Mathematics

15 pages, 325 KB

Open AccessFeature PaperEditor’s ChoiceArticle

by Martín Cera, Pedro García-Vázquez and Juan Carlos Valenzuela-Tripodoro

Mathematics 2026, 14(3), 466; https://doi.org/10.3390/math14030466 - 29 Jan 2026

Viewed by 866

A Roman dominating function for a (non-weighted) graph

G = (V, E)

is a function

f : V \to {0, 1, 2}

such that every vertex

u \in V

with

f (u) = 0

[...] Read more.

A Roman dominating function for a (non-weighted) graph

G = (V, E)

is a function

f : V \to {0, 1, 2}

such that every vertex

u \in V

with

f (u) = 0

has at least one neighbor

v \in V

such that

f (v) = 2

. The minimum weight

\sum_{v \in V} f (v)

of a Roman dominating function f on G is called the Roman domination number of G and is denoted by

γ_{R} (G)

. A graph

G = (V, E)

, together with a positive real-valued weight-function

w : V \to R^{> 0}

, is called a weighted graph and is denoted by

(G; w)

. The minimum weight

\sum_{v \in V} f (v) w (v)

of a Roman dominating function f on G is called the weighted Roman domination number of G and is denoted by

γ_{w R} (G)

. The domination and Roman domination numbers of unweighted graphs have been extensively studied, particularly for their applications in bioinformatics and computational biology. However, graphs used to model biomolecular structures often require weights to be biologically meaningful. In this paper, we initiate the study of the weighted Roman domination number in weighted graphs. We first establish several bounds for this parameter and present various realizability results. Furthermore, we determine the exact values for several well-known graph families and demonstrate an equivalence between the weighted Roman domination number and the differential of a weighted graph. Full article

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

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20 pages, 1262 KB

Open AccessFeature PaperEditor’s ChoiceArticle

An Adaptive Scheme for Neuron Center Selection to Design an Efficient Radial Basis Neural Network Using PSO

by Arshad Afzal

Mathematics 2026, 14(3), 469; https://doi.org/10.3390/math14030469 - 29 Jan 2026

Cited by 1 | Viewed by 576

Abstract

An adaptive and efficient particle swarm optimization (PSO)-based learning algorithm to determine neuron centers in the hidden layer of a radial basis neural network (RBNN) is developed in the present work for regression problems. The proposed PSO–RBNN algorithm searches the entire input domain [...] Read more.

An adaptive and efficient particle swarm optimization (PSO)-based learning algorithm to determine neuron centers in the hidden layer of a radial basis neural network (RBNN) is developed in the present work for regression problems. The proposed PSO–RBNN algorithm searches the entire input domain space to discover optimal neuron centers by solving an optimization problem and aims to overcome the limitation of center selection from the training data. The network is built in a sequential manner using optimal neuron centers until some specified criterion is met, and therefore, it exploits the concept of neuron significance during the learning process. The Gaussian function with a constant spread (also known as width) is chosen as the radial basis function for each neuron. To illustrate the effectiveness of the PSO–RBNN algorithm over the orthogonal least squares (OLS) method (a popular learning algorithm under a similar category, which selects the neuron center from training data), numerical simulations for different types of nonlinear problems of varying dimensions and complexities are conducted. Finally, a comparison with multiple existing algorithms for network design is made using available data. The results show that the RBNN architecture developed with the proposed learning algorithm exhibits superior convergence, displays good generalization ability, and requires a smaller number of neurons, resulting in an efficient and compact network architecture. Full article

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

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32 pages, 382 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Quantitative Modeling of Investment–Output Dynamics: A Panel NARDL and GMM-Arellano–Bond Approach with Evidence from the Circular Economy

by Dorin Jula, Nicolae-Marius Jula and Kamer-Ainur Aivaz

Mathematics 2026, 14(3), 463; https://doi.org/10.3390/math14030463 - 28 Jan 2026

Cited by 4 | Viewed by 1481

Abstract

This study develops an integrated panel econometric framework for modeling investment–output dynamics in circular economy sectors, explicitly addressing dynamic propagation, long-run equilibrium relationships, endogeneity, and nonlinear responses. Building on the Samuelson–Hicks Multiplier–Accelerator model, the analysis combines two complementary approaches. A dynamic panel specification [...] Read more.

This study develops an integrated panel econometric framework for modeling investment–output dynamics in circular economy sectors, explicitly addressing dynamic propagation, long-run equilibrium relationships, endogeneity, and nonlinear responses. Building on the Samuelson–Hicks Multiplier–Accelerator model, the analysis combines two complementary approaches. A dynamic panel specification estimated by the Generalized Method of Moments (Arellano–Bond) is employed to capture output inertia, intertemporal transmission of investment shocks, and stability properties of the dynamic system. In parallel, a nonlinear panel ARDL model estimated using the Pooled Mean Group (PMG/NARDL) methodology is used to identify cointegration and to distinguish between the long-run and short-run effects of positive and negative investment variations. The empirical analysis relies on a balanced panel of 28 European economies (EU-27 and the United Kingdom) over the period 2005–2023, using sectoral circular economy data, with gross value added as the output variable and gross private investment as the main regressor. The results indicate the existence of a stable cointegrated relationship between investment and output, characterized by significant asymmetries, with expansionary investment shocks exerting larger and more persistent effects than contractionary shocks. Dynamic GMM estimates further confirm delayed investment effects and a stable autoregressive structure. Overall, the paper contributes to mathematical economic modeling by providing a unified dynamic–equilibrium panel framework and by extending the empirical relevance of Multiplier–Accelerator dynamics to circular economy systems. Full article

(This article belongs to the Special Issue Quantitative Analysis and Mathematical Modeling in Economics and Financial Decision-Making)

20 pages, 1930 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Is Weniger’s Transformation Capable of Simulating the Stieltjes Function Branch Cut?

by Riccardo Borghi

Mathematics 2026, 14(2), 376; https://doi.org/10.3390/math14020376 - 22 Jan 2026

Viewed by 675

Abstract

The resummation of Stieltjes series remains a key challenge in mathematical physics, especially when Padé approximants fail, as in the case of superfactorially divergent series. Weniger’s

δ

-transformation, which incorporates a priori structural information on Stieltjes series, offers a superior framework with respect [...] Read more.

The resummation of Stieltjes series remains a key challenge in mathematical physics, especially when Padé approximants fail, as in the case of superfactorially divergent series. Weniger’s

δ

-transformation, which incorporates a priori structural information on Stieltjes series, offers a superior framework with respect to Padé. In the present work, the following fundamental question is addressed: Is the

δ

-transformation, once it is applied to a typical Stieltjes series, capable of correctly simulating the branch cut structure of the corresponding Stieltjes function? Here, it is proved that the intrinsic log-convexity of the Stieltjes moment sequence (guaranteed via the positivity of Hankel’s determinants) allows the necessary condition for

δ

to have all real poles to be satisfied. The same condition, however, is not sufficient to guarantee this. In attempting to bridge such a gap, we propose a mechanism rooted in the iterative action of a specific linear differential operator acting on a class of suitable auxiliary log-concave polynomials. To this end, we show that the denominator of the

δ

-approximants can always be recast as a high-order derivative of a log-concave polynomial. Then, on invoking the Gauss–Lucas theorem, a consistent geometrical justification of the

δ

pole positioning is proposed. Through such an approach, the pole alignment along the negative real axis can be viewed as the result of the progressive restriction of the convex hull under differentiation. Since a fully rigorous proof of this conjecture remains an open challenge, in order to substantiate it, a comprehensive numerical investigation across an extensive catalog of Stieltjes series is proposed. Our results provide systematic evidence of the potential

δ

-transformation ability to mimic the singularity structure of several target functions, including those involving superfactorial divergences. Full article

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

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34 pages, 495 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Rigidity and Toledo Invariant for Spin*(8)-Higgs Bundles

by Álvaro Antón-Sancho

Mathematics 2026, 14(2), 358; https://doi.org/10.3390/math14020358 - 21 Jan 2026

Viewed by 511

Abstract

In this paper, we study

{Spin}^{*} (8)

-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on

{SO}^{*} (8)

. The group

{Spin}^{*} (8)

is exceptional among classical real [...] Read more.

In this paper, we study

{Spin}^{*} (8)

-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on

{SO}^{*} (8)

. The group

{Spin}^{*} (8)

is exceptional among classical real forms, as its complexification

Spin (8, C)

admits triality, an outer automorphism of order 3, but triality does not preserve the real form

{Spin}^{*} (8)

. We establish the Toledo bound

| τ | \leq 4 (g - 1)

for semistable

{Spin}^{*} (8)

-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the

{SO}^{*} (8)

moduli space with discrete fibers parametrized by spin structures, and has a dimension of

15 (g - 1)

, one less than expected. Using Morse theory, we establish connectedness of moduli spaces for

τ = 0

and maximal

| τ |

. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into

{Spin}^{*} (8)

. We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form. Full article

(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)

21 pages, 918 KB

Open AccessEditor’s ChoiceArticle

A Numerical Approach for the Simultaneous Identification of a Source Term and a Robin Boundary Coefficient in Time-Fractional Reaction–Diffusion Equations

by Miglena N. Koleva

Mathematics 2026, 14(2), 324; https://doi.org/10.3390/math14020324 - 18 Jan 2026

Viewed by 935

Abstract

In the present study, we develop numerical approaches for the simultaneous determination of a time-dependent right-hand side and a Robin boundary coefficient in linear and quasilinear Caputo time-fractional reaction–diffusion problems based on boundary and interior observations. The well-posedness of the corresponding direct problems [...] Read more.

In the present study, we develop numerical approaches for the simultaneous determination of a time-dependent right-hand side and a Robin boundary coefficient in linear and quasilinear Caputo time-fractional reaction–diffusion problems based on boundary and interior observations. The well-posedness of the corresponding direct problems is established. A temporal semidiscretization is first constructed using the

L 2 - 1_{σ}

scheme, and the solution is decomposed with respect to the unknown functions. The correctness of the proposed method is proved. For the nonlinear diffusion problem, a quasilinearization technique is employed, and the spatial discretization is carried out using finite difference schemes. An iterative procedure is developed to solve the resulting inverse problem. Numerical simulations with noisy data are presented and discussed to demonstrate the efficiency of the method. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)

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25 pages, 1954 KB

Open AccessFeature PaperEditor’s ChoiceArticle

RMFGP: A Rotated Multi-Fidelity Gaussian Process Framework for Supervised Dimension Reduction

by Jiahao Zhang, Shiqi Zhang and Guang Lin

Mathematics 2026, 14(2), 325; https://doi.org/10.3390/math14020325 - 18 Jan 2026

Cited by 1 | Viewed by 950

Abstract

High-dimensional surrogate modeling with limited high-fidelity data poses a major challenge in uncertainty quantification. Classical supervised dimension reduction methods often fail in this setting due to insufficient accurate observations, while low-fidelity data are abundant but biased. In this work, we propose a Rotated [...] Read more.

High-dimensional surrogate modeling with limited high-fidelity data poses a major challenge in uncertainty quantification. Classical supervised dimension reduction methods often fail in this setting due to insufficient accurate observations, while low-fidelity data are abundant but biased. In this work, we propose a Rotated Multi-Fidelity Gaussian Process (RMFGP) framework that enables reliable dimension reduction and surrogate construction under severe data scarcity. The proposed method integrates nonlinear multi-fidelity Gaussian process regression with sliced average variance estimation (SAVE) to iteratively identify informative input directions. Low-fidelity data are first used to extract coarse structural information, which is exploited to rotate the input space prior to multi-fidelity model training. Predictions generated by the trained RMFGP surrogate are then used to refine the dimension reduction, allowing accurate estimation of the central sufficient dimension reduction subspace even when high-fidelity data are scarce. A Bayesian active learning strategy based on predictive uncertainty is further incorporated to adaptively select new high-fidelity samples. Numerical examples, including stochastic partial differential equations, demonstrate that RMFGP significantly improves prediction accuracy, convergence, and uncertainty propagation compared to existing Gaussian process-based dimension reduction approaches, while requiring substantially fewer high-fidelity evaluations. Full article

(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)

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8 pages, 248 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fermi Sea Topology and Boundary Geometry for Free Particles in One- and Two-Dimensional Lattices

by Guillermo R. Zemba

Mathematics 2026, 14(2), 303; https://doi.org/10.3390/math14020303 - 15 Jan 2026

Viewed by 924

Abstract

Free gases of spinless fermions moving on a lattice-symmetric geometric background are considered. Their topological properties at zero temperature can be used to classify their Fermi seas and associated boundaries. The flat orbifolds

R^{d} / Γ

, where

Γ

is the crystallographic [...] Read more.

Free gases of spinless fermions moving on a lattice-symmetric geometric background are considered. Their topological properties at zero temperature can be used to classify their Fermi seas and associated boundaries. The flat orbifolds

R^{d} / Γ

, where

Γ

is the crystallographic group of symmetry in d-dimensional momentum space, are used to accomplish this task. Two topological classes exist for

d = 1

: an interval, which is identified as a conductor, and a circumference, which corresponds to an insulator. The number of topological classes increases to 17 for

d = 2

: 8 have the topology of a disk, that are generally recognized as conductors, and 4 correspond to a two-sphere, matching insulators. Both sets eventually contain a finite number of conical singularities and reflection corners at the boundaries. The remaining cases in the listing relate to conductors (annulus, Möbius strip) and insulators (two-torus, real projective plane, Klein bottle). Examples that fall under this list are given, along with physical interpretations of the singularities. It is anticipated that the findings of this classification will be robust under perturbative interactions due to its topological character. Full article

(This article belongs to the Special Issue Effective Field Theories for Condensed Matter and Statistical Systems)

29 pages, 1782 KB

Open AccessEditor’s ChoiceArticle

Reinforcement Learning-Guided NSGA-II Enhanced with Gray Relational Coefficient for Multi-Objective Optimization: Application to NASDAQ Portfolio Optimization

by Zhiyuan Wang, Qinxu Ding, Ding Ding, Siying Zhu, Jing Ren, Yue Wang and Chong Hui Tan

Mathematics 2026, 14(2), 296; https://doi.org/10.3390/math14020296 - 14 Jan 2026

Cited by 3 | Viewed by 1938

Abstract

In modern financial markets, decision-makers increasingly rely on quantitative methods to navigate complex trade-offs among multiple, often conflicting objectives. This paper addresses constrained multi-objective optimization (MOO) with an application to portfolio optimization for minimizing risk and maximizing return. To this end, and to [...] Read more.

In modern financial markets, decision-makers increasingly rely on quantitative methods to navigate complex trade-offs among multiple, often conflicting objectives. This paper addresses constrained multi-objective optimization (MOO) with an application to portfolio optimization for minimizing risk and maximizing return. To this end, and to address existing gaps, we propose a novel reinforcement learning (RL)-guided non-dominated sorting genetic algorithm II (NSGA-II) enhanced with gray relational coefficients (GRC), termed RL-NSGA-II-GRC, which combines an RL agent controller and GRC-based selection to improve the convergence and diversity of the Pareto-optimal fronts. The agent adapts key evolutionary parameters online using population-level metrics of hypervolume, feasibility, and diversity, while the GRC-enhanced tournament operator ranks parents via a unified score simultaneously considering dominance rank, crowding distance, and geometric proximity to ideal reference. We evaluate the framework on the Kursawe and CONSTR benchmark problems and on a NASDAQ portfolio optimization application. On the benchmarks, RL-NSGA-II-GRC achieves convergence metric improvements of about 5.8% and 4.4% over the original NSGA-II, while preserving a well-distributed set of non-dominated solutions. In the portfolio application, the method produces a smooth and densely populated efficient frontier that supports the identification of the maximum Sharpe ratio portfolio (with annualized Sharpe ratio = 1.92), as well as utility-optimal portfolios for different risk-aversion levels. The main contributions of this work are three-fold: (1) we propose an RL-NSGA-II-GRC method that integrates an RL agent into the evolutionary framework to adaptively control key parameters using generational feedback; (2) we design a GRC-enhanced binary tournament selection operator that provides a comprehensive performance indicator to efficiently guide the search toward the Pareto-optimal front; (3) we demonstrate, on benchmark MOO problems and a NASDAQ portfolio case study, that the proposed method delivers improved convergence and well-populated efficient frontiers that support actionable investment insights. Full article

(This article belongs to the Special Issue Multi-Objective Evolutionary Algorithms and Their Applications)

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21 pages, 6222 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Weighted, Mixed ℓ_p Norm Regularization for Gaussian Noise-Based Denoising Method Extension

by Yuanmin Wang and Jinsong Leng

Mathematics 2026, 14(2), 298; https://doi.org/10.3390/math14020298 - 14 Jan 2026

Viewed by 823

Abstract

Many denoising methods model noise as Gaussian noise. However, the realistic noise captured by camera devices does not satisfy Gaussian distribution. Hence, those methods do not perform well when being applied to real-world image denoising tasks. In this work, we indicate that the [...] Read more.

Many denoising methods model noise as Gaussian noise. However, the realistic noise captured by camera devices does not satisfy Gaussian distribution. Hence, those methods do not perform well when being applied to real-world image denoising tasks. In this work, we indicate that the spatial correlation in noise and the variation of noise intensity are the main factors that impact the performance of Gaussian noise-based methods, and accordingly propose an extension of the method based on the weighted, mixed non-convex

ℓ_{p}

norm. The proposed method first strengthens the intensity of the noise pattern in the original denoising result through the Guided Filter, then removes the over-amplified frequency in the local area by the proposed regularization term. We prove that the optimal solution can be achieved through the sub-gradient-based iterative optimization scheme, and further reduce the computational cost by optimizing the initial values. Numerical experiments show that the proposed extending method can balance well texture preservation and noise removal, and the PSNR of the extending method’s results are greatly improved, even outperforming the recently proposed realistic noise removal methods which also include deep learning based methods. Full article

(This article belongs to the Special Issue Mathematical Methods for Image Processing and Computer Vision)

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18 pages, 16707 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Robust Trajectory Tracking for Omnidirectional Mobile Robots with Input Time Delay: An ADRC Approach

by Alberto Luviano-Juárez, Mario Ramírez-Neria and Jaime González-Sierra

Mathematics 2026, 14(2), 266; https://doi.org/10.3390/math14020266 - 10 Jan 2026

Cited by 2 | Viewed by 829

Abstract

In this article, the problem of control of the kinematic model of an omnidirectional robot with time delay in the control input is tackled through an Active Disturbance Rejection Control (ADRC) with a disturbance predictor-based scheme, which consists in predicting the generalized forward [...] Read more.

In this article, the problem of control of the kinematic model of an omnidirectional robot with time delay in the control input is tackled through an Active Disturbance Rejection Control (ADRC) with a disturbance predictor-based scheme, which consists in predicting the generalized forward disturbance input in order to cancel it and then using a feedforward linearization approach to control the system in trajectory tracking tasks. The novelties of the scheme are to demonstrate that using the proposed extended state disturbance estimation leads to a forward estimation following the Taylor series approximation, and, to avoid using additional pose predictions, a feedforward input as an exact linearization approach is used, in which the remaining dynamics can be lumped into the generalized disturbance input. Thus, the use of extended states in prediction improves the robustness of the predictor while increasing the prediction horizon for larger time delays. The stability of the proposal is demonstrated using the second method of Lyapunov, which shows the closed-loop estimation/tracking ultimate bound behavior. Additionally, numerical simulations and experimental tests validate the robustness of the approach in trajectory-tracking tasks. Full article

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

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27 pages, 2126 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Two Types of Proximal Connected-Image Contractions in b-Metric Spaces and Applications to Fractional Differential Models

by Khuanchanok Chaichana, Ben Wongsaijai, Adsadang Himakalasa and Phakdi Charoensawan

Mathematics 2026, 14(2), 262; https://doi.org/10.3390/math14020262 - 9 Jan 2026

Cited by 2 | Viewed by 536

Abstract

In this paper, we introduce a new class of proximal connected-image contractions in the framework of b-metric spaces endowed with a directed graph. By employing auxiliary functions, we establish several existence and uniqueness results for best proximity points and fixed points under [...] Read more.

In this paper, we introduce a new class of proximal connected-image contractions in the framework of b-metric spaces endowed with a directed graph. By employing auxiliary functions, we establish several existence and uniqueness results for best proximity points and fixed points under appropriate contractive conditions. To demonstrate the applicability of our theoretical findings, we apply the main results to a class of fractional differential equations, showing the effectiveness of the proposed approach. Full article

(This article belongs to the Special Issue Advanced Theoretical and Numerical Methods for Fractional Differential and Difference Equations)

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15 pages, 2743 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Constructing Two Edge-Disjoint Hamiltonian Cycles in BCube Data Center Networks for All-to-All Broadcasting

by Kung-Jui Pai

Mathematics 2026, 14(2), 232; https://doi.org/10.3390/math14020232 - 8 Jan 2026

Cited by 2 | Viewed by 1044

Abstract

The rapid growth in demand for diverse network application services has driven the continuous development and expansion of data centers. BCubes was proposed by Microsoft Research Asia for designing modular data centers, and it is a multi-layer recursively constructed network with many advantages. [...] Read more.

The rapid growth in demand for diverse network application services has driven the continuous development and expansion of data centers. BCubes was proposed by Microsoft Research Asia for designing modular data centers, and it is a multi-layer recursively constructed network with many advantages. This article shows that BCube is the existence of two edge-disjoint Hamiltonian cycles, abbreviated as two EDHCs, which provide two significant benefits in data center operations: (1) parallel data broadcast and (2) edge fault-tolerance in network communications. We present the following results in this paper: (1) By utilizing the network topology characteristics, we first provide construction algorithms for two EDHCs on low-dimensional BCubes. (2) Based on the algorithm and the recursive structure of BCubes, we prove that two EDHCs exist for all BCubes. (3) Considering all-to-all broadcasting using two EDHCs as transmission channels, we evaluate the performance of all-to-all broadcasting through simulations on low-dimensional BCubes. Full article

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45 pages, 588 KB

Open AccessEditor’s ChoiceArticle

Mixed Parity Variants of Apéry-Type Binomial Series and Level Four Colored Multiple Zeta Values

by Ce Xu and Jianqiang Zhao

Mathematics 2026, 14(2), 222; https://doi.org/10.3390/math14020222 - 7 Jan 2026

Cited by 1 | Viewed by 559

Abstract

In this paper, we study an Apéry-type series involving the central binomial coefficients

\sum_{n_{1} > \dots > n_{d} > 0} \frac{1}{4^{n_{1}}} (\binom{2 n_{1}}{n_{1}}) n_{1}^{- s_{1}} \dots n_{d}^{- s_{d}}

[...] Read more.

In this paper, we study an Apéry-type series involving the central binomial coefficients

\sum_{n_{1} > \dots > n_{d} > 0} \frac{1}{4^{n_{1}}} (\binom{2 n_{1}}{n_{1}}) n_{1}^{- s_{1}} \dots n_{d}^{- s_{d}}

and its variations where the summation indices may have mixed parities and some or all “>” are replaced by “≥”, as long as the series are defined. These sums have naturally appeared in the calculation of massive Feynman integrals by the work of Jegerlehner, Kalmykov, and Veretin. We show that all these sums can be expressed as

Q

-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level four, i.e., special values of multiple polylogarithms at fourth roots of unity. For example, our main theorem shows that when

n_{1}^{- s_{1}}

is replaced by

{(2 n_{1})}^{- s_{1}}

and other

n_{j}^{- s_{j}}

’s are replaced by either

{(2 n_{j})}^{- s_{j}}

or

{(2 n_{j} + 1)}^{- s_{j}}

, then all the colored multiple zeta values can be chosen to have the same weight

s_{1} + \dots + s_{d}

, but the weights of these values are only bounded by

s_{1} + \dots + s_{d}

for general variant Apéry-type series of mixed parities. We also show that the corresponding series where

(\binom{2 n_{1}}{n_{1}}) / 4^{n_{1}}

is replaced by

{(\binom{2 n_{1}}{n_{1}})}^{2} / 16^{n_{1}}

can be expressed in a similar way except for a possible extra factor of

1 / π

, with the weight of the colored multiple zeta values similarly bounded. Full article

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

40 pages, 1118 KB

Open AccessEditor’s ChoiceArticle

FORCE: Fast Outlier-Robust Correlation Estimation via Streaming Quantile Approximation for High-Dimensional Data Streams

by Sooyoung Jang and Changbeom Choi

Mathematics 2026, 14(1), 191; https://doi.org/10.3390/math14010191 - 4 Jan 2026

Cited by 4 | Viewed by 1877

Abstract

The estimation of correlation matrices in high-dimensional data streams presents a fundamental conflict between computational efficiency and statistical robustness. Moment-based estimators, such as Pearson’s correlation, offer linear

O (N)

complexity but lack robustness. In contrast, high-breakdown methods like the minimum covariance [...] Read more.

The estimation of correlation matrices in high-dimensional data streams presents a fundamental conflict between computational efficiency and statistical robustness. Moment-based estimators, such as Pearson’s correlation, offer linear

O (N)

complexity but lack robustness. In contrast, high-breakdown methods like the minimum covariance determinant (MCD) are computationally prohibitive (

O (N p^{2} + p^{3})

) for real-time applications. This paper introduces Fast Outlier-Robust Correlation Estimation (FORCE), a streaming algorithm that performs adaptive coordinate-wise trimming using the

P^{2}

algorithm for streaming quantile approximation, requiring only

O (p)

memory independent of stream length. We evaluate FORCE against six baseline algorithms—including exact trimmed methods (TP-Exact, TP-TER) that use

O (N log N)

sorting with

O (N p)

storage—across five benchmark datasets spanning synthetic, financial, medical, and genomic domains. FORCE achieves speedups of approximately 470× over FastMCD and 3.9× over Spearman’s rank correlation. On S&P 500 financial data, coordinate-wise trimmed methods substantially outperform FastMCD: TP-Exact achieves the best RMSE (0.0902), followed by TP-TER (0.0909) and FORCE (0.1186), compared to FastMCD’s 0.1606. This result demonstrates that coordinate-wise trimming better accommodates volatility clustering in financial time series than multivariate outlier exclusion. FORCE achieves 76% of TP-Exact’s accuracy while requiring

10^{4} \times

less memory, enabling robust estimation in true streaming environments where data cannot be retained for batch processing. We validate the 25% breakdown point shared by all IQR-based trimmed methods using the ODDS-satellite benchmark (31.7% contamination), confirming identical degradation for FORCE, TP-Exact, and TP-TER. For memory-constrained streaming applications with contamination below 25%, FORCE provides the only viable path to robust correlation estimation with bounded memory. Full article

(This article belongs to the Special Issue Modeling and Simulation for Optimizing Complex Dynamical Systems)

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19 pages, 4702 KB

Open AccessFeature PaperEditor’s ChoiceArticle

How Far Can We Trust Chaos? Extending the Horizon of Predictability

by Alexandros K. Angelidis, Georgios C. Makris, Evangelos Ioannidis, Ioannis E. Antoniou and Charalampos Bratsas

Mathematics 2025, 13(23), 3851; https://doi.org/10.3390/math13233851 - 1 Dec 2025

Cited by 1 | Viewed by 8813

Abstract

Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision that is inherent in all computational machines. Beyond the horizon of predictability, numerical computations accumulate [...] Read more.

Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision that is inherent in all computational machines. Beyond the horizon of predictability, numerical computations accumulate errors, often undetectable. We investigate the possibility of reliable (error-free) time series of chaos. We prove that this is feasible for two well-studied isomorphic chaotic maps, namely the Tent map and the Logistic map. The generated chaotic time series have an unlimited horizon of predictability. A new linear formula for the horizon of predictability of the Analytic Computation of the Logistic map, for any given precision and acceptable error, is obtained. Reliable (error-free) time series of chaos serve as the “gold standard” for chaos applications. The practical significance of our findings include: (i) the ability to compare the performance of neural networks that predict chaotic time series; (ii) the reliability and numerical accuracy of chaotic orbit computations in encryption, maintaining high cryptographic strength; and (iii) the reliable forecasting of future prices in chaotic economic and financial models. Full article

(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)

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19 pages, 5590 KB

Open AccessEditor’s ChoiceArticle

Out of Distribution Adaptation in Offline RL via Causal Normalizing Flows

by Minjae Cho and Chuangchuang Sun

Mathematics 2025, 13(23), 3835; https://doi.org/10.3390/math13233835 - 30 Nov 2025

Cited by 1 | Viewed by 1567

Abstract

Despite the success of reinforcement learning (RL), the common assumption of online interaction prevents its widespread adoption. Offline RL has emerged as an alternative that learns a policy from precollected data. However, this learning paradigm introduces a new challenge called “distributional shift”, degrading [...] Read more.

Despite the success of reinforcement learning (RL), the common assumption of online interaction prevents its widespread adoption. Offline RL has emerged as an alternative that learns a policy from precollected data. However, this learning paradigm introduces a new challenge called “distributional shift”, degrading the performance of the policy when evaluated on out-of-distribution (OOD) scenarios (i.e., outside of the training data). Most existing works resolve this by policy regularization to optimize a policy within the support of the data. However, this overlooks the potential for high-reward regions outside of the data. This motivates offline policy optimization that is capable of finding high-reward regions outside of the data. In this paper, we devise a causality-based model architecture to accurately capture the OOD scenarios wherein the policy can be optimized without performance degradation. Specifically, we adapt causal normalizing flows (CNFs) to learn the transition dynamics and reward function for data generation and augmentation in offline policy learning. Based on the physics-based qualitative causal graph and precollected data, we develop a model-based offline OOD-adapting causal RL (MOOD-CRL) algorithm to learn the quantitative structural causal model. Consequently, MOOD-CRL can exercise counterfactual reasoning for sequential decision-making, revealing a high potential for OOD adaptation. The effectiveness is validated through extensive empirical evaluations with ablations including data quality and algorithmic sensitivity. Our results show that MOOD-CRL achieves comparable results with its online counterparts and consistently outperforms state-of-the-art model-free and model-based baselines by a significant margin. Full article

(This article belongs to the Section D: Statistics and Operational Research)

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27 pages, 437 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Polarities of Exceptional Geometries of Type

E_{6}

by Vincent Batens and Hendrik Van Maldeghem

Mathematics 2025, 13(23), 3804; https://doi.org/10.3390/math13233804 - 27 Nov 2025

Viewed by 1023

Abstract

A polarity of an exceptional geometry of type

E_{6}

is called regularif its fixed structure, viewed as a simplicial complex, is a building. Polarities that do not act trivially on the underlying field were classified a long time ago by Jacques Tits. [...] Read more.

A polarity of an exceptional geometry of type

E_{6}

is called regularif its fixed structure, viewed as a simplicial complex, is a building. Polarities that do not act trivially on the underlying field were classified a long time ago by Jacques Tits. In the present paper, we classify the regular polarities of exceptional geometries of type

E_{6}

that act trivially on the underlying (arbitrary) field. As a result, we discover new subgeometries of the exceptional geometry of type

E_{6}

. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

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32 pages, 3820 KB

Open AccessFeature PaperEditor’s ChoiceArticle

FAS-XAI: Fuzzy and Explainable AI for Interpretable Vetting of Kepler Exoplanet Candidates

by Gabriel Marín Díaz

Mathematics 2025, 13(23), 3796; https://doi.org/10.3390/math13233796 - 26 Nov 2025

Cited by 3 | Viewed by 1601

Abstract

The detection of exoplanets in space-based photometry relies on identifying periodic transit signatures in stellar light curves. The Kepler Threshold Crossing Events (TCE) catalog collects all periodic dimming signals detected by the pipeline, while the Kepler Objects of Interest (KOI) catalog provides vetted [...] Read more.

The detection of exoplanets in space-based photometry relies on identifying periodic transit signatures in stellar light curves. The Kepler Threshold Crossing Events (TCE) catalog collects all periodic dimming signals detected by the pipeline, while the Kepler Objects of Interest (KOI) catalog provides vetted dispositions (CONFIRMED, CANDIDATE, FALSE POSITIVE). However, the pathway from raw TCE detections to KOI classifications remains ambiguous in many borderline cases. We introduce FAS-XAI, a framework that integrates Fuzzy C-Means (FCM) clustering, supervised learning, and explainable AI (XAI) to improve transparency in exoplanet candidate classification. FCM applied to TCE parameters (period, duration, depth, and SNR) reveals three meaningful regimes in the transit-signal space and quantifies ambiguity through fuzzy memberships. Linking these clusters to KOI dispositions highlights a progressive consolidation of confirmed planets within the high-SNR, medium-duration regime. A supervised XGBoost classifier trained on KOI labels and augmented with fuzzy memberships achieves strong performance (Accuracy = 0.73, Macro F1 = 0.69, ROC–AUC = 0.855), clearly separating CONFIRMED and FALSE POSITIVE objects while appropriately reflecting the transitional nature of CANDIDATES. SHAP, LIME, and ELI5 provide consistent global and local attributions, identifying period, duration, depth, SNR, and fuzzy ambiguity as the key explanatory features. Finally, stellar parameters from Kepler DR25 validate the physical plausibility of the detected regimes, demonstrating that FAS-XAI captures astrophysically meaningful patterns rather than purely statistical structures. Overall, the framework illustrates how fuzzy logic and explainable AI can jointly enhance the interpretability and scientific rigor of exoplanet vetting pipelines. Full article

(This article belongs to the Special Issue Fuzzy Logic and Explainable AI in Mathematical Decision-Making)

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19 pages, 7441 KB

Open AccessFeature PaperEditor’s ChoiceArticle

All for One or One for All? A Comparative Study of Grouped Data in Mixed-Effects Additive Bayesian Networks

by Magali Champion, Matteo Delucchi and Reinhard Furrer

Mathematics 2025, 13(22), 3649; https://doi.org/10.3390/math13223649 - 14 Nov 2025

Viewed by 1089

Abstract

Additive Bayesian networks (ABNs) provide a flexible framework for modeling complex multivariate dependencies among variables of different distributions, including Gaussian, Poisson, binomial, and multinomial. This versatility makes ABNs particularly attractive in clinical research, where heterogeneous data are frequently collected across distinct groups. However, [...] Read more.

Additive Bayesian networks (ABNs) provide a flexible framework for modeling complex multivariate dependencies among variables of different distributions, including Gaussian, Poisson, binomial, and multinomial. This versatility makes ABNs particularly attractive in clinical research, where heterogeneous data are frequently collected across distinct groups. However, standard applications either pool all data together, ignoring group-specific variability, or estimate separate models for each group, which may suffer from limited sample sizes. In this work, we extend ABNs to a mixed-effect framework that accounts for group structure through partial pooling, and we evaluate its performance in a large-scale simulation study. We compare three strategies—partial pooling, complete pooling, and no pooling—cross a wide range of network sizes, sparsity levels, group configurations, and sample sizes. Performance is assessed in terms of structural accuracy, parameter estimation accuracy, and predictive performance. Our results demonstrate that partial pooling consistently yields superior structural and parametric accuracy while maintaining robust predictive performance across all evaluated settings for grouped data structures. These findings highlight the potential of mixed-effect ABNs as a versatile approach for learning probabilistic graphical models from grouped data with diverse distributions in real-world applications. Full article

(This article belongs to the Special Issue Bayesian Networks: Parameter and Structure Learning with Their Real-World Applications for Decision Making)

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19 pages, 965 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Discrete Time Scattering and Wold’s Decomposition in Pictures

by Rafi Rizqy Firdaus and Serge Richard

Mathematics 2025, 13(22), 3634; https://doi.org/10.3390/math13223634 - 13 Nov 2025

Viewed by 815

Abstract

Based on explicit computations, various concepts of discrete time scattering theory are reviewed, discussed, and illustrated. The dynamics take place on a discrete half-space. All operators are represented graphically. The expressions obtained for the wave operators lead to an easily visualized interpretation of [...] Read more.

Based on explicit computations, various concepts of discrete time scattering theory are reviewed, discussed, and illustrated. The dynamics take place on a discrete half-space. All operators are represented graphically. The expressions obtained for the wave operators lead to an easily visualized interpretation of Wold’s decomposition, a seminal result of operator theory. This work has a clear pedagogical orientation, with the aim of providing explicit formulas and graphical representations for operators which are usually only known to exist. Full article

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

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20 pages, 4034 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Preserving Multiple Conserved Quantities of Stochastic Differential Equations via Projection Technique

by Xuliang Li, Zhenyu Wang and Xiaohua Ding

Mathematics 2025, 13(22), 3614; https://doi.org/10.3390/math13223614 - 11 Nov 2025

Viewed by 934

Abstract

Stochastic differential equations (SDEs) with multiple conserved quantities are ubiquitous in scientific fields, modeling systems from molecular dynamics to celestial mechanics. While geometric numerical integrators that preserve single invariants are well-established, constructing efficient and high-order numerical schemes for SDEs with multiple conserved quantities [...] Read more.

Stochastic differential equations (SDEs) with multiple conserved quantities are ubiquitous in scientific fields, modeling systems from molecular dynamics to celestial mechanics. While geometric numerical integrators that preserve single invariants are well-established, constructing efficient and high-order numerical schemes for SDEs with multiple conserved quantities remains a challenge. Existing approaches often suffer from high computational costs or lack desirable numerical properties like symmetry. This paper introduces two novel classes of projection-based numerical methods tailored for SDEs with multiple conserved quantities. The first method projects the increments of an underlying numerical scheme onto a discrete tangent space, ensuring all invariants are preserved by construction. The second method leverages a local coordinates approach, transforming the SDE onto the manifold defined by the invariants, solving it numerically, and then projecting back, guaranteeing the solution evolves on the correct manifold. We prove that both methods inherit the mean-square convergence order of their underlying schemes. Furthermore, we propose a simplified strategy that reduces computational expense by redefining the multiple invariants into a single one, offering a practical trade-off between exact preservation and efficiency. Numerical experiments confirm the theoretical findings and demonstrate the superior efficiency and structure-preserving capabilities of our methods. Full article

(This article belongs to the Special Issue Advances in Computational Mathematics and Applied Mathematics, 2nd Edition)

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28 pages, 12813 KB

Open AccessEditor’s ChoiceArticle

Training-Free Few-Shot Image Classification via Kernel Density Estimation with CLIP Embeddings

by Marcos Sergio Pacheco dos Santos Lima Junior, Juan Miguel Ortiz-de-Lazcano-Lobato and Ezequiel López-Rubio

Mathematics 2025, 13(22), 3615; https://doi.org/10.3390/math13223615 - 11 Nov 2025

Cited by 1 | Viewed by 2448

Abstract

Few-shot image classification aims to recognize novel classes from only a handful of labeled examples, a challenge in domains where data collection is costly or impractical. Existing solutions often rely on meta learning, fine tuning, or data augmentation, introducing computational overhead, risk of [...] Read more.

Few-shot image classification aims to recognize novel classes from only a handful of labeled examples, a challenge in domains where data collection is costly or impractical. Existing solutions often rely on meta learning, fine tuning, or data augmentation, introducing computational overhead, risk of overfitting, or are not highly efficient. This paper introduces ProbaCLIP, a simple training-free approach that leverages Kernel Density Estimation (KDE) within the embedding space of Contrastive Language-Image Pre-training (CLIP). Unlike other CLIP-based methods, the proposed approach operates solely on visual embeddings and does not require text labels. Class-conditional probability densities were estimated from few-shot support examples, and queries were classified by likelihood evaluation, where Principal Component Analysis (PCA) was used for dimensionality reduction, compressing the dissimilarities between classes on each episode. We further introduced an optional bandwidth optimization strategy and a consensus decision mechanism through cross-validation, while addressing the special case of one-shot classification with distance-based measures. Extensive experiments on multiple datasets demonstrated that our method achieved competitive or superior accuracy compared to the state-of-the-art few-shot classifiers, reaching up to 98.37% accuracy in five-shot tasks and up to 99.80% in a 16-shot framework with ViT-L/14@336px. We proved our methodology by achieving high performance without gradient-based training, text supervision, or auxiliary meta-training datasets, emphasizing the effectiveness of combining pre-trained embeddings with statistical density estimation for data-scarce classification. Full article

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

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34 pages, 1584 KB

Open AccessEditor’s ChoiceArticle

Cost Optimization in a GI/M/2/N Queue with Heterogeneous Servers, Working Vacations, and Impatient Customers via the Bat Algorithm

by Abdelhak Guendouzi and Salim Bouzebda

Mathematics 2025, 13(21), 3559; https://doi.org/10.3390/math13213559 - 6 Nov 2025

Cited by 3 | Viewed by 1203

Abstract

This paper analyzes a finite-capacity

G I / M / 2 / N

queue with two heterogeneous servers operating under a multiple working-vacation policy, Bernoulli feedback, and customer impatience. Using the supplementary-variable technique in tandem with a tailored recursive scheme, we derive the [...] Read more.

This paper analyzes a finite-capacity

G I / M / 2 / N

queue with two heterogeneous servers operating under a multiple working-vacation policy, Bernoulli feedback, and customer impatience. Using the supplementary-variable technique in tandem with a tailored recursive scheme, we derive the stationary distributions of the system size as observed at pre-arrival instants and at arbitrary epochs. From these, we obtain explicit expressions for key performance metrics, including blocking probability, average reneging rate, mean queue length, mean sojourn time, throughput, and server utilizations. We then embed these metrics in an economic cost function and determine service-rate settings that minimize the total expected cost via the Bat Algorithm. Numerical experiments implemented in R validate the analysis and quantify the managerial impact of the vacation, feedback, and impatience parameters through sensitivity studies. The framework accommodates general renewal arrivals (

G I

), thereby extending classical (

M / M / 2 / N

) results to more realistic input processes while preserving computational tractability. Beyond methodological interest, the results yield actionable design guidance: (i) they separate Palm and time-stationary viewpoints cleanly under non-Poisson input, (ii) they retain heterogeneity throughout all formulas, and (iii) they provide a cost–optimization pipeline that can be deployed with routine numerical effort. Methodologically, we (i) characterize the generator of the augmented piecewise–deterministic Markov process and prove the existence/uniqueness of the stationary law on the finite state space, (ii) derive an explicit Palm–time conversion formula valid for non-Poisson input, (iii) show that the boundary-value recursion for the Laplace–Stieltjes transforms runs in linear time

O (N)

and is numerically stable, and (iv) provide influence-function (IPA) sensitivities of performance metrics with respect to

(μ_{1}, μ_{2}, ν, α, ϕ, β)

. Full article

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

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41 pages, 3112 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Bird’s-Eye View on a New Stochastic Interpretation of Quantum Mechanics

by Olavo L. Silva Filho and Marcello Ferreira

Mathematics 2025, 13(21), 3571; https://doi.org/10.3390/math13213571 - 6 Nov 2025

Cited by 4 | Viewed by 1626

Abstract

Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism [...] Read more.

Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities. Full article

(This article belongs to the Special Issue Advances in Mathematics for Quantum Mechanics)

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22 pages, 934 KB

Open AccessEditor’s ChoiceArticle

AORO: Auto-Optimizing Reasoning Order for Multi-Hop Question Answering

by Shaobo Li, Ziyi Cao, Kun Bu and Zhenzhou Ji

Mathematics 2025, 13(21), 3489; https://doi.org/10.3390/math13213489 - 1 Nov 2025

Viewed by 1312

Abstract

Answering multi-hop questions requires first retrieving a sequence of supporting facts, and the order in which these facts are retrieved significantly affects retriever performance. To achieve a clearer reasoning order, it is beneficial to address the easier facts first then move to the [...] Read more.

Answering multi-hop questions requires first retrieving a sequence of supporting facts, and the order in which these facts are retrieved significantly affects retriever performance. To achieve a clearer reasoning order, it is beneficial to address the easier facts first then move to the more difficult ones. However, current orders are usually pre-defined during data construction or specified manually, which restricts the model’s reasoning potential. This paper proposes Auto-Optimizing Reasoning Order (AORO), a method to automatically optimize the reasoning order for each sample, where difficulty is determined by a retrieval model trained with carefully curated data. First, a retriever is trained using data that encompasses all combinations of the possible reasoning orders. The trained retriever is then used to assess the difficulty of each fact, placing the fact with the least difficulty at the beginning of the sequence. Next, the retrieval model is retrained based on these optimized sequences, which are empirically better suited to its capabilities. This process creates an iterative self-debiasing paradigm, and these steps are repeated until all facts are reordered. Experiments conducted on two multi-hop QA benchmarks, QASC and MultiRC, demonstrate the effectiveness of AORO, which outperforms strong baselines using the same PTM, and further enables advanced PTMs to achieve improvements of up to 1.6 points in Recall@10 and 3.7 points in F1 score. Additional case analyses reveal empirical patterns in the optimal reasoning order: the pattern appears independent of the dataset and the underlying pre-trained model; and the sequence proceeds by confirming the truth of the question, answering the question, and filling in any gaps, which aligns with human reasoning. Full article

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38 pages, 1461 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Mixed ABMs for NDC Pension Schemes in the Presence of Demographic and Economic Uncertainty

by Jacopo Giacomelli and Massimiliano Menzietti

Mathematics 2025, 13(21), 3454; https://doi.org/10.3390/math13213454 - 29 Oct 2025

Viewed by 1059

Abstract

The crisis of pension systems based on pay-as-you-go (PAYG) financing has led to the introduction in some countries, including Italy, of so-called notional defined contribution (NDC) pension accounts. These systems mimic the functioning of defined contribution systems in benefit calculations while remaining based [...] Read more.

The crisis of pension systems based on pay-as-you-go (PAYG) financing has led to the introduction in some countries, including Italy, of so-called notional defined contribution (NDC) pension accounts. These systems mimic the functioning of defined contribution systems in benefit calculations while remaining based on PAYG financing. Despite many appealing features, NDC accounts cannot automatically guarantee a system’s financial sustainability in the presence of demographic or economic fluctuations. The literature proposes automatic balance mechanisms (ABMs) of the notional rate applied to notional accounts and an indexation rate applied to pensions. ABMs may be based on two indicators: the liquidity ratio or the solvency ratio. Such ABMs may strengthen a system’s financial sustainability but may produce significant fluctuations in the adjusted notional rate, thereby undermining the social adequacy of the system. In this work, we introduce a mixed ABM based on both the liquidity ratio and solvency ratio and identify the optimal combination that guarantees financial sustainability of the system and, at the same time, maximizes the return paid to the participants at fixed levels of confidence. The numerical results show the advantages of a mixed mechanism over those based on a single indicator. Indeed, although the results depend on the system’s initial conditions and the different ABM configurations tested (16 in total), some common patterns emerge across the solutions. A solvency ratio-based ABM maximizes social utility, while a liquidity ratio-based one ensures financial stability. Although not optimal for either criterion, the ABM that mixes the liquidity ratio and solvency ratio in proportions ranging from 60–40% to 50–50% emerges from our numerical simulations as the best compromise to achieve these two objectives jointly. Full article

(This article belongs to the Special Issue Modern Trends in Mathematics, Probability and Statistics for Finance)

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17 pages, 294 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Approximate Fiber Products of Schemes and Their Étale Homotopical Invariants

by Dongfang Zhao

Mathematics 2025, 13(21), 3448; https://doi.org/10.3390/math13213448 - 29 Oct 2025

Viewed by 1042

Abstract

The classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme,

ϕ : X \to S

and

ψ : Y \to S

, coincide exactly. This condition of strict equality, however, is insufficient [...] Read more.

The classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme,

ϕ : X \to S

and

ψ : Y \to S

, coincide exactly. This condition of strict equality, however, is insufficient for describing many real-world applications, such as the geometric structure of semantic spaces in modern large language models whose foundational architecture is the Transformer neural network: The token spaces of these models are fundamentally approximate, and recent work has revealed complex geometric singularities, challenging the classical manifold hypothesis. This paper develops a new framework to study and quantify the nature of approximate alignment between morphisms in the context of arithmetic geometry, using the tools of étale homotopy theory. We introduce the central object of our work, the étale mismatch torsor, which is a sheaf of torsors over the product scheme

X \times_{S} Y

. The structure of this sheaf serves as a rich, intrinsic, and purely algebraic object amenable to both qualitative classification and quantitative analysis of the global relationship between the two morphisms. Our main results are twofold. First, we provide a complete classification of these structures, establishing a bijection between their isomorphism classes and the first étale cohomology group

H_{é t}^{1} (X \times_{S} Y, \underset{̲}{π_{1}^{é t} (S)})

. Second, we construct a canonical filtration on this classifying cohomology group based on the theory of infinitesimal neighborhoods. This filtration induces a new invariant, which we term the order of mismatch, providing a hierarchical, algebraic measure for the degree of approximation between the morphisms. We apply this framework to the concrete case of generalized Howe curves over finite fields, demonstrating how both the characteristic class and its order reveal subtle arithmetic properties. Full article

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

13 pages, 474 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Necessary and Sufficient Reservoir Condition for Universal Reservoir Computing

by Shuhei Sugiura, Ryo Ariizumi, Toru Asai and Shun-ichi Azuma

Mathematics 2025, 13(21), 3440; https://doi.org/10.3390/math13213440 - 28 Oct 2025

Viewed by 1689

Abstract

We discuss necessary and sufficient conditions for universal approximation using reservoir computing. Reservoir computing is a machine learning method used to train a dynamical system model by tuning only the static part of the model. The universality is the ability of the model [...] Read more.

We discuss necessary and sufficient conditions for universal approximation using reservoir computing. Reservoir computing is a machine learning method used to train a dynamical system model by tuning only the static part of the model. The universality is the ability of the model to approximate any dynamical system with any precision. In the previous studies, we provided two sufficient conditions for the universality. We employed the universality definition that has been discussed since the earliest studies on reservoir computing. In this present paper, we prove that these two conditions and the universality are equivalent to one another. Using this equivalence, we show that a universal model must have a “pathological” property that can only be achieved or approached by chaotic reservoirs. Full article

(This article belongs to the Special Issue Machine Learning: Mathematical Foundations and Applications)

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15 pages, 294 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Conics and Transformations Defined by the Parallelians of a Triangle

by Helena Koncul, Boris Odehnal and Ivana Božić Dragun

Mathematics 2025, 13(21), 3424; https://doi.org/10.3390/math13213424 - 27 Oct 2025

Cited by 1 | Viewed by 1697

Abstract

For any point P in the Euclidean plane of a triangle

Δ

, the six parallelians of P lie on a single conic, which shall be called the parallelian conic of P with respect to

Δ

. We provide a synthetic and an [...] Read more.

For any point P in the Euclidean plane of a triangle

Δ

, the six parallelians of P lie on a single conic, which shall be called the parallelian conic of P with respect to

Δ

. We provide a synthetic and an analytic proof of this fact. Then, we studied the shape of this particular conic, depending on the choice of the pivot point P. This led to the finding that the only circular parallelian conic is the first Lemoine circle. Points on the Steiner inellipse produce parabolae, and those on a certain central line yield equilateral hyperbolae. The hexagon built by the parallelians has an inconic

I

and the tangents of

P

at the parallelians define some triangles and hexagons with several circum- and inconics. Certain pairings of conics, together with in- and circumscribed polygons, give rise to different kinds of porisms. Further, the inconics and circumconics of the triangles and hexagons span exponential pencils of conics in which any pair of subsequent conics defines a new conic as the polar image of the inconic with regard to the circumconic. This allows us to construct chains of nested porisms. The trilinear representations of the centers of the appearing conics, as well as the perspectors of some deduced triangles, depending on the indeterminate coordinates of P, define some algebraic transformations that establish algebraic relations between well- and lesser-known triangle centers. We completed our studies by compiling a list of possible porisms between any pair of conics. Further, we describe the possible loci of pivot points so that the mentioned conics allow for porisms of polygons with arbitrary numbers of vertices. Full article

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

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37 pages, 4383 KB

Open AccessFeature PaperEditor’s ChoiceArticle

The Spatial Regime Conversion Method

by Charles G. Cameron, Cameron A. Smith and Christian A. Yates

Mathematics 2025, 13(21), 3406; https://doi.org/10.3390/math13213406 - 26 Oct 2025

Viewed by 1199

Abstract

We present the spatial regime conversion method (SRCM), a novel hybrid modelling framework for simulating reaction–diffusion systems that adaptively combines stochastic discrete and deterministic continuum representations. Extending the regime conversion method (RCM) to spatial settings, the SRCM employs a discrete reaction–diffusion master equation [...] Read more.

We present the spatial regime conversion method (SRCM), a novel hybrid modelling framework for simulating reaction–diffusion systems that adaptively combines stochastic discrete and deterministic continuum representations. Extending the regime conversion method (RCM) to spatial settings, the SRCM employs a discrete reaction–diffusion master equation (RDME) representation in regions of low concentration and continuum partial differential equations (PDEs) where concentrations are high, dynamically switching based on local thresholds. This is an advancement over the existing methods in the literature, requiring no fixed spatial interfaces, enabling efficient and accurate simulation of systems in which stochasticity plays a key role but is not required uniformly across the domain. We specify the full mathematical formulation of the SRCM, including conversion reactions, hybrid kinetic rules, and consistent numerical updates. The method is validated across several one-dimensional test systems, including simple diffusion from a region of high concentration, the formation of a morphogen gradient, and the propagation of FKPP travelling waves. The results show that the SRCM captures key stochastic features while offering substantial gains in computational efficiency over fully stochastic models. Full article

(This article belongs to the Special Issue Stochastic Models in Mathematical Biology, 2nd Edition)

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25 pages, 782 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Degenerate Fractals: A Formal and Computational Framework for Zero-Dimension Attractors

by Ion Andronache

Mathematics 2025, 13(21), 3407; https://doi.org/10.3390/math13213407 - 26 Oct 2025

Viewed by 2104

Abstract

This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the degenerate case

D = 0

[...] Read more.

This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the degenerate case

D = 0

has been little explored. Starting from the question “what happens to a fractal when its complexity collapses completely?”, Moran’s similarity equation becomes tautological (

r^{s} = 1

with solution

s = {dim}_{M} = 0

) and that only the Hausdorff and box-counting definitions allow an exact calculation. Based on Banach’s fixed point theorem and these definitions, we prove that the attractor of a degenerate IFS is a singleton with

{dim}_{H} = {dim}_{B} = 0

. We develop a reproducible computational methodology to visualize the collapse in dimensions 1–3 (the Iterated Line Contraction—1D/Iterated Square Contraction—2D/Iterated Cube Contraction—3D families), including deterministic and stochastic variants, and we provide a Python script 3.9. The theoretical and numerical results show that the covering box-counting retains unity across all generations, confirming the zero-dimension element and the stability of the phenomenon under moderate perturbations. We conclude that degenerate fractals are an indispensable benchmark for validating fractal dimension estimators and for studying transitions to attractors with positive dimensions. Full article

(This article belongs to the Special Issue Advances in Fractal Geometry and Applications)

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9 pages, 227 KB

Open AccessEditor’s ChoiceArticle

Green’s Functions for Neumann Boundary Conditions

by Jerrold Franklin

Mathematics 2025, 13(21), 3399; https://doi.org/10.3390/math13213399 - 25 Oct 2025

Cited by 2 | Viewed by 2025

Abstract

Green’s functions for Neumann boundary conditions have been considered in Math, Physics, and Electromagnetism textbooks, but often with mistakes of omission and commission. Special constraints and other properties required for Neumann boundary conditions have generally not been noticed or treated correctly. In this [...] Read more.

Green’s functions for Neumann boundary conditions have been considered in Math, Physics, and Electromagnetism textbooks, but often with mistakes of omission and commission. Special constraints and other properties required for Neumann boundary conditions have generally not been noticed or treated correctly. In this paper, we derive appropriate Neumann Green’s functions with these properties properly incorporated. Full article

23 pages, 356 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion

by Yoshiyuki Kitaoka

Mathematics 2025, 13(21), 3401; https://doi.org/10.3390/math13213401 - 25 Oct 2025

Cited by 1 | Viewed by 1056

Abstract

Let

f (x)

be a monic integral polynomial of degree n and p a prime number, for which

f (x)

is fully decomposable modulo p. Let

r_{1}, \dots, r_{n}

be the roots of [...] Read more.

Let

f (x)

be a monic integral polynomial of degree n and p a prime number, for which

f (x)

is fully decomposable modulo p. Let

r_{1}, \dots, r_{n}

be the roots of

f (x) mod p

with

0 \leq r_{1} \leq \dots \leq r_{n} < p

. We have conjectured that the sequence of

(r_{1}, \dots, r_{n}) / p

is uniformly distributed in some sense. We provide a clear explanation of this and generalize the Weyl criterion. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

18 pages, 1825 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fast Deep Belief Propagation: An Efficient Learning-Based Algorithm for Solving Constraint Optimization Problems

by Shufeng Kong, Feifan Chen, Zijie Wang and Caihua Liu

Mathematics 2025, 13(20), 3349; https://doi.org/10.3390/math13203349 - 21 Oct 2025

Viewed by 1919

Abstract

Belief Propagation (BP) is a fundamental heuristic for solving Constraint Optimization Problems (COPs), yet its practical applicability is constrained by slow convergence and instability in loopy factor graphs. While Damped BP (DBP) improves convergence by using manually tuned damping factors, its reliance on [...] Read more.

Belief Propagation (BP) is a fundamental heuristic for solving Constraint Optimization Problems (COPs), yet its practical applicability is constrained by slow convergence and instability in loopy factor graphs. While Damped BP (DBP) improves convergence by using manually tuned damping factors, its reliance on labor-intensive hyperparameter optimization limits scalability. Deep Attentive BP (DABP) addresses this by automating damping through recurrent neural networks (RNNs), but introduces significant memory overhead and sequential computation bottlenecks. To reduce memory usage and accelerate deep belief propagation, this paper introduces Fast Deep Belief Propagation (FDBP), a deep learning framework that improves COP solving through online self-supervised learning and graphics processing unit (GPU) acceleration. FDBP decouples the learning of damping factors from BP message passing, inferring all parameters for an entire BP iteration in a single step, and leverages mixed precision to further optimize GPU memory usage. This approach substantially improves both the efficiency and scalability of BP optimization. Extensive evaluations on synthetic and real-world benchmarks highlight the superiority of FDBP, especially for large-scale instances where DABP fails due to memory constraints. Moreover, FDBP achieves an average speedup of 2.87× over DABP with the same restart counts. Because BP for COPs is a mathematically grounded GPU-parallel message-passing framework that bridges applied mathematics, computing, and machine learning, and is widely applicable across science and engineering, our work offers a promising step toward more efficient solutions to these problems. Full article

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

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15 pages, 645 KB

Open AccessFeature PaperEditor’s ChoiceArticle

GPU-Accelerated Pseudospectral Methods for Optimal Control Problems

by Yilin Zou and Fanghua Jiang

Mathematics 2025, 13(20), 3252; https://doi.org/10.3390/math13203252 - 11 Oct 2025

Viewed by 1972

Abstract

Pseudospectral methods are effective tools for solving optimal control problems, but they result in large-scale nonlinear programming (NLP) problems that are computationally demanding. A major bottleneck is the repeated evaluation of the objective function, system dynamics, path constraints, and their derivatives. This paper [...] Read more.

Pseudospectral methods are effective tools for solving optimal control problems, but they result in large-scale nonlinear programming (NLP) problems that are computationally demanding. A major bottleneck is the repeated evaluation of the objective function, system dynamics, path constraints, and their derivatives. This paper presents an approach to accelerating these computations using Graphics Processing Units (GPUs). We offload the evaluation of the NLP functions and their first and second derivatives to the GPU by developing custom CUDA kernels that exploit the parallelism in the discretized problem structure. The effectiveness of this method is demonstrated on a low-thrust interplanetary trajectory optimization problem. A comparison with a CPU implementation shows that the GPU-accelerated approach reduces the overall computational time. This work demonstrates the potential of GPU acceleration and provides a foundation for future research into fully GPU-native optimal control solvers. Full article

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16 pages, 3838 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Metric Morphological Interpretation of 3D Structures by Gray–Scott Model Simulation Utilising 2D Multifractal Analysis

by Akira Takahara and Yoshihiro Sato

Mathematics 2025, 13(19), 3234; https://doi.org/10.3390/math13193234 - 9 Oct 2025

Viewed by 877

Abstract

Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure [...] Read more.

Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure of three-dimensional slices. The proposed methods were verified using Monte Carlo and Gray–Scott simulations; the pixel-existence probability (PEP)-averaging method, which averages the pixel-existence probability in the slice direction, was confirmed to be the most suitable for analysing three-dimensional structures in two dimensions. This method enables a stable quantitative evaluation, regardless of the direction from which the three-dimensional structure is observed. Full article

(This article belongs to the Special Issue Advances in Fractal Geometry and Applications)

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28 pages, 567 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fine-Tune LLMs for PLC Code Security: An Information-Theoretic Analysis

by Ping Chen, Xiaojing Liu and Yi Wang

Mathematics 2025, 13(19), 3211; https://doi.org/10.3390/math13193211 - 7 Oct 2025

Cited by 4 | Viewed by 4703

Abstract

Programmable Logic Controllers (PLCs), widely used in industrial automation, are often programmed in IEC 61131-3 Structured Text (ST), which is prone to subtle logic vulnerabilities. Traditional tools like static analysis and fuzzing struggle with the complexity and domain-specific semantics of ST. This work [...] Read more.

Programmable Logic Controllers (PLCs), widely used in industrial automation, are often programmed in IEC 61131-3 Structured Text (ST), which is prone to subtle logic vulnerabilities. Traditional tools like static analysis and fuzzing struggle with the complexity and domain-specific semantics of ST. This work explores Large Language Models (LLMs) for PLC vulnerability detection, supported by both theoretical insights and empirical validation. Theoretically, we prove that control flow features carry the most vulnerability-relevant information, establish a feature informativeness hierarchy, and derive sample complexity bounds. We also propose an optimal synthetic data mixing strategy to improve learning with limited supervision. Empirically, we build a dataset combining real-world and synthetic ST code with five vulnerability types. We fine-tune open-source LLMs (CodeLlama, Qwen2.5-Coder, Starcoder2) using LoRA, demonstrating significant gains in binary and multi-class classification. The results confirm our theoretical predictions and highlight the promise of LLMs for PLC security. Our work provides a principled and practical foundation for LLM-based analysis of cyber-physical systems, emphasizing the role of domain knowledge, efficient adaptation, and formal guarantees. Full article

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32 pages, 7592 KB

Open AccessEditor’s ChoiceArticle

Backstepping Sliding Mode Control of Quadrotor UAV Trajectory

by Yohannes Lisanewerk Mulualem, Gang Gyoo Jin, Jaesung Kwon and Jongkap Ahn

Mathematics 2025, 13(19), 3205; https://doi.org/10.3390/math13193205 - 6 Oct 2025

Cited by 7 | Viewed by 2287

Abstract

Unmanned Aerial Vehicles (UAVs), commonly known as drones, have become widely used in many fields, ranging from agriculture to military operations, due to recent advances in technology and decreases in costs. Quadrotors are particularly important UAVs, but their complex, coupled dynamics and sensitivity [...] Read more.

Unmanned Aerial Vehicles (UAVs), commonly known as drones, have become widely used in many fields, ranging from agriculture to military operations, due to recent advances in technology and decreases in costs. Quadrotors are particularly important UAVs, but their complex, coupled dynamics and sensitivity to outside disturbances make them challenging to control. This paper introduces a new control method for quadrotors called Backstepping Sliding Mode Control (BSMC), which combines the strengths of two established techniques: Backstepping Control (BC) and Sliding Mode Control (SMC). Its primary goal is to improve trajectory tracking while also reducing chattering, a common problem with SMC that causes rapid, high-frequency oscillations. The BSMC method achieves this by integrating the SMC switching gain directly into the BC through a process of differential iteration. Herein, a Lyapunov stability analysis confirms the system’s asymptotic stability; a genetic algorithm is used to optimize controller parameters; and the proposed control strategy is evaluated under diverse payload conditions and dynamic wind disturbances. The simulation results demonstrated its capability to handle payload variations ranging from 0.5 kg to 18 kg in normal environments, and up to 12 kg during gusty wind scenarios. Furthermore, the BSMC effectively minimized chattering and achieved a superior performance in tracking accuracy and robustness compared to the traditional SMC and BC. Full article

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

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17 pages, 333 KB

Open AccessFeature PaperEditor’s ChoiceArticle

The Next Terracini Loci of Segre–Veronese Varieties and Their Maximal Weights

by Edoardo Ballico

Mathematics 2025, 13(19), 3166; https://doi.org/10.3390/math13193166 - 2 Oct 2025

Viewed by 838

Abstract

We describe all Terracini loci of Segre–Veronese varieties with at most roughly double the points of the minimal one. In this range we compute the maximum of all weights of the Terracini sets. To prove these results we use cohomological tools (residual exact [...] Read more.

We describe all Terracini loci of Segre–Veronese varieties with at most roughly double the points of the minimal one. In this range we compute the maximum of all weights of the Terracini sets. To prove these results we use cohomological tools (residual exact sequences) applied to some critical schemes associated with a Terracini set and containing all of its points. We expect that these critical schemes will be a very useful tool for other related problems. Full article

40 pages, 476 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Regularity of Generalized Mean-Field G-SDEs

by Karl-Wilhelm Georg Bollweg and Thilo Meyer-Brandis

Mathematics 2025, 13(19), 3099; https://doi.org/10.3390/math13193099 - 27 Sep 2025

Viewed by 768

Abstract

We study the regularity properties of the unique solution of a generalized mean-field G-SDE. More precisely, we consider a generalized mean-field G-SDE with a square-integrable random initial condition, establish its first- and second-order Fréchet differentiability in the stochastic initial condition, and [...] Read more.

We study the regularity properties of the unique solution of a generalized mean-field G-SDE. More precisely, we consider a generalized mean-field G-SDE with a square-integrable random initial condition, establish its first- and second-order Fréchet differentiability in the stochastic initial condition, and specify the G-SDEs of the respective Fréchet derivatives. The first- and second-order Fréchet derivatives are obtained for locally Lipschitz coefficients admitting locally Lipschitz first- and second-order Fréchet derivatives respectively. Our approach heavily relies on the Grönwall inequality, which leverages the Lipschitz continuity of the coefficients. Full article

(This article belongs to the Special Issue Applications of Differential Equations in Sciences)

17 pages, 915 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Solutions for Linear Fractional Differential Equations with Multiple Constraints Using Fractional B-Poly Bases

by Md. Habibur Rahman, Muhammad I. Bhatti and Nicholas Dimakis

Mathematics 2025, 13(19), 3084; https://doi.org/10.3390/math13193084 - 25 Sep 2025

Viewed by 1024

Abstract

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and [...] Read more.

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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20 pages, 424 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Exploiting Generalized Cyclic Symmetry to Find Fast Rectangular Matrix Multiplication Algorithms Easier

by Charlotte Vermeylen, Nico Vervliet, Lieven De Lathauwer and Marc Van Barel

Mathematics 2025, 13(19), 3064; https://doi.org/10.3390/math13193064 - 23 Sep 2025

Viewed by 1419

Abstract

The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the ‘optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated [...] Read more.

The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the ‘optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated as a non-convex optimization problem—more specifically, as a challenging tensor decomposition problem. In this work, we build upon a state-of-the-art augmented Lagrangian algorithm, which formulates the FMM problem as a constrained least squares problem, by incorporating a new, generalized cyclic symmetric (CS) structure in the decomposition. This structure decreases the number of variables, thereby reducing the large search space and the computational cost per iteration. The constraints are used to find practical solutions, i.e., decompositions with simple coefficients, which yield fast algorithms when implemented in hardware. For the FMM problem, usually a very large number of starting points are necessary to converge to a solution. Extensive numerical experiments for different problem sizes demonstrate that including this structure yields more ‘unique’ practical decompositions for a fixed number of starting points. Uniqueness is defined relative to the known scale and trace invariance transformations that hold for all FMM decompositions. Making it easier to find practical decompositions may lead to the discovery of faster FMM algorithms when used in combination with sufficient computational power. Lastly, we show that the CS structure reduces the cost of multiplying a matrix by itself. Full article

(This article belongs to the Special Issue Spectral Theory of Tensors, Tensor (Rank) Decompositions, and Their Applications)

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18 pages, 316 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Weak Convergence of Robust Functions on Topological Groups

by Víctor Ayala, Heriberto Román-Flores and Adriano Da Silva

Mathematics 2025, 13(18), 3004; https://doi.org/10.3390/math13183004 - 17 Sep 2025

Viewed by 804

Abstract

This paper introduces weak variants of level convergence (L-convergence) and epigraph convergence (E-convergence) for nets of level functions on general topological spaces, extending the classical metric and real-valued frameworks to ordered codomains and generalized minima. We show that L-convergence implies E-convergence and that [...] Read more.

This paper introduces weak variants of level convergence (L-convergence) and epigraph convergence (E-convergence) for nets of level functions on general topological spaces, extending the classical metric and real-valued frameworks to ordered codomains and generalized minima. We show that L-convergence implies E-convergence and that the two notions coincide when the limit function is level-continuous, mirroring the relationship between strong and weak variational convergence. In Hausdorff topological groups, we define robust level functions and prove that every level function can be approximated by robust ones via convolution-type operations, enabling perturbation-resilient modeling. These results both generalize and connect to

Γ

-convergence: they recover the classical metric, lower semicontinuous case, and extend the scope for optimization on Lie groups, fuzzy systems, and mechanics in non-Euclidean spaces. An explicit nonmetrizable example demonstrates the relevance of our theory beyond the reach of

Γ

-convergence. Full article

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

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11 pages, 277 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Strong Gelfand Pairs of the Symplectic Group Sp₄(q) Where q Is Even

by Stephen P. Humphries and Joseph E. Marrow

Mathematics 2025, 13(18), 2977; https://doi.org/10.3390/math13182977 - 15 Sep 2025

Viewed by 1530

Abstract

A strong Gelfand pair

(G, H)

is a finite group G together with a subgroup H such that every irreducible character of H induces to a multiplicity-free character of G. We classify the strong Gelfand pairs of the symplectic [...] Read more.

A strong Gelfand pair

(G, H)

is a finite group G together with a subgroup H such that every irreducible character of H induces to a multiplicity-free character of G. We classify the strong Gelfand pairs of the symplectic groups

{Sp}_{4} (q)

for even q. Full article

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

21 pages, 1190 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Fractional Integration Model and Testing Procedure with Roots Within the Unit Circle

by Guglielmo Maria Caporale and Luis Alberiko Gil-Alana

Mathematics 2025, 13(18), 2978; https://doi.org/10.3390/math13182978 - 15 Sep 2025

Viewed by 1211

Abstract

In this paper we propose a statistical model that combines both autoregressions and fractional differentiation in a unified treatment. However, instead of imposing that the roots are strictly on the unit circle, we also allow them to be within the unit circle. This [...] Read more.

In this paper we propose a statistical model that combines both autoregressions and fractional differentiation in a unified treatment. However, instead of imposing that the roots are strictly on the unit circle, we also allow them to be within the unit circle. This permits a higher degree of flexibility in the specification of the model, with rates of dependence combining exponential with hyperbolic decays. Monte Carlo experiments and empirical applications to climatological and financial data show that the proposed approach performs well. Full article

(This article belongs to the Special Issue Applied Time Series and Artificial Intelligence in Economics and Finance)

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32 pages, 1551 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Free Vibration Analysis of Porous FGM Plates on Elastic Foundations with Temperature-Dependent Material Properties

by Aleksandar Radaković, Dragan Čukanović, Aleksandar Nešović, Petar Knežević, Milan T. Djordjević and Gordana Bogdanović

Mathematics 2025, 13(18), 2957; https://doi.org/10.3390/math13182957 - 12 Sep 2025

Cited by 4 | Viewed by 1661

Abstract

This study investigates the free vibration behaviors of functionally graded (FGM) plates with a porous structure, resting on a Kerr-type elastic foundation, while accounting for thermal effects and complex material property distributions. Within the framework of higher-order shear deformation theory (HSDT), two novel [...] Read more.

This study investigates the free vibration behaviors of functionally graded (FGM) plates with a porous structure, resting on a Kerr-type elastic foundation, while accounting for thermal effects and complex material property distributions. Within the framework of higher-order shear deformation theory (HSDT), two novel shape functions are introduced to accurately model transverse shear deformation across the plate thickness without employing shear correction factors. These functions are constructed to satisfy shear stress boundary conditions and capture nonlinear effects induced by material gradation and porosity. A variational formulation is developed to describe the dynamic response of FGM plates in a thermo-mechanical environment, incorporating temperature-dependent material properties and three porosity distributions: uniform, linear, and trigonometric. Numerical solutions are obtained using in-house MATLAB codes, allowing complete control over the formulation and interpretation of the results. The model is validated through detailed comparisons with existing literature, demonstrating high accuracy. The findings reveal that the porosity distribution pattern and gradient intensity significantly influence natural frequencies and mode shapes. The trigonometric porosity distribution exhibits favorable dynamic performance due to preserved stiffness in the surface regions. Additionally, the Kerr-type elastic foundation enables fine tuning of the dynamic response, depending on its specific parameters. The proposed approach provides a reliable and efficient tool for analyzing FGM structures under complex loading conditions and lays the groundwork for future extensions involving nonlinear, time-dependent, and multiphysics analyses. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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19 pages, 408 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Critical Parameters of Branching Random Walks

by Daniela Bertacchi and Fabio Zucca

Mathematics 2025, 13(18), 2962; https://doi.org/10.3390/math13182962 - 12 Sep 2025

Cited by 2 | Viewed by 1860

Abstract

Given a discrete spatial structure X, we define continuous-time branching processes

{η_{t}}_{t \geq 0}

that model a population breeding and dying on X. These processes are usually called branching random walks, and

η_{t} (x)

[...] Read more.

Given a discrete spatial structure X, we define continuous-time branching processes

{η_{t}}_{t \geq 0}

that model a population breeding and dying on X. These processes are usually called branching random walks, and

η_{t} (x)

denotes the number of individuals alive at site x at time t. They are characterised by breeding rates

k_{x y}

(governing the rate at which individuals at x send offspring to y) and by a multiplicative speed parameter

λ

. These processes also serve as models for epidemic spreading, where

λ k_{x y}

represents the infection rate from x to y. In this context,

η_{t} (x)

represents the number of infected individuals at x at time t, and the removal of an individual is due to either death or recovery. Two critical parameters of interest are the global critical parameter

λ_{w}

, related to global survival, and the local critical parameter

λ_{s}

, related to survival within finite sets (with

λ_{w} \leq λ_{s}

). In disease or pest control, the primary goal is to lower

λ

so that the process dies out, at least locally. Nevertheless, a process that survives globally can still pose a threat, especially if sudden changes cause global survival to transition into local survival. In fact, local modifications to the rates can affect the values of both critical parameters, making it important to understand when and how they can be increased. Using results on the comparison of the extinction probabilities for a single branching random walk across different sets, we extend the analysis to the extinction probabilities and critical parameters of pairs of branching random walks whose rates coincide outside a fixed set

A \subseteq X

. We say that two branching random walks are equivalent if their rates coincide everywhere except on a finite subset of X. Given an equivalence class of branching random walks, we prove that if one process has

λ_{w}^{*} \neq λ_{s}^{*}

, then

λ_{w}^{*}

is the maximal possible value of this parameter within the class. We describe the possible configurations for the critical parameters within these equivalence classes. Full article

(This article belongs to the Special Issue Applied Probability, Statistics and Operational Research)

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19 pages, 314 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Supercyclic Weighted Composition Operators on the Space of Smooth Functions

by Juan Bès and Christopher Foster

Mathematics 2025, 13(18), 2944; https://doi.org/10.3390/math13182944 - 11 Sep 2025

Viewed by 1259

Abstract

A weighted composition operator on the space of scalar-valued smooth functions on an open subset of a d-dimensional Euclidean space is supercyclic if and only if it is weakly mixing, and it is strongly supercyclic if and only if it is mixing. Every [...] Read more.

A weighted composition operator on the space of scalar-valued smooth functions on an open subset of a d-dimensional Euclidean space is supercyclic if and only if it is weakly mixing, and it is strongly supercyclic if and only if it is mixing. Every such mixing operator is chaotic. In the one-dimensional case, it is supercyclic if and only if it is mixing and if and only if it is chaotic. Full article

(This article belongs to the Section C3: Real Analysis)

17 pages, 344 KB

Open AccessEditor’s ChoiceArticle

On Some Classes of Enriched Cyclic Contractive Self-Mappings and Their Boundedness and Convergence Properties

by Manuel De la Sen

Mathematics 2025, 13(18), 2948; https://doi.org/10.3390/math13182948 - 11 Sep 2025

Cited by 2 | Viewed by 881

Abstract

This paper focuses on dealing with several types of enriched cyclic contractions defined in the union of a set of non-empty closed subsets of normed or metric spaces. In general, any finite number

p \geq 2

of subsets is permitted in the cyclic [...] Read more.

This paper focuses on dealing with several types of enriched cyclic contractions defined in the union of a set of non-empty closed subsets of normed or metric spaces. In general, any finite number

p \geq 2

of subsets is permitted in the cyclic arrangement. The types of examined single-valued enriched cyclic contractions are, in general, less stringent from the point of view of constraints on the self-mappings compared to

p

-cyclic contractions while the essential properties of these last ones are kept. The convergence of distances is investigated as well as that of sequences generated by the considered enriched cyclic mappings. It is proved that, both in normed spaces and in simple metric spaces, the distances of sequences of points in adjacent subsets converge to the distance between such subsets under weak extra conditions compared to the cyclic contractive case, which is simply that the contractive constant be less than one. It is also proved that if the metric space is a uniformly convex Banach space and one of the involved subsets is convex then all the sequences between adjacent subsets converge to a unique set of best proximity points, one of them per subset which conform a limit cycle, although the sets of best proximity points are not all necessarily singletons in all the subsets. Full article

(This article belongs to the Topic Fixed Point Theory and Measure Theory)

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