregate output as determined by aggregate supply of capital and education investment. We will consider the effectiveness of education including its quality for the growth of the National Product. It is surprising that small changes in the quality of education have a considerable long-term impact on economic growth. Secondly, we consider the positive and negative influences of chaotic fluctuations of capital investments caused by hype cycles or erratic policies. Finally, we introduce a continuous control by consumption on education investments. In this three-dimensional macro-economic model, a tipping point exists where an increase in consumption affecting the amount of education and innovation leads to a decline in economic growth. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 3rd Edition)

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23 pages, 649 KB

Open AccessEditor’s ChoiceArticle

Manifold Causal Conditional Deep Networks for Heterogeneous Treatment Effect Estimation and Policy Evaluation

by Jong-Min Kim

Mathematics 2026, 14(4), 738; https://doi.org/10.3390/math14040738 - 22 Feb 2026

Cited by 1 | Viewed by 728

Abstract

We present a comprehensive framework for estimating heterogeneous treatment effects and evaluating decision-making policies in high-dimensional settings. Our approach combines nonlinear manifold learning techniques—UMAP, t-SNE, and Isomap—with a Causal Conditional Deep Network (CCDN) to model complex nonlinear interactions among covariates, treatments, and outcomes. [...] Read more.

We present a comprehensive framework for estimating heterogeneous treatment effects and evaluating decision-making policies in high-dimensional settings. Our approach combines nonlinear manifold learning techniques—UMAP, t-SNE, and Isomap—with a Causal Conditional Deep Network (CCDN) to model complex nonlinear interactions among covariates, treatments, and outcomes. Within this framework, we assess five treatment assignment policies—Greedy, Thompson Sampling, Epsilon-Greedy, Random, and a novel LLM-guided Thompson policy—across simulated and real-world datasets, including Adult, Wine Quality, and Boston Housing. Empirical results reveal a fundamental trade-off: exploitative policies like Greedy minimize cumulative regret but underperform in recovering heterogeneous treatment effects, whereas exploratory policies, particularly Random and LLM-Thompson, achieve a lower Conditional Average Treatment Effect Root Mean Squared Error (CATE RMSE) by providing broader coverage of the action–covariate space. Notably, LLM-Thompson consistently delivers strong performance across noisy, real-world datasets, highlighting the advantage of uncertainty-aware exploration in capturing treatment heterogeneity. Overall, the framework demonstrates that integrating manifold-informed deep networks with principled exploration strategies enhances both policy optimization and individualized treatment effect estimation in high-dimensional, complex environments. Full article

(This article belongs to the Special Issue Statistical Data Modeling and Machine Learning with Applications, 4th Edition)

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18 pages, 636 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Directional Quaternion Step Differentiation and a Bicomplex Double-Step Calculus for Cancellation-Free First and Second Derivatives

by Ji Eun Kim

Mathematics 2026, 14(4), 728; https://doi.org/10.3390/math14040728 - 20 Feb 2026

Viewed by 550

Abstract

Accurate derivative information is central to sensitivity analysis and optimization, yet standard finite differences can lose many digits when the step size is small because of subtractive cancellation. Complex-step differentiation largely resolves this issue for first derivatives, but robust second derivatives and mixed [...] Read more.

Accurate derivative information is central to sensitivity analysis and optimization, yet standard finite differences can lose many digits when the step size is small because of subtractive cancellation. Complex-step differentiation largely resolves this issue for first derivatives, but robust second derivatives and mixed partials remain delicate: several practical complex-step variants for

f^{″}

still subtract nearly equal quantities, and quaternion-step rules are often presented as separate constructions. We develop a unified slice-based framework that extracts first and second derivatives from a single evaluation by projecting algebraic coefficients in commutative subalgebras of the complexified quaternions. First, we formulate a directional quaternion-steprule parameterized by an arbitrary unit pure quaternion u and provide an explicit projection operator that makes the underlying complex slice

C_{u} ≃ C

transparent; the resulting first-derivative formula is rotation invariant and recovers classical j-step and planar

(j, k)

-step rules as special cases. Second, we construct a bicomplex double-step calculus in the commuting imaginary units

i

and u and show that one evaluation at

z + (i + u) h

separates derivative information into distinct coefficients, with the

i u

-component equal to

h^{2} f^{″} (z) + O (h^{4})

, giving a subtraction-free

O (h^{2})

approximation of

f^{″}

. For bivariate analytic functions we additionally derive one-shot identities for

f_{x}

,

f_{y}

, and

f_{x y}

from

f (x + u h, y + i h)

and supply practical extraction identities, step-size guidance for

h^{2}

-scaled coefficients, and branch-consistency diagnostics for non-entire functions. The “cancellation-free” property here refers to avoiding the subtraction of nearly equal real quantities at the level of the differentiation formula; in floating-point arithmetic, coefficient extraction and the

1 / h^{2}

scaling for second-order quantities still interact with roundoff, and we quantify the resulting stable regimes numerically. Full article

(This article belongs to the Special Issue New Advances in Complex Analysis and Functional Analysis)

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21 pages, 15000 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Hierarchically Coupled Biochemical Switches for Stem Cell Differentiation

by Nikolaos K. Voulgarakis

Mathematics 2026, 14(4), 678; https://doi.org/10.3390/math14040678 - 14 Feb 2026

Viewed by 497

Abstract

In multicellular organisms, the development of diverse cell types relies on stem cell differentiation through a hierarchy of fate decisions. Pluripotent stem cells first give rise to multipotent progenitors, which then undergo successive fate decisions to generate specialized cells within their respective lineages. [...] Read more.

In multicellular organisms, the development of diverse cell types relies on stem cell differentiation through a hierarchy of fate decisions. Pluripotent stem cells first give rise to multipotent progenitors, which then undergo successive fate decisions to generate specialized cells within their respective lineages. Waddington used the metaphor of a marble rolling down a hill through hierarchically branching valleys that represent the various states of cell differentiation, with the final valleys at the bottom symbolizing the specialized cells. Mathematically, specialized cells are seen as stable attractors in a complex dynamical system that displays multistability. However, this framework does not necessarily describe the hierarchical branching of stem cell differentiation. In a recent study, we addressed this issue by assuming that each gene regulatory network (GRN) consists of hierarchically coupled gene subnetworks (modules) that are self-regulated due to epigenetic factors. Each module was modeled using the normal form of relevant bifurcations. Overall, this approach captures both multistability and hierarchical branching in differentiation. Here, the normal forms of bifurcations are replaced by realistic biochemical switches. Theoretical analysis and numerical simulations demonstrated that hierarchically coupled biochemical switches can depict the three fundamental aspects of Waddington’s epigenetic landscape: (a) differentiation trajectories exhibit hierarchical branching, (b) attractors are robust to perturbations (homeorhesis), and (c) the proportions of specialized cells are preserved. It was further shown that appropriate external interventions can induce either probabilistic cellular reprogramming or highly predictable reprogramming outcomes. The incorporation of biochemical switches, rather than purely abstract normal forms, can contribute to more biologically grounded mathematical models of stem cell differentiation. This work also highlights the importance of normal forms for qualitatively understanding cell state dynamics and for building realistic modular GRNs. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine, 2nd Edition)

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18 pages, 235 KB

Open AccessEditor’s ChoiceArticle

Solving of a Variational Inequality Problem Under the Presence of Computational Errors

by Alexander J. Zaslavski

Mathematics 2026, 14(4), 664; https://doi.org/10.3390/math14040664 - 13 Feb 2026

Cited by 2 | Viewed by 449

Abstract

W. Takahashi and M. Toyoda (2003) proved weak convergence of an iteration process of solving a variational inequality problem for an inverse strongly-monotone mapping. In our recent work we showed that, for the same iterative process, most of its exact iterates are approximate [...] Read more.

W. Takahashi and M. Toyoda (2003) proved weak convergence of an iteration process of solving a variational inequality problem for an inverse strongly-monotone mapping. In our recent work we showed that, for the same iterative process, most of its exact iterates are approximate solutions of the variational inequality. In this paper, we show that the iteration process for solving a variational inequality problem for an inverse strongly monotone mapping generates an approximate solution in the presence of small computational errors. We also estimate a number of iterates needed in order to obtain such an approximate solution. Full article

(This article belongs to the Special Issue Variational Problems and Applications, 3rd Edition)

16 pages, 300 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

by Liqun Qi, Chunfeng Cui and Yi Xu

Mathematics 2026, 14(4), 635; https://doi.org/10.3390/math14040635 - 11 Feb 2026

Cited by 3 | Viewed by 445

Abstract

We study positive semi-definite (PSD) biquadratic forms and their sum-of-squares (SOS) representations. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is an SOS. This extends [...] Read more.

We study positive semi-definite (PSD) biquadratic forms and their sum-of-squares (SOS) representations. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is an SOS. This extends the known result for fully symmetric biquadratic forms. Furthermore, we describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For

m \geq 2

, we provide a explicit example to show the lower bound for sos rank of

m \times 2

biquadratic forms is

m + 1

, and show that previously proved results indicating that a

2 \times 2

PSD biquadratic form can be expressed as the sum of three squares and a

3 \times 2

PSD biquadratic form can be expressed as the sum of four squares are tight. We also present an

3 \times 3

SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares. Moreover, we establish a universal upper bound

m n - 1

for any

m \times n

SOS biquadratic form, which improves the trivial bound

m n

. Full article

14 pages, 1049 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fractional Fuzzy Force-Position Control of Constrained Robots

by Aldo Jonathan Muñoz-Vázquez, Mohamed Gharib, Juan Diego Sánchez-Torres and Anh-Tu Nguyen

Mathematics 2026, 14(3), 565; https://doi.org/10.3390/math14030565 - 4 Feb 2026

Viewed by 603

Abstract

Modern robotic tasks often require interaction with the surrounding elements in the workspace. In some high-precision tasks, it is essential to stabilize the contact force on a smooth yet rigid surface, which can be modeled as a unilateral constraint. This challenge becomes increasingly [...] Read more.

Modern robotic tasks often require interaction with the surrounding elements in the workspace. In some high-precision tasks, it is essential to stabilize the contact force on a smooth yet rigid surface, which can be modeled as a unilateral constraint. This challenge becomes increasingly complex in the presence of disturbances. This study addresses these issues using a robust fuzzy force-position controller that combines the approximation capabilities of fuzzy inference systems with the nonlocal properties of fractional operators. The proposed approach extends the error integration to include proportional-integral-derivative (PID) components of the position error, along with the integral of the contact force error. This formulation leverages the orthogonality between force and velocity subspaces to achieve accurate force-position stabilization. Additionally, an adaptive mechanism enhances closed-loop performance and robustness. The effectiveness of the proposed controller is validated through analytical derivations and simulations, thereby demonstrating its reliability in constrained environments. Full article

(This article belongs to the Special Issue Recent Advances in Adaptive Control Theory and Applications for Nonlinear Systems)

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