Axioms

10 pages, 258 KB

Open AccessArticle

Some New Applications of the Mellin Transform Involving the Lambert Transforms and Implications for the Riemann Hypothesis

by Hari M. Srivastava, Jeetendrasingh Maan and Emilio R. Negrín

Axioms 2025, 14(11), 814; https://doi.org/10.3390/axioms14110814 (registering DOI) - 31 Oct 2025

Abstract

This work investigates the interplay between the Mellin transform and Lambert transforms to derive several novel results. In particular, we establish new inversion formulae for the Lambert transforms along with a Plancherel-type identity. Additionally, we explore the implications of these findings, highlighting their [...] Read more.

This work investigates the interplay between the Mellin transform and Lambert transforms to derive several novel results. In particular, we establish new inversion formulae for the Lambert transforms along with a Plancherel-type identity. Additionally, we explore the implications of these findings, highlighting their relevance to Salem’s equivalence and potential connections with the Riemann hypothesis. Full article

(This article belongs to the Section Algebra and Number Theory)

35 pages, 37988 KB

Open AccessArticle

Shock Mach Number Effect on Instability Evolution at a Light–Heavy Fluid Interface: A Numerical Investigation

by Salman Saud Alsaeed, Satyvir Singh and Nahar F. Alshammari

Axioms 2025, 14(11), 813; https://doi.org/10.3390/axioms14110813 (registering DOI) - 31 Oct 2025

Abstract

Shock–accelerated interfaces between fluids of different densities are prone to Richtmyer–Meshkov-type instabilities, whose evolution is strongly influenced by the incident shock Mach number. In this study, we present a systematic numerical investigation of the Mach number effect on the instability growth at a [...] Read more.

Shock–accelerated interfaces between fluids of different densities are prone to Richtmyer–Meshkov-type instabilities, whose evolution is strongly influenced by the incident shock Mach number. In this study, we present a systematic numerical investigation of the Mach number effect on the instability growth at a light–heavy fluid layer. The governing dynamics are modeled using the compressible multi-species Euler equations, and the simulations are performed with a high-order modal discontinuous Galerkin method. This approach provides accurate resolution of sharp interfaces, shock waves, and small-scale vortical structures. A series of two-dimensional simulations is carried out for a range of shock Mach numbers impinging on a sinusoidally perturbed light–heavy fluid interface. The results highlight the distinct stages of instability evolution, from shock–interface interaction and baroclinic vorticity deposition to nonlinear roll-up and interface deformation. Quantitative diagnostics—including circulation, enstrophy, vorticity extrema, and mixing width—are employed to characterize the instability dynamics and to isolate the role of Mach number in enhancing or suppressing growth. Particular attention is given to the mechanisms of vorticity generation through baroclinic torque and compressibility effects. Moreover, the analysis of controlling parameters, including Atwood number, layer thickness, and initial perturbation amplitude, broadens the parametric understanding of shock-driven instabilities. The results reveal that increasing shock Mach number markedly enhances vorticity generation and accelerates interface growth, while the resulting nonlinear morphology remains strongly sensitive to variations in Atwood number and perturbation amplitude. Full article

(This article belongs to the Special Issue Applied Mathematics and Mathematical Modeling)

16 pages, 552 KB

Open AccessArticle

Novel Data-Driven Shrinkage Ridge Parameters for Handling Multicollinearity in Regression Models with Environmental and Chemical Data Applications

by Muteb Faraj Alharthi

Axioms 2025, 14(11), 812; https://doi.org/10.3390/axioms14110812 (registering DOI) - 31 Oct 2025

Abstract

Multicollinearity among predictor variables is a common challenge in modeling chemical and environmental datasets in physical sciences, often leading to collinearity issues and unreliable parameter estimates when fitting regression models. Ridge regression method emerges as an effective solution to this problem by introducing [...] Read more.

Multicollinearity among predictor variables is a common challenge in modeling chemical and environmental datasets in physical sciences, often leading to collinearity issues and unreliable parameter estimates when fitting regression models. Ridge regression method emerges as an effective solution to this problem by introducing a penalty term ( Full article

(This article belongs to the Section Mathematical Analysis)

15 pages, 227 KB

Open AccessArticle

Evaluating Some Improper Sine and Cosine Integrals

by Russell A. Gordon

Axioms 2025, 14(11), 811; https://doi.org/10.3390/axioms14110811 (registering DOI) - 31 Oct 2025

Abstract

We consider four infinite collections of improper integrals involving the sine and cosine functions and provide explicit values for the integrals in each of the collections. The methods used are elementary and involve the sine integral function and the cosine integral function. All [...] Read more.

We consider four infinite collections of improper integrals involving the sine and cosine functions and provide explicit values for the integrals in each of the collections. The methods used are elementary and involve the sine integral function and the cosine integral function. All needed results are included in this paper, but the details are a little tedious at times. Full article

89 pages, 1735 KB

Open AccessArticle

Quantum Field Theory of 3+1 Dimensional BTZ Gravity: Graviton Self-Energy, Axion Interactions, and Dark Matter in the Ultrahyperfunction Framework

by Hameeda Mir, Angelo Plastino, Behnam Pourhassan and Mario Carlos Rocca

Axioms 2025, 14(11), 810; https://doi.org/10.3390/axioms14110810 (registering DOI) - 31 Oct 2025

Abstract

We present a comprehensive quantum field theoretical analysis of graviton self-energy and mass generation in 3+1 dimensional BTZ black hole spacetime, incorporating axion interactions within the framework of dark matter theory. Using a novel mathematical approach based on ultrahyperfunctions, generalizations of Schwartz tempered [...] Read more.

We present a comprehensive quantum field theoretical analysis of graviton self-energy and mass generation in 3+1 dimensional BTZ black hole spacetime, incorporating axion interactions within the framework of dark matter theory. Using a novel mathematical approach based on ultrahyperfunctions, generalizations of Schwartz tempered distributions to the complex plane, we derive exact quantum relativistic expressions for graviton and axion self-energies without requiring ad hoc regularization procedures. Our approach extends the Gupta–Feynman quantization framework to BTZ gravity while introducing a new constraint that eliminates unitarity violations inherent in previous formulations, thereby avoiding the need for ghost fields. Through systematic application of generalized Feynman parameters, we evaluate both bradyonic and tachyonic graviton modes, revealing distinct quantum correction patterns that depend critically on momentum, energy, and mass parameters. Key findings include (1) natural graviton mass generation through cosmological constant interactions, yielding

m^{2} = 2 | Λ | / κ (1 - κ)

; (2) qualitatively different quantum behaviors between bradyonic and tachyonic modes, with bradyonic corrections reaching amplitudes 6 times larger than their tachyonic counterparts; (3) the discovery of momentum-dependent quantum dissipation effects that provide natural ultraviolet regulation; and (4) the first explicit analytical expressions and graphical representations for 17 distinct graviton self-energy contributions. The ultrahyperfunction formalism proves essential for handling the non-renormalizable nature of the theory, providing mathematically rigorous treatment of highly singular integrals while maintaining Lorentz invariance. Our results suggest observable consequences in gravitational wave propagation through frequency-dependent dispersive effects and modifications to black hole thermodynamics, potentially bridging theoretical quantum gravity with experimental constraints. Full article

67 pages, 732 KB

Open AccessReview

New Perspectives on Kac–Moody Algebras Associated with Higher-Dimensional Manifolds

by Rutwig Campoamor-Stursberg, Alessio Marrani and Michel Rausch de Traubenberg

Axioms 2025, 14(11), 809; https://doi.org/10.3390/axioms14110809 (registering DOI) - 31 Oct 2025

Abstract

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle

S^{1}

, we extend the approach to compact and non-compact group manifolds, [...] Read more.

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle

S^{1}

, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, we present the construction of generalized current algebras

g (M)

, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and we illustrate the framework through explicit realizations on

S U (2)

,

S U (2) / U (1)

, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications. Full article

(This article belongs to the Special Issue New Perspectives in Lie Algebras, 2nd Edition)

25 pages, 878 KB

Open AccessArticle

Stability and Controllability of Nonlinear Dynamic Systems with Neural Networks: An Application to Financial Data

by Lamiae Seddati, Touria Karite, Ahmed Aberqi and Nuno R. O. Bastos

Axioms 2025, 14(11), 808; https://doi.org/10.3390/axioms14110808 - 30 Oct 2025

Abstract

This paper presents a novel approach to the controllability of nonlinear dynamic systems using recurrent neural networks (RNNs). We develop a comprehensive theoretical framework that integrates controllability analysis, stability verification via Lyapunov functions, and the derivation of optimal control laws based on Pontryagin’s [...] Read more.

This paper presents a novel approach to the controllability of nonlinear dynamic systems using recurrent neural networks (RNNs). We develop a comprehensive theoretical framework that integrates controllability analysis, stability verification via Lyapunov functions, and the derivation of optimal control laws based on Pontryagin’s Maximum Principle. Our methodology not only ensures theoretical soundness but also offers practical effectiveness. To demonstrate its applicability, we conduct simulations using real-world data from the AAPL stock database. The proposed RNN-based control framework significantly reduces the deviation between predicted system outputs and actual observations. We further enhance performance through two complementary strategies, a direct control method and a parameter optimization approach, both of which contribute to the accuracy and adaptability of the control system. These results confirm the potential of neural network-based control in managing complex nonlinear dynamics. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences, 2nd Edition)

24 pages, 419 KB

Open AccessArticle

A Systematic Study on Distributivity of Threshold-Generated Implications over Uninorms

by Zhihong Yi

Axioms 2025, 14(11), 807; https://doi.org/10.3390/axioms14110807 (registering DOI) - 30 Oct 2025

Abstract

The distributivity of implications over fuzzy operators is a desirable property for fuzzy systems and can be employed in the elimination of the explosion of if–then rules. In this paper, we try to explore the relationship between the distributivity over the uninorms-related fuzzy [...] Read more.

The distributivity of implications over fuzzy operators is a desirable property for fuzzy systems and can be employed in the elimination of the explosion of if–then rules. In this paper, we try to explore the relationship between the distributivity over the uninorms-related fuzzy connectives and the distributivity over uninorms in the threshold generation method, i.e., the distributive equations

I (u, U_{1} (v, w)) = U_{2} (I (u, v), I (u, w))

and

I (U_{1} (u, v), w)) = U_{2} (I (u, w), I (v, w))

with I being the threshold-generated implication. Consequently, we find that if the uninorms are restricted to special classes, then the distributivity property by the first equation can be preserved between the original and threshold-generated implications; under certain constraints on the threshold-generated implication, the distributivity property by the second equation becomes trivial. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

22 pages, 2661 KB

Open AccessArticle

An Energy Minimization-Based Deep Learning Approach with Enhanced Stability for the Allen-Cahn Equation

by Xianghong He, Yuhan Wang, Rentao Wu, Jidong Gao and Rongpei Zhang

Axioms 2025, 14(11), 806; https://doi.org/10.3390/axioms14110806 - 30 Oct 2025

Abstract

The Allen-Cahn equation is a fundamental model in materials science for describing phase separation phenomena. This paper introduces an Energy-Stabilized Scaled Deep Neural Network (ES-ScaDNN) framework to solve the Allen-Cahn equation by energy minimization. Unlike traditional numerical methods, our approach directly approximates the [...] Read more.

The Allen-Cahn equation is a fundamental model in materials science for describing phase separation phenomena. This paper introduces an Energy-Stabilized Scaled Deep Neural Network (ES-ScaDNN) framework to solve the Allen-Cahn equation by energy minimization. Unlike traditional numerical methods, our approach directly approximates the solution of steady-state solution the Allen-Cahn equation by minimizing the associated energy functional using a deep neural network. ES-ScaDNN incorporates two key innovations. The first is a scaling layer designed to map the network output to the physical range of the Allen-Cahn phase variable. The second is a variance-based regularization term designed to promote clear phase separation. We demonstrate the accuracy and efficiency of ES-ScaDNN through comprehensive numerical experiments in both one and two dimensions. Our results show that ReLU activation functions are particularly well-suited for one-dimensional cases, while tanh functions are more suitable for two-dimensional problems due to their superior ability to maintain solution smoothness. Furthermore, we investigate how training epochs and the interface parameter ε influence the behavior of the solution. ES-ScaDNN provides a novel, accurate, and efficient deep learning framework for solving the Allen-Cahn equation, paving the way for tackling more complex phase-field problems. Full article

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22 pages, 14103 KB

Open AccessArticle

The Fourier Regularization for Solving a Cauchy Problem for the Laplace Equation with Uncertainty

by Xiaoya Liu, Yiliang He and Hong Yang

Axioms 2025, 14(11), 805; https://doi.org/10.3390/axioms14110805 - 30 Oct 2025

Abstract

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in [...] Read more.

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in the data can lead to significant errors in the solution and lack stability. Secondly, the determination of the parameters of the classical Laplace equation is difficult to adapt to the requirements of complex applications. For this purpose, in this paper, the Laplace equation with uncertain parameters is defined, and the uncertainty is represented by fuzzy numbers. In the case of granular differentiability, it is transformed into a granular differential equation, proving its serious ill-posedness. To overcome the ill-posedness, the Fourier regularization method is used to stabilize the numerical solution, and the stability estimation and error analysis between the regularization solution and the exact solution are given. Finally, numerical examples are given to illustrate the effectiveness and practicability of this method. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

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24 pages, 288 KB

Open AccessArticle

Classification of Four-Dimensional Complex Poisson Algebras

by Hani Abdelwahab and José María Sánchez

Axioms 2025, 14(11), 804; https://doi.org/10.3390/axioms14110804 - 30 Oct 2025

Abstract

This paper provides a complete classification of four-dimensional complex Poisson algebras, extending previous work on lower-dimensional cases. We develop a systematic approach that combines the classification of four-dimensional complex commutative associative algebras with the classification of four-dimensional complex Lie algebras, allowing us to [...] Read more.

This paper provides a complete classification of four-dimensional complex Poisson algebras, extending previous work on lower-dimensional cases. We develop a systematic approach that combines the classification of four-dimensional complex commutative associative algebras with the classification of four-dimensional complex Lie algebras, allowing us to determine all compatible Poisson structures up to isomorphism. Our main result presents 71 distinct types: 60 isolated non-isomorphic Poisson algebras with non-trivial commutative product and 11 one-parameter families of non-isomorphic Poisson algebras. In addition to these, we include the known four-dimensional complex Lie algebras, viewed as Poisson algebras with trivial commutative product. This purely algebraic classification provides a foundation for future studies, including the geometric classification of Poisson algebras, their degenerations, and the analysis of orbit closures in the corresponding algebraic variety. Full article

(This article belongs to the Section Algebra and Number Theory)

38 pages, 527 KB

Open AccessArticle

Stone and Flat Topologies on the Minimal Prime Spectrum of a Commutator Lattice

by George Georgescu, Leonard Kwuida and Claudia Mureşan

Axioms 2025, 14(11), 803; https://doi.org/10.3390/axioms14110803 - 30 Oct 2025

Abstract

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not [...] Read more.

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not even need to be enforced on their congruence lattices, let alone on those of the members of the variety they generate. Commutator lattices, defined by Czelakowski in 2008, are commutative multiplicative lattices having as prototype the algebraic structure of the congruence lattice of a such an algebra. Considering the prime elements with respect to the commutator operation, we obtain algebraic characterizations for minimal primes, then study the Stone and flat topologies on the set of minimal primes in a commutator lattice. We also prove abstract versions of congruence extension properties, actually of the general case of arbitrary morphisms instead of algebra embeddings, by means of complete join–semilattice morphisms between commutator lattices. We thus obtain abstractions for our results on congruence lattices and generalizations for results on frames and quantales, but also further cases in which these results hold. Furthermore, we investigate the lattice structures of these topologies as sublattices of the power sets of the sets of (minimal) primes. Full article

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

15 pages, 459 KB

Open AccessArticle

On Order Degree Problem for Moore Bound

by Himihami Mudiyanselage Chinthaka Wijerathne, Jayantha Lanel, Kissani Perera and Chathura Wanigasekara

Axioms 2025, 14(11), 802; https://doi.org/10.3390/axioms14110802 - 30 Oct 2025

Abstract

The degree diameter problem is a quest to determine the largest graph in terms of vertices satisfying given degree and diameter constraints. The largest possible graphs that can exist and that are subject to degree and diameter constraints are called Moore graphs. Since [...] Read more.

The degree diameter problem is a quest to determine the largest graph in terms of vertices satisfying given degree and diameter constraints. The largest possible graphs that can exist and that are subject to degree and diameter constraints are called Moore graphs. Since Moore graphs are rare, researchers are eager to build graphs closer to Moore graphs. This paper discusses the possibility of constructing graphs closer to Moore graphs, keeping a fixed order and minimizing the number of vertex pairs that break the diameter constraint, and suggests a new general relative index that measures the closeness to optimality. Based on the proposed index, it is highlighted that some of the graphs constructed in this work are closer to Moore graphs than the existing best results in the degree diameter problem. Furthermore, a fitness landscape analysis is conducted to identify the nature and the difficulty of the problem. This new method can be considered a new approach to constructing graphs closer to Moore graphs. Full article

(This article belongs to the Special Issue Mathematics and Its Applications in Other Disciplines)

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26 pages, 1188 KB

Open AccessArticle

Estimation of the Inverse Power Lindley Distribution Parameters Using Ranked Set Sampling with an Application to Failure Time Data

by Ghadah Alomani, Sid Ahmed Benchiha and Amer Ibrahim Al-Omari

Axioms 2025, 14(11), 801; https://doi.org/10.3390/axioms14110801 - 30 Oct 2025

Abstract

In this paper, the ranked set sampling method (RSS) is considered for estimating the inverse power Lindley distribution (IPLD) parameters and compared with the commonly simple random sampling. Different estimation methods are investigated including the commonly maximum likelihood, minimum distance estimation methods (Anderson [...] Read more.

In this paper, the ranked set sampling method (RSS) is considered for estimating the inverse power Lindley distribution (IPLD) parameters and compared with the commonly simple random sampling. Different estimation methods are investigated including the commonly maximum likelihood, minimum distance estimation methods (Anderson Darling (AD), right tail Anderson Darling, left tail Anderson Darling, AD left tail second order, Cramér-von Mises), methods of maximum and minimum spacing distance (maximum product spacing distance, minimum spacing distance), methods of ordinary and weighted least squares, and the Kolmogorov–Smirnov method. A simulation study is conducted to compare these methods using RSS and SRS based on the same number of measured units in terms of mean squared error, bias, efficiency, and mean relative estimation error. A failure data set is fitted to the IPLD and the proposed estimation methods are applied to the data. Full article

(This article belongs to the Section Mathematical Analysis)

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27 pages, 1008 KB

Open AccessArticle

Hybrid Euler–Lagrange Approach for Fractional-Order Modeling of Glucose–Insulin Dynamics

by Muflih Alhazmi, Safa M. Mirgani and Sayed Saber

Axioms 2025, 14(11), 800; https://doi.org/10.3390/axioms14110800 - 30 Oct 2025

Abstract

We propose a hybrid Caputo–Lagrange Discretization Method (CLDM) for the fractional-order modeling of glucose–insulin dynamics. The model incorporates key physiological mechanisms such as glucose suppression, insulin activation, and delayed feedback with memory effects captured through Caputo derivatives. Analytical results establish positivity, boundedness, existence, [...] Read more.

We propose a hybrid Caputo–Lagrange Discretization Method (CLDM) for the fractional-order modeling of glucose–insulin dynamics. The model incorporates key physiological mechanisms such as glucose suppression, insulin activation, and delayed feedback with memory effects captured through Caputo derivatives. Analytical results establish positivity, boundedness, existence, uniqueness, and Hyers–Ulam stability. Numerical simulations confirm that the proposed method improves accuracy and efficiency compared with the Residual Power Series Method and the fractional Runge–Kutta method. Sensitivity analysis highlights fractional order

θ

as a biomarker for metabolic memory. The findings demonstrate that CLDM offers a robust and computationally efficient framework for biomedical modeling with potential applications in diabetes research and related physiological systems. Full article

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

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11 pages, 219 KB

Open AccessArticle

Double Wronskian Representation of the Nth-Order Solutions to the Davey–Stewartson Equations

by Pierre Gaillard

Axioms 2025, 14(11), 799; https://doi.org/10.3390/axioms14110799 - 30 Oct 2025

Abstract

Solutions to the Davey–Stewartson equations (DS) in terms of the determinants of order

2 N

depending on

4 N

real parameters are constructed. The Darboux transformation is used to construct solutions of order N to this equation. We obtain a representation of the [...] Read more.

Solutions to the Davey–Stewartson equations (DS) in terms of the determinants of order

2 N

depending on

4 N

real parameters are constructed. The Darboux transformation is used to construct solutions of order N to this equation. We obtain a representation of the solutions to the equations (DS) in terms of two Wronskians. We obtain what we call a double Wronskian representation of the solutions to the (DS) equations. With this method, some particular explicit solutions for the first orders are constructed. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems: Theory and Applications, 2nd Edition)

10 pages, 259 KB

Open AccessArticle

Scator Holomorphic Functions

by Jan L. Cieśliński, Zbigniew Hasiewicz and Artur Kobus

Axioms 2025, 14(11), 798; https://doi.org/10.3390/axioms14110798 - 29 Oct 2025

Abstract

Scators form a linear space equipped with a specific non-distributive product. In the elliptic case they can be interpreted as a kind of hypercomplex number. The requirement that the scator partial derivatives are direction-independent leads to a generalization of the Cauchy–Riemann equation and [...] Read more.

Scators form a linear space equipped with a specific non-distributive product. In the elliptic case they can be interpreted as a kind of hypercomplex number. The requirement that the scator partial derivatives are direction-independent leads to a generalization of the Cauchy–Riemann equation and to scator holomorphic functions. In this paper we find a complete set of

C^{2}

-solutions to the generalized Cauchy–Riemann system in the

(1 + n)

-dimensional elliptic scator space. For any

n ⩾ 2

this set consists of three classes: components exponential functions (already known), a new class of affine linear functions, and some exceptional solutions parameterized by arbitrary functions of one variable. We show, however, that the last class of solutions is not scator holomorphic and the generalized Cauchy–Riemann system should be supplemented with additional constraints to avoid such spurious solutions. The obtained family of scator holomorphic functions, although relatively narrow, is greater than that of analogous functions in quaternionic or Clifford analysis. Full article

(This article belongs to the Section Mathematical Analysis)

32 pages, 2990 KB

Open AccessArticle

Enhancing Classification Results of Slope Entropy Using Downsampling Schemes

by Vicent Moltó-Gallego, David Cuesta-Frau and Mahdy Kouka

Axioms 2025, 14(11), 797; https://doi.org/10.3390/axioms14110797 - 29 Oct 2025

Abstract

Entropy calculation provides meaningful insight into the dynamics and complexity of temporal signals, playing a crucial role in classification tasks. These measures are able to describe intrinsic characteristics of temporal series, such as regularity, complexity or predictability. Depending on the characteristics of the [...] Read more.

Entropy calculation provides meaningful insight into the dynamics and complexity of temporal signals, playing a crucial role in classification tasks. These measures are able to describe intrinsic characteristics of temporal series, such as regularity, complexity or predictability. Depending on the characteristics of the signal under study, the performance of entropy as a feature for classification may vary, and not any kind of entropy calculation technique may be suitable for that specific signal. Therefore, we aim to increase entropy’s classification accuracy performance, specially in the case of Slope Entropy (SlpEn), by enhancing the information content of the patterns present in the data before calculating the entropy, with downsampling techniques. More specifically, we will be using both uniform downsampling (UDS) and non-uniform downsampling techniques. In the case of non-uniform downsapling, the technique used is known as Trace Segmentation (TS), which is a non-uniform downsampling scheme that is able to enhance the most prominent patterns present in a temporal series while discarding the less relevant ones. SlpEn is a novel method recently proposed in the field of time series entropy estimation that in general outperforms other methods in classification tasks. We will combine it both with TS or UDS. In addition, since both techniques reduce the number of samples that the entropy will be calculated on, it can significantly decrease the computation time. In this work, we apply TS or UDS to the data before calculating SlpEn to assess how downsampling can impact the behaviour of SlpEn in terms of performance and computational cost, experimenting on different kinds of datasets. In addition, we carry out a comparison between SlpEn and one of the most commonly used entropy calculation methods: Permutation Entropy (PE). Results show that both uniform and non-uniform downsampling are able to enhance the performance of both SlpEn and PE when used as the only features in classification tasks, gaining up to 13% and 22% in terms of accuracy, respectively, when using TS and up to 10% and 21% when using UDS. In addition, when downsampling to 50% of the original data, we obtain a speedup around ×2 with individual entropy calculations, while, when incorporating the downsampling algorithms into time count, speedups with UDS are between ×1.2 and ×1.7, depending on the dataset. With TS, these speedups are above ×2, while maintaining accuracy levels similar to those obtained when using the 100% of the original data. Our findings suggest that most temporal series, specially medical ones, have been measured using a sampling frequency above the optimal threshold, thus obtaining unnecessary information for classification tasks, which is then discarded when performing downsampling. Downsampling techniques are potentially beneficial to any kind of entropy calculation technique, not only those used in the paper. It is able to enhance entropy’s performance in classification tasks while reducing its computation time, thus resulting in a win-win situation. We recommend to downsample to percentages between 20% and 45% of the original data to obtain the best results in terms of accuracy in classification tasks. Full article

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22 pages, 1270 KB

Open AccessArticle

A Novel Family of CDF Estimators Under PPS Sampling: Computational, Theoretical, and Applied Perspectives

by Salman Shah, Eisa Mahmoudi, Hasnain Iftikhar, Paulo Canas Rodrigues, Ronny Ivan Gonzales Medina and Javier Linkolk López-Gonzales

Axioms 2025, 14(11), 796; https://doi.org/10.3390/axioms14110796 - 29 Oct 2025

Abstract

Accurate estimation of population distribution characteristics is a fundamental task in survey sampling and statistical inference. This paper introduces a new family of estimators for the cumulative distribution function (CDF) under probability proportional to size (PPS) sampling, incorporating auxiliary information to enhance efficiency. [...] Read more.

Accurate estimation of population distribution characteristics is a fundamental task in survey sampling and statistical inference. This paper introduces a new family of estimators for the cumulative distribution function (CDF) under probability proportional to size (PPS) sampling, incorporating auxiliary information to enhance efficiency. The proposed approach employs dual auxiliary variables in the estimation phase, while the sampling design relies on a single auxiliary variable. Theoretical properties, including bias and mean squared error (MSE), are rigorously derived to establish the efficiency of the new class. An extensive empirical evaluation using three distinct populations—fisheries data, wine chemistry data, and demographic records—demonstrates the superiority of the proposed estimators. In terms of accuracy, the best-performing proposed estimator achieves an MSE of 0.0012, compared to 0.0127 for the widely used GK estimator. Percentage relative efficiency (PRE) values further underscore these improvements, with gains ranging from 123% to over 328% across the three populations. Graphical comparisons confirm these trends, illustrating that the proposed estimators consistently dominate conventional approaches. Overall, the findings highlight both the theoretical soundness and practical utility of the proposed family, offering robust and computationally efficient improvements for CDF estimation in complex survey designs. Full article

(This article belongs to the Special Issue Computational Statistics and Its Applications, 2nd Edition)

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