Mathematics

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

Viewed by 2026

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

Viewed by 1192

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 721

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

Viewed by 1115

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 839

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 642

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 694

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 1669

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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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 3 | Viewed by 1233

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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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 1 | Viewed by 846

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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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 1001

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 777

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 784

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 1348

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 1364

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 901

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 1711

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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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

Viewed by 595

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)

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 1 | Viewed by 1464

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

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 1382

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 1479

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 657

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 2 | Viewed by 3620

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 1 | Viewed by 1581

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 658

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 569

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 795

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 1073

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 575

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 1048

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 914

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 1241

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 1 | Viewed by 1395

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 1066

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 637

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)

10 pages, 1488 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Electromigration of Aquaporins Controls Water-Driven Electrotaxis

by Pablo Sáez and Sohan Kale

Mathematics 2025, 13(18), 2936; https://doi.org/10.3390/math13182936 - 10 Sep 2025

Viewed by 797

Abstract

Cell motility is a process central to life and is undoubtedly influenced by mechanical and chemical signals. Even so, other stimuli are also involved in controlling cell migration in vivo and in vitro. Among these, electric fields have been shown to provide a [...] Read more.

Cell motility is a process central to life and is undoubtedly influenced by mechanical and chemical signals. Even so, other stimuli are also involved in controlling cell migration in vivo and in vitro. Among these, electric fields have been shown to provide a powerful and programmable cue to manipulate cell migration. There is now a clear consensus that the electromigration of membrane components represents the first response to an external electric field, which subsequently activates downstream signals responsible for controlling cell migration. Here, we focus on a specific mode of electrotaxis: frictionless, amoeboid-like migration. We used the Finite Element Method to solve an active gel model coupled with a mathematical model of the electromigration of aquaporins and investigate the effect of electric fields on ameboid migration. We demonstrate that an electric field can polarize aquaporins in a cell and, consequently, that the electromigration of aquaporins can be exploited to regulate water flux across the cell membrane. Our findings indicate that controlling these fluxes allows modulation of cell migration velocity, thereby reducing the cell’s migratory capacity. Our work provides a mechanistic framework to further study the impact of electrotaxis and to add new insights into specific modes by which electric fields modify cell motility. Full article

(This article belongs to the Special Issue Advances in Biological Systems with Mathematics)

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27 pages, 2240 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Hybrid Entropy-Based Metrics for k-Hop Environment Analysis in Complex Networks

by Csaba Biró

Mathematics 2025, 13(17), 2902; https://doi.org/10.3390/math13172902 - 8 Sep 2025

Cited by 1 | Viewed by 1038

Abstract

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when [...] Read more.

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when classical indices (e.g., degree or clustering) are similar. The metrics are formally defined and shown to be bounded, isomorphism-invariant, and stable under small edge edits. Their behavior is assessed on representative topologies (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, random geometric graphs, and the Zephyr quantum architecture). Across these settings, EWR and NED display predominantly negative correlation with degree and provide information largely orthogonal to standard centralities; vertices with identical degree can differ by factors of two to three in the proposed scores, revealing bridges and heterogeneous regions. These properties indicate utility for vulnerability assessment, topology-aware optimization, and layout heuristics in engineered and quantum networks. Full article

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

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20 pages, 302 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Unified Approach to Implicit Fractional Differential Equations with Anti-Periodic Boundary Conditions

by Ricardo Almeida

Mathematics 2025, 13(17), 2890; https://doi.org/10.3390/math13172890 - 7 Sep 2025

Cited by 2 | Viewed by 1126

Abstract

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order

α \in (0, 1)

and

α \in (1, 2)

, [...] Read more.

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order

α \in (0, 1)

and

α \in (1, 2)

, both defined with respect to a general kernel function. The existence and uniqueness of solutions are established using Banach’s and Schaefer’s fixed-point theorems under suitable Lipschitz conditions. Furthermore, Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated for each problem. Examples are provided to illustrate the applicability of the main results. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

10 pages, 467 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Local Splitting into Incoming and Outgoing Waves and the Integral Representation of Regular Scalar Waves

by Didier Felbacq and Emmanuel Rousseau

Mathematics 2025, 13(17), 2875; https://doi.org/10.3390/math13172875 - 5 Sep 2025

Viewed by 865

Abstract

The problem of the integral representation over a bounded surface of a regular field satisfying the Helmholtz equation in all space is investigated. This problem is equivalent to local splitting into an incoming field and an outgoing field. This splitting is not possible [...] Read more.

The problem of the integral representation over a bounded surface of a regular field satisfying the Helmholtz equation in all space is investigated. This problem is equivalent to local splitting into an incoming field and an outgoing field. This splitting is not possible in general. Full article

(This article belongs to the Special Issue Analytical Methods in Wave Scattering and Diffraction, 3rd Edition)

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23 pages, 345 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Certain Subclasses of Analytic Functions Associated with a Symmetric q-Differential Operator

by Vasile-Aurel Caus

Mathematics 2025, 13(17), 2860; https://doi.org/10.3390/math13172860 - 4 Sep 2025

Cited by 1 | Viewed by 1085

Abstract

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities [...] Read more.

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities involving their coefficients. Additionally, we establish several inclusion relations between these subclasses, obtained by varying the defining parameters. In the second part, we focus on differential subordination and superordination for functions transformed by the operator. We provide sufficient conditions under which such functions are subordinate or superordinate to univalent functions, and we determine the best dominant and best subordinant in specific cases. These results are complemented by several corollaries that highlight particular instances of the main theorems. Furthermore, we present a sandwich-type result that brings together the subordination and superordination frameworks in a unified analytic statement. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

13 pages, 286 KB

Open AccessFeature PaperEditor’s ChoiceReview

Role of Qubits in Quantum Entanglement and Quantum Teleportation

by Laure Gouba

Mathematics 2025, 13(17), 2857; https://doi.org/10.3390/math13172857 - 4 Sep 2025

Viewed by 2680

Abstract

A qubit is an exhibition of quantum entanglement and a key element in the quantum teleportation process. In this paper, we review the role of qubits in quantum entanglement and quantum teleportation. Full article

(This article belongs to the Special Issue Recent Advances in Quantum Theory and Its Applications)

36 pages, 7369 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Ontology-Driven Digital Twin Framework for Aviation Maintenance and Operations

by Igor Kabashkin

Mathematics 2025, 13(17), 2817; https://doi.org/10.3390/math13172817 - 2 Sep 2025

Cited by 4 | Viewed by 3616

Abstract

This paper presents a novel ontology-driven digital twin framework specifically designed for aviation maintenance and operations that addresses these challenges through semantic reasoning and explainable decision support. The proposed framework integrates seven interconnected ontologies—structural, functional, behavioral, monitoring, maintenance, lifecycle, and environmental. It collectively [...] Read more.

This paper presents a novel ontology-driven digital twin framework specifically designed for aviation maintenance and operations that addresses these challenges through semantic reasoning and explainable decision support. The proposed framework integrates seven interconnected ontologies—structural, functional, behavioral, monitoring, maintenance, lifecycle, and environmental. It collectively provides a comprehensive semantic representation of aircraft systems and their operational context. Each ontology is mathematically formalized using description logics and graph theory, creating a unified knowledge graph that enables transparent, traceable reasoning from sensor observations to maintenance decisions. The digital twin is formally defined as a 6-tuple that incorporates semantic transformation engines, cross-ontology mappings, and dynamic reasoning mechanisms. Unlike traditional data-driven approaches that operate as black boxes, the ontology-driven framework provides explainable inference capabilities essential for regulatory compliance and safety certification in aviation. The semantic foundation enables causal reasoning, rule-based validation, and context-aware maintenance recommendations while supporting standardization and interoperability across manufacturers, airlines, and regulatory bodies. The research contributes a mathematically grounded, semantically transparent framework that bridges the gap between domain knowledge and operational data in aviation maintenance. This work establishes the foundation for next-generation cognitive maintenance systems that can support intelligent, adaptive, and trustworthy operations in modern aviation ecosystems. Full article

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20 pages, 309 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Refraction Laws in Temporal Media

by Cristian E. Gutiérrez and Eric Stachura

Mathematics 2025, 13(17), 2777; https://doi.org/10.3390/math13172777 - 29 Aug 2025

Cited by 1 | Viewed by 1469

Abstract

The time-dependent Maxwell system in the sense of distributions is considered in the context of temporal interfaces. Just as with spatial interfaces, electromagnetic waves at temporal interfaces scatter and create a transmitted and reflected wave. A rigorous derivation of boundary conditions for the [...] Read more.

The time-dependent Maxwell system in the sense of distributions is considered in the context of temporal interfaces. Just as with spatial interfaces, electromagnetic waves at temporal interfaces scatter and create a transmitted and reflected wave. A rigorous derivation of boundary conditions for the electric and magnetic fields at temporal interfaces is provided with precise assumptions of the material parameters. In turn, this is used to obtain a general Snell’s Law at such interfaces. From this, explicit formulas for the reflection and transmission coefficients are obtained. Unlike previous works, there is no simplifying ansatz on the solution to the Maxwell system made, nor is it assumed that the fields are smooth. Material parameters which are not necessarily constant on either side of the temporal interface are also considered. Full article

(This article belongs to the Special Issue Mathematical Analysis: Theory, Methods and Applications)

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21 pages, 1434 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Estimating Skewness and Kurtosis for Asymmetric Heavy-Tailed Data: A Regression Approach

by Joseph H. T. Kim and Heejin Kim

Mathematics 2025, 13(16), 2694; https://doi.org/10.3390/math13162694 - 21 Aug 2025

Cited by 8 | Viewed by 4337

Abstract

Estimating skewness and kurtosis from real-world data remains a long-standing challenge in actuarial science and financial risk management, where these higher-order moments are critical for capturing asymmetry and tail risk. Traditional moment-based estimators are known to be highly sensitive to outliers and often [...] Read more.

Estimating skewness and kurtosis from real-world data remains a long-standing challenge in actuarial science and financial risk management, where these higher-order moments are critical for capturing asymmetry and tail risk. Traditional moment-based estimators are known to be highly sensitive to outliers and often fail when the assumption of normality is violated. Despite numerous extensions—from robust moment-based methods to quantile-based measures—being proposed over the decades, no universally satisfactory solution has been reported, and many existing methods exhibit limited effectiveness, particularly under challenging distributional shapes. In this paper we propose a novel method that jointly estimates skewness and kurtosis based on a regression adaptation of the Cornish–Fisher expansion. By modeling the empirical quantiles as a cubic polynomial of the standard normal variable, the proposed approach produces a reliable and efficient estimator that better captures distributional shape without strong parametric assumptions. Our comprehensive simulation studies show that the proposed method performs much better than existing estimators across a wide range of distributions, especially when the data are skewed or heavy-tailed, as is typical in actuarial and financial applications. Full article

(This article belongs to the Special Issue Actuarial Statistical Modeling and Applications)

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40 pages, 725 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Upper and Lower Bounds of Performance Metrics in Hybrid Systems with Setup Time

by Ken’ichi Kawanishi and Yuki Ino

Mathematics 2025, 13(16), 2685; https://doi.org/10.3390/math13162685 - 20 Aug 2025

Viewed by 836

Abstract

To address the increasing demand for computational and communication resources, modern networked systems often rely on heterogeneous servers, including those requiring setup times, such as virtual machines or servers, and others that are always active. In this paper, we model and analyze the [...] Read more.

To address the increasing demand for computational and communication resources, modern networked systems often rely on heterogeneous servers, including those requiring setup times, such as virtual machines or servers, and others that are always active. In this paper, we model and analyze the performance of such hybrid systems using a level-dependent quasi-birth-and-death (LDQBD) process. Building upon an existing queueing model, we extend the analysis by considering scalable approximation methods. Since matrix analytic methods become computationally expensive in large-scale settings, we propose a stochastic bounding approach that derives upper and lower bounds for the stationary distribution, thereby significantly reducing computational cost. This approach further provides bounds on the performance metrics of the hybrid system. Full article

(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models, 2nd Edition)

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12 pages, 244 KB

Open AccessFeature PaperEditor’s ChoiceArticle

New Sufficient Conditions for Moment Determinacy via Probability Density Tails

by Gwo Dong Lin and Jordan M. Stoyanov

Mathematics 2025, 13(16), 2671; https://doi.org/10.3390/math13162671 - 19 Aug 2025

Cited by 1 | Viewed by 872

Abstract

One of the ways to characterize a probability distribution is to show that it is moment-determinate, uniquely determined by knowing all its moments. The uniqueness, in the absolutely continuous case, depends entirely on the behaviour of the tails of the probability density function [...] Read more.

One of the ways to characterize a probability distribution is to show that it is moment-determinate, uniquely determined by knowing all its moments. The uniqueness, in the absolutely continuous case, depends entirely on the behaviour of the tails of the probability density function f. We find and exploit a condition, (D), in terms only of f which is of a ‘general’ form and easy to check. Condition (D), showing the ‘speed’ for f to tend to zero, is sufficient to conclude the moment determinacy. We establish a series of theorems and corollaries in both Stieltjes and Hamburger cases and provide an interesting illustrative example. The results in this paper are either new or extend some recently published results. Full article

36 pages, 2144 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Dynamic Portfolio Optimization Using Information from a Crisis Indicator

by Victor Gonzalo, Markus Wahl and Rudi Zagst

Mathematics 2025, 13(16), 2664; https://doi.org/10.3390/math13162664 - 19 Aug 2025

Viewed by 1595

Abstract

Investors face the challenge of how to incorporate economic and financial forecasts into their investment strategy, especially in times of financial crisis. To model this situation, we consider a financial market consisting of a risk-free asset with a constant interest rate as well [...] Read more.

Investors face the challenge of how to incorporate economic and financial forecasts into their investment strategy, especially in times of financial crisis. To model this situation, we consider a financial market consisting of a risk-free asset with a constant interest rate as well as a risky asset whose drift and volatility is influenced by a stochastic process indicating the probability of potential market downturns. We use a dynamic portfolio optimization approach in continuous time to maximize the expected utility of terminal wealth and solve the corresponding HJB equations for the general class of HARA utility functions. The resulting optimal strategy can be obtained in closed form. It corresponds to a CPPI strategy with a stochastic multiplier that depends on the information from the crisis indicator. In addition to the theoretical results, a performance analysis of the derived strategy is implemented. The specified model is fitted using historic market data and the performance is compared to the optimal portfolio strategy obtained in a Black–Scholes framework without crisis information. The new strategy clearly dominates the BS-based CPPI strategy with respect to the Sharpe Ratio and Adjusted Sharpe Ratio. Full article

(This article belongs to the Special Issue Latest Advances in Mathematical Economics)

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14 pages, 2664 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On S_n Iteration for Fixed Points of (E)-Operators with Numerical Analysis and Polynomiography

by Cristian Ciobanescu

Mathematics 2025, 13(16), 2625; https://doi.org/10.3390/math13162625 - 15 Aug 2025

Cited by 1 | Viewed by 939

Abstract

The first part of this study is related to the search of fixed points for (E)-operators (Garcia-Falset operators), in the Banach setting, by means of a three-step iteration procedure. The main results reveal some conclusions related to weak and strong convergence [...] Read more.

The first part of this study is related to the search of fixed points for (E)-operators (Garcia-Falset operators), in the Banach setting, by means of a three-step iteration procedure. The main results reveal some conclusions related to weak and strong convergence of the considered iterative scheme toward a fixed point. On the other hand, the usefulness of the S_n iterative scheme is once again revealed by demonstrating through numerical simulations the advantages of using it for solving the problem of the maximum modulus of complex polynomials compared to standard algorithms, such as Newton, Halley, or Kalantary’s so-called B₄ iteration. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications: 3rd Edition)

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18 pages, 1152 KB

Open AccessEditor’s ChoiceArticle

Coordinated Truck Loading and Routing Problem: A Forestry Logistics Case Study

by Cristian Oliva, Manuel Cepeda and Sebastián Muñoz-Herrera

Mathematics 2025, 13(15), 2537; https://doi.org/10.3390/math13152537 - 7 Aug 2025

Cited by 1 | Viewed by 1295

Abstract

This study addresses a real-world logistics problem in forestry operations: the distribution of plants from cultivation centers to planting sites under strict delivery time windows and limited depot resources. We introduce the Coordinated Truck Loading and Routing Problem (CTLRP), an extension of the [...] Read more.

This study addresses a real-world logistics problem in forestry operations: the distribution of plants from cultivation centers to planting sites under strict delivery time windows and limited depot resources. We introduce the Coordinated Truck Loading and Routing Problem (CTLRP), an extension of the classical Vehicle Routing Problem with Time Windows (VRPTW) that integrates routing decisions with truck loading schedules at a single depot with constrained capacity. To solve this NP-hard problem, we develop a metaheuristic algorithm based on Ant Colony Optimization (ACO), enhanced with a global memory system and a novel stochastic return rule that allows trucks to return to the depot when additional deliveries are suboptimal. Parameter calibration experiments are conducted to determine optimal values for the return probability and ant population size. The algorithm is tested on a real forestry dispatch scenario over six working days. The results show that an Ant Colony System (ACS–CTLRP) algorithm reduces total distance traveled by 23%, travel time by 22%, and the number of trucks used by 13 units, while increasing fleet utilization from 54% to 83%. These findings demonstrate that the proposed method significantly outperforms current company planning and offers a transferable framework for depot-constrained routing problems in time-sensitive distribution environments. Full article

(This article belongs to the Special Issue New Advances and Applications of Operational Research and Scheduling Theory)

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14 pages, 302 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group

by Giovanni Calvaruso and Lorenzo Pellegrino

Mathematics 2025, 13(15), 2529; https://doi.org/10.3390/math13152529 - 6 Aug 2025

Cited by 1 | Viewed by 822

Abstract

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally [...] Read more.

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. Full article

(This article belongs to the Special Issue Recent Studies in Differential Geometry and Its Applications)

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