Mathematics

25 pages, 1931 KB

Open AccessArticle

A Methodological Comparison of Forecasting Models Using KZ Decomposition and Walk-Forward Validation

by Khawla Al-Saeedi, Diwei Zhou, Andrew Fish, Katerina Tsakiri and Antonios Marsellos

Mathematics 2025, 13(21), 3410; https://doi.org/10.3390/math13213410 (registering DOI) - 26 Oct 2025

The accurate forecasting of surface air temperature (T2M) is crucial for climate analysis, agricultural planning, and energy management. This study proposes a novel forecasting framework grounded in structured temporal decomposition. Using the Kolmogorov–Zurbenko (KZ) filter, all predictor variables are decomposed into three physically [...] Read more.

The accurate forecasting of surface air temperature (T2M) is crucial for climate analysis, agricultural planning, and energy management. This study proposes a novel forecasting framework grounded in structured temporal decomposition. Using the Kolmogorov–Zurbenko (KZ) filter, all predictor variables are decomposed into three physically interpretable components: long-term, seasonal, and short-term variations, forming an expanded multi-scale feature space. A central innovation of this framework lies in training a single unified model on the decomposed feature set to predict the original target variable, thereby enabling the direct learning of scale-specific driver–response relationships. We present the first comprehensive benchmarking of this architecture, demonstrating that it consistently enhances the performance of both regularized linear models (Ridge and Lasso) and tree-based ensemble methods (Random Forest and XGBoost). Under rigorous walk-forward validation, the framework substantially outperforms conventional, non-decomposed approaches—for example, XGBoost improves the coefficient of determination (

R^{2}

) from 0.80 to 0.91. Furthermore, temporal decomposition enhances interpretability by enabling Ridge and Lasso models to achieve performance levels comparable to complex ensembles. Despite these promising results, we acknowledge several limitations: the analysis is restricted to a single geographic location and time span, and short-term components remain challenging to predict due to their stochastic nature and the weaker relevance of predictors. Additionally, the framework’s effectiveness may depend on the optimal selection of KZ parameters and the availability of sufficiently long historical datasets for stable walk-forward validation. Future research could extend this approach to multiple geographic regions, longer time series, adaptive KZ tuning, and specialized short-term modeling strategies. Overall, the proposed framework demonstrates that temporal decomposition of predictors offers a powerful inductive bias, establishing a robust and interpretable paradigm for surface air temperature forecasting. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods for Prediction and Optimisation in Artificial Intelligence)

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31 pages, 649 KB

Open AccessArticle

Qualitative Analysis of Delay Stochastic Systems with Generalized Memory Effects

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Mathematics 2025, 13(21), 3409; https://doi.org/10.3390/math13213409 (registering DOI) - 26 Oct 2025

Abstract

Fractional stochastic differential equations (FSDEs) are powerful tools for modeling real-world phenomena, as they incorporate both memory effects and stochastic noise. A central focus in their analysis is establishing the well-posedness and regularity of solutions. Moreover, the averaging principle offers a systematic approach [...] Read more.

Fractional stochastic differential equations (FSDEs) are powerful tools for modeling real-world phenomena, as they incorporate both memory effects and stochastic noise. A central focus in their analysis is establishing the well-posedness and regularity of solutions. Moreover, the averaging principle offers a systematic approach to simplify complex dynamical systems by approximating their behavior through time-averaged models. In this paper, we develop a theoretical framework for a class of FSDEs involving the Hilfer–Katugampola derivative. Our main contributions include proving the well-posedness and regularity of solutions, establishing a generalized averaging principle, and demonstrating real-life applications solved via the Euler–Maruyama method. All numerical simulations were conducted using the Python programming language (version 3.11). These results are formulated for the

\tilde{P}

th moment, providing a unified analysis that extends existing findings. Full article

(This article belongs to the Special Issue Fractional Calculus and Fractional Differential Equations: Theory and Applications)

31 pages, 716 KB

Open AccessArticle

Optimizing the Mean Shift Algorithm for Efficient Clustering

by Rustam Mussabayev, Alexander Krassovitskiy and Meruyert Aristombayeva

Mathematics 2025, 13(21), 3408; https://doi.org/10.3390/math13213408 (registering DOI) - 26 Oct 2025

Abstract

Mean Shift is a flexible, non-parametric clustering algorithm that identifies dense regions in data through gradient ascent on a kernel density estimate. Its ability to detect arbitrarily shaped clusters without requiring prior knowledge of the number of clusters makes it widely applicable across [...] Read more.

Mean Shift is a flexible, non-parametric clustering algorithm that identifies dense regions in data through gradient ascent on a kernel density estimate. Its ability to detect arbitrarily shaped clusters without requiring prior knowledge of the number of clusters makes it widely applicable across diverse domains. However, its quadratic computational complexity restricts its use on large or high-dimensional datasets. Numerous acceleration techniques, collectively referred to as Fast Mean Shift strategies, have been developed to address this limitation while preserving clustering quality. This paper presents a systematic theoretical analysis of these strategies, focusing on their computational impact, pairwise combinability, and mapping onto distinct stages of the Mean Shift pipeline. Acceleration methods are categorized into seed reduction, neighborhood search acceleration, adaptive bandwidth selection, kernel approximation, and parallelization, with their algorithmic roles examined in detail. A pairwise compatibility matrix is proposed to characterize synergistic and conflicting interactions among strategies. Building on this analysis, we introduce a decision framework for selecting suitable acceleration strategies based on dataset characteristics and computational constraints. This framework, together with the taxonomy, combinability analysis, and scenario-based recommendations, establishes a rigorous foundation for understanding and systematically applying Fast Mean Shift methods. Full article

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

26 pages, 784 KB

Open AccessFeature PaperArticle

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 (registering DOI) - 26 Oct 2025

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)

37 pages, 4383 KB

Open AccessFeature PaperArticle

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 (registering DOI) - 26 Oct 2025

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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24 pages, 1524 KB

Open AccessArticle

Counting Tree-like Multigraphs with a Given Number of Vertices and Multiple Edges

by Muhammad Ilyas, Seemab Hayat and Naveed Ahmed Azam

Mathematics 2025, 13(21), 3405; https://doi.org/10.3390/math13213405 (registering DOI) - 26 Oct 2025

Abstract

The enumeration of chemical graphs plays a crucial role in cheminformatics and bioinformatics, especially in the search for novel drug discovery. These graphs are usually tree-like multigraphs, or they consist of tree-like multigraphs attached to a central core. In both configurations, the tree-like [...] Read more.

The enumeration of chemical graphs plays a crucial role in cheminformatics and bioinformatics, especially in the search for novel drug discovery. These graphs are usually tree-like multigraphs, or they consist of tree-like multigraphs attached to a central core. In both configurations, the tree-like components play a key role in determining the properties and activities of chemical compounds. In this work, we propose a dynamic programming approach to precisely count the number of tree-like multigraphs with a given number of n vertices and

Δ

multiple edges. Our method transforms multigraphs into rooted forms by designating their unicentroid or bicentroid as the root and then defining a canonical representation based on the maximal subgraphs rooted at the root’s children. This canonical form ensures that each multigraph is counted only once. Recursive formulas are then established based on the number of vertices and multiple edges in the largest subgraphs rooted at the root’s children. The resulting algorithm achieves a time complexity of

O (n^{2} (n + Δ (n + Δ^{2} \cdot min {n, Δ})))

and space complexity of

O (n^{2} (Δ^{3} + 1))

. Extensive experiments demonstrate that the proposed method scales efficiently, being able to count multigraphs with up to 200 vertices (e.g., (200, 26)) and up to 50 multiple edges (e.g., (90, 50)) in under 15 min. In contrast, the available state-of-the-art tool Nauty runs out of memory beyond moderately sized instances, as it relies on explicit generation of all candidate multigraphs. These results highlight the practical advantage and strong potential of the proposed method as a scalable tool for chemical graph enumeration in drug discovery applications. Full article

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

17 pages, 313 KB

Open AccessFeature PaperArticle

On the Concept of Algebraic Crystallography

by Dominique Bourn

Mathematics 2025, 13(21), 3404; https://doi.org/10.3390/math13213404 (registering DOI) - 26 Oct 2025

Abstract

Category Theory provides us with a clear notion of what is an internal algebraic structure. This will allow us to focus our attention on a certain kind of relationship between context and structure; namely on categories

E

(context) in which, [...] Read more.

Category Theory provides us with a clear notion of what is an internal algebraic structure. This will allow us to focus our attention on a certain kind of relationship between context and structure; namely on categories

E

(context) in which, on any object X, there is, at most, one algebraic structure of some type S. Full article

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

25 pages, 48579 KB

Open AccessArticle

Parametric Surfaces for Elliptic and Hyperbolic Geometries

by László Szirmay-Kalos, András Fridvalszky, László Szécsi and Márton Vaitkus

Mathematics 2025, 13(21), 3403; https://doi.org/10.3390/math13213403 (registering DOI) - 25 Oct 2025

Abstract

Background/Objectives: In computer graphics, virtual worlds are constructed and visualized through algorithmic processes. These environments are typically populated with objects defined by mathematical models, traditionally based on Euclidean geometry. However, there is increasing interest in exploring non-Euclidean geometries, which require adaptations of [...] Read more.

Background/Objectives: In computer graphics, virtual worlds are constructed and visualized through algorithmic processes. These environments are typically populated with objects defined by mathematical models, traditionally based on Euclidean geometry. However, there is increasing interest in exploring non-Euclidean geometries, which require adaptations of the modeling techniques used in Euclidean spaces. Methods: This paper focuses on defining parametric curves and surfaces within elliptic and hyperbolic geometries. We explore free-form splines interpreted as hierarchical motions along geodesics. Translation, rotation, and ruling are managed through supplementary curves to generate surfaces. We also discuss how to compute normal vectors, which are essential for animation and lighting. The rendering approach we adopt aligns with physical principles, assuming that light follows geodesic paths. Results: We extend the Kochanek–Bartels spline to both elliptic and hyperbolic geometries using a sequence of geodesic-based interpolations. Simple recursive formulas are introduced for derivative calculations. With well-defined translation and rotation in these curved spaces, we demonstrate the creation of ruled, extruded, and rotational surfaces. These results are showcased through a virtual reality application designed to navigate and visualize non-Euclidean spaces. Full article

(This article belongs to the Special Issue Image Processing and Generation, Pattern Recognition, and Data Visualization)

20 pages, 611 KB

Open AccessFeature PaperArticle

Efficient Evaluation of Sobol’ Sensitivity Indices via Polynomial Lattice Rules and Modified Sobol’ Sequences

by Venelin Todorov and Petar Zhivkov

Mathematics 2025, 13(21), 3402; https://doi.org/10.3390/math13213402 (registering DOI) - 25 Oct 2025

Abstract

Accurate and efficient estimation of Sobol’ sensitivity indices is a cornerstone of variance-based global sensitivity analysis, providing critical insights into how uncertainties in input parameters affect model outputs. This is particularly important for large-scale environmental, engineering, and financial models, where understanding parameter influence [...] Read more.

Accurate and efficient estimation of Sobol’ sensitivity indices is a cornerstone of variance-based global sensitivity analysis, providing critical insights into how uncertainties in input parameters affect model outputs. This is particularly important for large-scale environmental, engineering, and financial models, where understanding parameter influence is essential for improving model reliability, guiding calibration, and supporting informed decision-making. However, computing Sobol’ indices requires evaluating high-dimensional integrals, presenting significant numerical and computational challenges. In this study, we present a comparative analysis of two of the best available Quasi-Monte Carlo (QMC) techniques: polynomial lattice rules (PLRs) and modified Sobol’ sequences. The performance of both approaches is systematically assessed in terms of performance and accuracy. Extensive numerical experiments demonstrate that the proposed PLR-based framework achieves superior precision for several sensitivity measures, while modified Sobol’ sequences remain competitive for lower-dimensional indices. Our results show that IPLR-α3 outperforms traditional QMC methods in estimating both dominant and weak sensitivity indices, offering a robust framework for high-dimensional models. These findings provide practical guidelines for selecting optimal QMC strategies, contributing to more reliable sensitivity analysis and enhancing the predictive power of complex computational models. Full article

23 pages, 354 KB

Open AccessFeature PaperArticle

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 (registering DOI) - 25 Oct 2025

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)

16 pages, 505 KB

Open AccessArticle

Estimating the Number of Junta Variables for Optimizing Boolean Functions in Quantum Memories

by Abdulaziz Alotaibi, Samar AbdelAzim, Sattam Saleem Alharbi and Mohamed Darwish

Mathematics 2025, 13(21), 3400; https://doi.org/10.3390/math13213400 (registering DOI) - 25 Oct 2025

Abstract

Optimizing Boolean function components to have the minimum number of inputs in order to reduce the memory space required during these functions in computing devices is a significant demand. This paper proposes a quantum computation approach based on the degree-of-entanglement quantum computation model [...] Read more.

Optimizing Boolean function components to have the minimum number of inputs in order to reduce the memory space required during these functions in computing devices is a significant demand. This paper proposes a quantum computation approach based on the degree-of-entanglement quantum computation model to estimate the number of junta variables of an unknown Boolean function presented through an oracle. The time complexity of the developed quantum approach is independent of the number of inputs and depends on an allowable assigned error

ϵ

. Thus, the time complexity of the developed algorithm is

O (ϵ^{- 2})

, compared to

O (2^{n + 1})

in the traditional approach. Also, the memory space of the developed approach is linear,

O (2 n + 4)

, in terms of the number of inputs compared to the exponential memory space

O (2^{n + 1})

using the traditional approach. Therefore, the developed quantum approach has exponential supremacy in comparison to the traditional approach. The developed approach was implemented practically using both the Qiskit simulator and the IBM real quantum computer. The obtained results expose high statistical fidelities between the empirical and theoretical results. Full article

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

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9 pages, 227 KB

Open AccessArticle

Green’s Functions for Neumann Boundary Conditions

by Jerrold Franklin

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

Abstract

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

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

16 pages, 14135 KB

Open AccessArticle

Underwater Image Enhancement with a Hybrid U-Net-Transformer and Recurrent Multi-Scale Modulation

by Zaiming Geng, Jiabin Huang, Xiaotian Wang, Yu Zhang, Xinnan Fan and Pengfei Shi

Mathematics 2025, 13(21), 3398; https://doi.org/10.3390/math13213398 (registering DOI) - 25 Oct 2025

Abstract

The quality of underwater imagery is inherently degraded by light absorption and scattering, a challenge that severely limits its application in critical domains such as marine robotics and archeology. While existing enhancement methods, including recent hybrid models, attempt to address this, they often [...] Read more.

The quality of underwater imagery is inherently degraded by light absorption and scattering, a challenge that severely limits its application in critical domains such as marine robotics and archeology. While existing enhancement methods, including recent hybrid models, attempt to address this, they often struggle to restore fine-grained details without introducing visual artifacts. To overcome this limitation, this work introduces a novel hybrid U-Net-Transformer (UTR) architecture that synergizes local feature extraction with global context modeling. The core innovation is a Recurrent Multi-Scale Feature Modulation (R-MSFM) mechanism, which, unlike prior recurrent refinement techniques, employs a gated modulation strategy across multiple feature scales within the decoder to iteratively refine textural and structural details with high fidelity. This approach effectively preserves spatial information during upsampling. Extensive experiments demonstrate the superiority of the proposed method. On the EUVP dataset, UTR achieves a PSNR of 28.347 dB, a significant gain of +3.947 dB over the state-of-the-art UWFormer. Moreover, it attains a top-ranking UIQM score of 3.059 on the UIEB dataset, underscoring its robustness. The results confirm that UTR provides a computationally efficient and highly effective solution for underwater image enhancement. Full article

(This article belongs to the Special Issue Artificial Intelligence, Algorithms, and Databases: Innovations and Cross-Disciplinary Impact)

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40 pages, 2911 KB

Open AccessArticle

A Vehicle Routing Problem Based on a Long-Distance Transportation Network with an Exact Optimization Algorithm

by Toygar Emre and Rızvan Erol

Mathematics 2025, 13(21), 3397; https://doi.org/10.3390/math13213397 (registering DOI) - 24 Oct 2025

Abstract

In vehicle routing problems, long-distance transportation poses a significant challenge to the optimization of transportation costs while adhering to regulations. This study investigates a special type of logistics problem that focuses on liquid transportation systems involving full truckload delivery and the rest–break–drive periods [...] Read more.

In vehicle routing problems, long-distance transportation poses a significant challenge to the optimization of transportation costs while adhering to regulations. This study investigates a special type of logistics problem that focuses on liquid transportation systems involving full truckload delivery and the rest–break–drive periods of truck drivers over long distances according to the regulations of the United States. Based on an exact solution algorithm, this work combines a long-distance full truckload fluid transportation problem with the concept of truck driver schedules for the first time. The goal is to optimize transportation expenses while managing challenges related to the rest–break–drive periods of truck drivers, time windows, trailer varieties, customer segments, food and non-food products, a diverse fleet, starting locations, and the diverse tasks of vehicles. In order to reach optimality, a construction heuristic and the column generation method were employed, supplemented by several acceleration strategies. Performance analysis, carried out with artificial input sets mirroring real-life scenarios, indicates that low optimality gaps can be obtained in an appropriate amount of time for large-scale long-haul liquid transportation. Full article

(This article belongs to the Special Issue Optimization and Mathematical Modelling in Transport and Logistics Network)

25 pages, 1288 KB

Open AccessArticle

An Analysis of Implied Volatility, Sensitivity, and Calibration of the Kennedy Model

by Dalma Tóth-Lakits, Miklós Arató and András Ványolos

Mathematics 2025, 13(21), 3396; https://doi.org/10.3390/math13213396 (registering DOI) - 24 Oct 2025

Abstract

The Kennedy model provides a flexible and mathematically consistent framework for modeling the term structure of interest rates, leveraging Gaussian random fields to capture the dynamics of forward rates. Building upon our earlier work, where we developed both theoretical results—including novel proofs of [...] Read more.

The Kennedy model provides a flexible and mathematically consistent framework for modeling the term structure of interest rates, leveraging Gaussian random fields to capture the dynamics of forward rates. Building upon our earlier work, where we developed both theoretical results—including novel proofs of the martingale property, connections between the Kennedy and HJM frameworks, and parameter estimation theory—and practical calibration methods, using maximum likelihood, Radon–Nikodym derivatives, and numerical optimization (stochastic gradient descent) on simulated and real par swap rate data, this study extends the analysis in several directions. We derive detailed formulas for the volatilities implied by the Kennedy model and investigate their asymptotic properties. A comprehensive sensitivity analysis is conducted to evaluate the impact of key parameters on derivative prices. We implement an industry-standard Monte Carlo method, tailored to the conditional distribution of the Kennedy field, to efficiently generate scenarios consistent with observed initial forward curves. Furthermore, we present closed-form pricing formulas for various interest rate derivatives, including zero-coupon bonds, caplets, floorlets, swaplets, and the par swap rate. A key advantage of these results is that the formulas are expressed explicitly in terms of the initial forward curve and the original parameters of the Kennedy model, which ensures both analytical tractability and consistency with market-observed data. These closed-form expressions can be directly utilized in calibration procedures, substantially accelerating multidimensional nonlinear optimization algorithms. Moreover, given an observed initial forward curve, the model provides significantly more accurate pricing formulas, enhancing both theoretical precision and practical applicability. Finally, we calibrate the Kennedy model to market-observed caplet prices. The findings provide valuable insights into the practical applicability and robustness of the Kennedy model in real-world financial markets. Full article

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

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30 pages, 1382 KB

Open AccessArticle

Asymptotic Analysis of the Bias–Variance Trade-Off in Subsampling Metropolis–Hastings

by Shuang Liu

Mathematics 2025, 13(21), 3395; https://doi.org/10.3390/math13213395 (registering DOI) - 24 Oct 2025

Abstract

Markov chain Monte Carlo (MCMC) methods are fundamental to Bayesian inference but are often computationally prohibitive for large datasets, as the full likelihood must be evaluated at each iteration. Subsampling-based approximate Metropolis–Hastings (MH) algorithms offer a popular alternative, trading a manageable bias for [...] Read more.

Markov chain Monte Carlo (MCMC) methods are fundamental to Bayesian inference but are often computationally prohibitive for large datasets, as the full likelihood must be evaluated at each iteration. Subsampling-based approximate Metropolis–Hastings (MH) algorithms offer a popular alternative, trading a manageable bias for a significant reduction in per-iteration cost. While this bias–variance trade-off is empirically understood, a formal theoretical framework for its optimization has been lacking. Our work establishes such a framework by bounding the mean squared error (MSE) as a function of the subsample size (m), the data size (n), and the number of epochs (E). This analysis reveals two optimal asymptotic scaling laws: the optimal subsample size is

m^{★} = O (E^{1 / 2})

, leading to a minimal MSE that scales as

{MSE}^{★} = O (E^{- 1 / 2})

. Furthermore, leveraging the large-sample asymptotic properties of the posterior, we show that when augmented with a control variate, the approximate MH algorithm can be asymptotically more efficient than the standard MH method under ideal conditions. Experimentally, we first validate the two optimal asymptotic scaling laws. We then use Bayesian logistic regression and Softmax classification models to highlight a key difference in convergence behavior: the exact algorithm starts with a high MSE that gradually decreases as the number of epochs increases. In contrast, the approximate algorithm with a practical control variate maintains a consistently low MSE that is largely insensitive to the number of epochs. Full article

20 pages, 2260 KB

Open AccessArticle

Null Space Properties of Neural Networks with Applications to Image Steganography

by Xiang Li and Kevin M. Short

Mathematics 2025, 13(21), 3394; https://doi.org/10.3390/math13213394 (registering DOI) - 24 Oct 2025

Abstract

This paper advances beyond adversarial neural network methods by considering whether the underlying mathematics of neural networks contains inherent properties that can be exploited to fool neural networks. In broad terms, this paper will consider a neural network to be composed of a [...] Read more.

This paper advances beyond adversarial neural network methods by considering whether the underlying mathematics of neural networks contains inherent properties that can be exploited to fool neural networks. In broad terms, this paper will consider a neural network to be composed of a series of linear transformations between layers of the network, interspersed with nonlinear stages that serve to compress outliers. The input layer of the network is typically extremely high-dimensional, yet the final classification is in a space of a much lower dimension. This dimensional reduction leads to the existence of a null space, and this paper will explore how that can be exploited. Specifically, this paper explores the null space properties of neural networks by extending the null space definition from linear to nonlinear maps and discussing the presence of a null space in neural networks. The null space of a neural network characterizes the component of input data that makes no contribution to the final prediction so that we can exploit it to trick the neural network. One application described here leads to a method of image steganography. Through experiments on image data sets such as MNIST, it has been shown that the null space components can be used to force the neural network to choose a selected hidden image class, even though the overall image can be made to look like a completely different image. The paper concludes with comparisons between what a human viewer would see and the part of the image that the neural network is actually using to make predictions, hence showing that what the neural network “sees” is completely different than what we would expect. Full article

27 pages, 344 KB

Open AccessArticle

𝒜-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications

by Amani Baazeem, Silvestru Sever Dragomir and Kais Feki

Mathematics 2025, 13(21), 3393; https://doi.org/10.3390/math13213393 (registering DOI) - 24 Oct 2025

Abstract

Consider a complex Hilbert space

(X, ⟨ \cdot, \cdot ⟩)

equipped with a positive (semidefinite) bounded linear operator

A

on

X

. The

A

-joint numerical radius for two

A

-bounded operators

T

and

S

is defined as [...] Read more.

Consider a complex Hilbert space

(X, ⟨ \cdot, \cdot ⟩)

equipped with a positive (semidefinite) bounded linear operator

A

on

X

. The

A

-joint numerical radius for two

A

-bounded operators

T

and

S

is defined as

ω_{A, e} (T, S) = {sup}_{{∥ x ∥}_{A} = 1} \sqrt{{|{⟨ T x, x ⟩}_{A}|}^{2} + {|{⟨ S x, x ⟩}_{A}|}^{2}} .

Among other results, in this study we demonstrate that

ω_{e, A}^{2} (T, S) \leq 2^{1 - \frac{1}{r}} {[max \{{∥T∥}_{A}^{2 r}, {∥S∥}_{A}^{2 r}\} + ω_{A}^{r} (S^{♯_{A}} T)]}^{\frac{1}{r}}

for

r \geq 1

and that

ω_{e, A}^{2} (T, S) \geq \frac{1}{2} ω_{A} (T^{2} + S^{2}) + \frac{1}{2} max \{ω_{A} (S + T), ω_{A} (S - T)\} |ω_{A} (S) - ω_{A} (T)|

. Additionally, we provide several inequalities related to the

A

-numerical radius and

A

-seminorm. Full article

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

31 pages, 1338 KB

Open AccessArticle

An Enhanced Discriminant Analysis Approach for Multi-Classification with Integrated Machine Learning-Based Missing Data Imputation

by Autcha Araveeporn and Atid Kangtunyakarn

Mathematics 2025, 13(21), 3392; https://doi.org/10.3390/math13213392 (registering DOI) - 24 Oct 2025

Abstract

This study addresses the challenge of accurate classification under missing data conditions by integrating multiple imputation strategies with discriminant analysis frameworks. The proposed approach evaluates six imputation methods (Mean, Regression, KNN, Random Forest, Bagged Trees, MissRanger) across several discriminant techniques. Simulation scenarios varied [...] Read more.

This study addresses the challenge of accurate classification under missing data conditions by integrating multiple imputation strategies with discriminant analysis frameworks. The proposed approach evaluates six imputation methods (Mean, Regression, KNN, Random Forest, Bagged Trees, MissRanger) across several discriminant techniques. Simulation scenarios varied in sample size, predictor dimensionality, and correlation structure, while the real-world application employed the Cirrhosis Prediction Dataset. The results consistently demonstrate that ensemble-based imputations, particularly regression, KNN, and MissRanger, outperform simpler approaches by preserving multivariate structure, especially in high-dimensional and highly correlated settings. MissRanger yielded the highest classification accuracy across most discriminant analysis methods in both simulated and real data, with performance gains most pronounced when combined with flexible or regularized classifiers. Regression imputation showed notable improvements under low correlation, aligning with the theoretical benefits of shrinkage-based covariance estimation. Across all methods, larger sample sizes and high correlation enhanced classification accuracy by improving parameter stability and imputation precision. Full article

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

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Mathematics

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