Axioms

13 pages, 291 KiB

Open AccessArticle

Toeplitz Operators with Radial Symbols on Weighted Pluriharmonic Bergman Spaces over Reinhardt Domains

by Zhi-Ling Sun, Feng Qi and Wei-Shih Du

Axioms 2025, 14(6), 478; https://doi.org/10.3390/axioms14060478 - 19 Jun 2025

Viewed by 235

In this paper, we design an operator A restricted to a weighted pluriharmonic Bergman space

b_{μ}^{2} (Ω)

over the Reinhardt domains, with an isometric isomorphism between

b_{μ}^{2} (Ω)

and the subset of [...] Read more.

In this paper, we design an operator A restricted to a weighted pluriharmonic Bergman space

b_{μ}^{2} (Ω)

over the Reinhardt domains, with an isometric isomorphism between

b_{μ}^{2} (Ω)

and the subset of

l_{2} (Z^{n})

. Furthermore, we show that Toeplitz operators

T_{a}

with radial symbols are unitary to the multiplication operators

γ_{a} I

on sequence space

l_{2}

by using the operator A. The Wick function of a Toeplitz operator with a radial symbol provides some features to the operator, establishing its spectral decomposition. Finally, we specify the obtained results on the Reinhardt domains for the unit ball. Full article

(This article belongs to the Section Mathematical Analysis)

10 pages, 674 KiB

Open AccessArticle

Abundant Exact Traveling-Wave Solutions for Stochastic Graphene Sheets Model

by Wael W. Mohammed, Taher S. Hassan, Rabeb Sidaoui, Hijyah Alshammary and Mohamed S. Algolam

Axioms 2025, 14(6), 477; https://doi.org/10.3390/axioms14060477 - 19 Jun 2025

Viewed by 189

Abstract

Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result [...] Read more.

Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result with a solution of stochastic ordinary differential equations. By applying the extended tanh function method, we obtain the soliton solutions for the deterministic counterparts of the graphene sheets model. Because graphene sheets are important in many fields, such as electronics, photonics, and energy storage, the solutions of the stochastic graphene sheets model are beneficial for understanding several fascinating scientific phenomena. Using the MATLAB program, we exhibit several 3D graphs that illustrate the impact of multiplicative noise on the exact solutions of the SGSM. By incorporating stochastic elements into the equations that govern the evolution of graphene sheets, researchers can gain insights into how these fluctuations impact the behavior of the material over time. Full article

(This article belongs to the Special Issue Advances in Partial Differential Equations: Qualitative Analysis and Numerical Methods)

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19 pages, 299 KiB

Open AccessArticle

A Bayesian Approach to Step-Stress Partially Accelerated Life Testing for a Novel Lifetime Distribution

by Mervat K. Abd Elaal, Hebatalla H. Mohammad, Zakiah I. Kalantan, Abeer A. EL-Helbawy, Gannat R. AL-Dayian, Sara M. Behairy and Reda M. Refaey

Axioms 2025, 14(6), 476; https://doi.org/10.3390/axioms14060476 - 19 Jun 2025

Viewed by 198

Abstract

In lifetime testing, the failure times of highly reliable products under normal usage conditions are often impractically long, making direct reliability assessment impractical. To overcome this, step-stress partially accelerated life testing is employed to reduce testing time while preserving data quality. This paper [...] Read more.

In lifetime testing, the failure times of highly reliable products under normal usage conditions are often impractically long, making direct reliability assessment impractical. To overcome this, step-stress partially accelerated life testing is employed to reduce testing time while preserving data quality. This paper develops a Bayesian model based on Type II censored data, assuming that item lifetimes follow the Topp–Leone inverted Kumaraswamy distribution, a flexible alternative to classical lifetime models due to its ability to capture various hazard rate shapes and to model bounded and skewed lifetime data more effectively than traditional models observed in real-world reliability data. Bayes estimators of the model parameters and acceleration factor are derived under both symmetric (balanced squared error) and asymmetric (balanced linear exponential) loss functions using informative priors. The novelty of this work lies in the integration of the Topp–Leone inverted Kumaraswamy distribution within the Bayesian step-stress partially accelerated life testing framework, which has not been explored previously, offering improved modeling capability for complex lifetime data. The proposed method is validated through comprehensive simulation studies under various censoring schemes, demonstrating robustness and superior estimation performance compared to traditional models. A real-data application involving COVID-19 mortality data further illustrates the practical relevance and improved fit of the model. Overall, the results highlight the flexibility, efficiency, and applicability of the proposed Bayesian approach in reliability analysis. Full article

24 pages, 1596 KiB

Open AccessArticle

Convergence and ω²-Stability Analysis of a Hybrid-Type Iterative Scheme with Application to Functional Delay Differential Equations

by Safeer Hussain Khan, Hina Dilawer, Hira Iqbal and Mujahid Abbas

Axioms 2025, 14(6), 475; https://doi.org/10.3390/axioms14060475 - 19 Jun 2025

Viewed by 157

Abstract

The purpose of this article is to analyze a hybrid-type iterative algorithm for a class of generalized non-expansive mappings satisfying the Garcia-Falset property in uniformly convex Banach spaces. Some existing results for such mappings have been obtained using the given algorithm. The [...] Read more.

The purpose of this article is to analyze a hybrid-type iterative algorithm for a class of generalized non-expansive mappings satisfying the Garcia-Falset property in uniformly convex Banach spaces. Some existing results for such mappings have been obtained using the given algorithm. The

ω^{2}

-stability of the iterative process is also studied. Using some examples, numerical experiments are conducted by comparing this iterative algorithm with different well-known iterative schemes. It is concluded that this iterative algorithm converges faster to the fixed point and is preferable over the previously known iterative schemes using the Garcia-Falset property. A weak solution of the Volterra–Stieltjes-type delay functional differential equation is presented to demonstrate the significance of the proposed results. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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19 pages, 330 KiB

Open AccessArticle

On the Existence of (p,q)-Solutions for the Post-Quantum Langevin Equation: A Fixed-Point-Based Approach

by Mohammed Jasim Mohammed, Ali Ghafarpanah, Sina Etemad, Sotiris K. Ntouyas and Jessada Tariboon

Axioms 2025, 14(6), 474; https://doi.org/10.3390/axioms14060474 - 19 Jun 2025

Viewed by 267

Abstract

The two-parameter

(p, q)

-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation [...] Read more.

The two-parameter

(p, q)

-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation using two-parameter

(p, q)

-Caputo derivatives. For this new Langevin equation, equivalently, we obtain the solution structure as a post-quantum integral equation and then conduct an existence analysis via a fixed-point-based approach. The use of theorems such as the Krasnoselskii and Leray–Schauder fixed-point theorems will guarantee the existence of solutions to this equation, whose uniqueness is later proven by Banach’s contraction principle. Finally, we provide three examples in different structures and validate the results numerically. Full article

(This article belongs to the Special Issue Fractional Order Functional Differential Equations and Fixed Point Theory)

24 pages, 20406 KiB

Open AccessArticle

Single-Mode Richtmyer–Meshkov Instability in Light Fluid Layer: Insights from Numerical Simulations

by Ahmed Hussein Msmali, Satyvir Singh and Mutum Zico Meetei

Axioms 2025, 14(6), 473; https://doi.org/10.3390/axioms14060473 - 19 Jun 2025

Viewed by 286

Abstract

This study presents high-fidelity numerical simulations of the shock-accelerated single-mode Richtmyer–Meshkov instability (RMI) in a light helium layer confined between two interfaces and surrounded by nitrogen gas. A high-order modal discontinuous Galerkin method is employed to solve the two-dimensional compressible Euler equations, enabling [...] Read more.

This study presents high-fidelity numerical simulations of the shock-accelerated single-mode Richtmyer–Meshkov instability (RMI) in a light helium layer confined between two interfaces and surrounded by nitrogen gas. A high-order modal discontinuous Galerkin method is employed to solve the two-dimensional compressible Euler equations, enabling detailed investigation of interface evolution, vorticity dynamics, and flow structure development under various physical conditions. The effects of helium layer thickness, initial perturbation amplitude, and incident shock Mach number are systematically explored by analyzing interface morphology, vorticity generation, enstrophy, and kinetic energy. The results show that increasing the helium layer thickness enhances vorticity accumulation and interface deformation by delaying interaction with the second interface, allowing more sustained instability growth. Larger initial perturbation amplitudes promote earlier onset of nonlinear deformation and stronger baroclinic vorticity generation, while higher shock strengths intensify pressure gradients across the interface, accelerating instability amplification and mixing. These findings highlight the critical interplay between layer confinement, perturbation strength, and shock strength in governing the nonlinear evolution of RMI in light fluid layers. Full article

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15 pages, 298 KiB

Open AccessFeature PaperArticle

The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals

by Antanas Laurinčikas

Axioms 2025, 14(6), 472; https://doi.org/10.3390/axioms14060472 - 17 Jun 2025

Viewed by 254

Abstract

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where [...] Read more.

In this paper, we obtain approximation theorems of classes of analytic functions by shifts

L (λ, α, s + i τ)

of the Lerch zeta-function for

τ \in [T, T + H]

where

H \in [T^{27 / 82}, T^{1 / 2}]

. The cases of all parameters,

λ, α \in (0, 1]

, are considered. If the set

{\log (m + α) : m \in N_{0}}

is linearly independent over

Q

, then every analytic function in the strip

{s = σ + i t \in C : σ \in (1 / 2, 1)}

is approximated by the above shifts. Full article

19 pages, 437 KiB

Open AccessArticle

Diversity and Semiconvergents in Pythagorean Tuning

by Rafael Cubarsi

Axioms 2025, 14(6), 471; https://doi.org/10.3390/axioms14060471 - 17 Jun 2025

Viewed by 282

Abstract

Several approaches to building generalized Pythagorean scales provide interpretation for the rational approximation of an irrational number. Generally, attention is paid to the convergents of the continued fraction expansions. The present paper focuses on the sequences of semiconvergents corresponding to the alternating best [...] Read more.

Several approaches to building generalized Pythagorean scales provide interpretation for the rational approximation of an irrational number. Generally, attention is paid to the convergents of the continued fraction expansions. The present paper focuses on the sequences of semiconvergents corresponding to the alternating best one-sided approximations. These sequences are interpreted as scale lineages organized as a kinship. Their properties are studied in terms of the two types of tones and elementary intervals, since each scale contains the tones of the previous scale plus the newly added tones, i.e., the generic diatones and accidentals. For the last scale of a lineage, the octave is regularly subdivided by sections, separated by a single elementary interval of the other type. Lineages are therefore related to the scale diversity with regard to their generic diatones and accidentals, which is analyzed from the Shannon diversity index, either for tone abundance or interval occupancy. Full article

(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)

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Graphical abstract

15 pages, 310 KiB

Open AccessArticle

Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination

by Zhi-Wei Xu and Gui-Lai Zhang

Axioms 2025, 14(6), 470; https://doi.org/10.3390/axioms14060470 - 16 Jun 2025

Viewed by 203

Abstract

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive [...] Read more.

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results. Full article

(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Edition)

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19 pages, 1015 KiB

Open AccessArticle

Cloud Platform Selection Using Extended Multi-Attribute Decision-Making Methods with Interval Type-2 Fuzzy Sets

by Ivana Spasenić, Danijela Tadić, Milan Čabarkapa, Dragan Marinković and Nikola Komatina

Axioms 2025, 14(6), 469; https://doi.org/10.3390/axioms14060469 - 16 Jun 2025

Viewed by 357

Abstract

The selection of an appropriate cloud platform represents a highly important strategic decision for any IT company. In pursuit of business optimization, cost reduction, improved reliability, and enhanced market competitiveness, selecting the most suitable cloud platform has become a major practical challenge. This [...] Read more.

The selection of an appropriate cloud platform represents a highly important strategic decision for any IT company. In pursuit of business optimization, cost reduction, improved reliability, and enhanced market competitiveness, selecting the most suitable cloud platform has become a major practical challenge. This paper proposes a novel two-stage multi-attribute decision-making (MADM) model, enhanced through the use of interval type-2 fuzzy sets (IT2FMADM). This was demonstrated through a case study in an IT company based in Serbia. In the first stage, three experts from the company were surveyed to assess the relative importance of the attributes, and their evaluations were aggregated using the fuzzy harmonic mean operator. As a result, unified fuzzy weight vectors were obtained. In the second stage, two MADM methods extended with interval type-2 fuzzy sets, namely COmplex PRoportional Assessment (IT2FCOPRAS) and Evaluation based on Distance from Average Solution (IT2FEDAS), were applied to support the selection of the most suitable cloud platform. Each platform was evaluated by decision-makers (DMs), who reached a consensus in their assessments, supported by data from company records. A comparative analysis of the results revealed that different methods may produce varying rankings of alternatives, particularly when the alternatives are objectively similar in their characteristics. Nevertheless, the proposed model can serve as a highly useful decision-support tool for company management. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic and Computational Intelligence)

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26 pages, 306 KiB

Open AccessArticle

Osculating Mate of a Curve in Minkowski 3-Space

by İskender Öztürk, Hasan Çakır and Mustafa Özdemir

Axioms 2025, 14(6), 468; https://doi.org/10.3390/axioms14060468 - 16 Jun 2025

Viewed by 179

Abstract

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We [...] Read more.

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as Bertrand, Mannheim, and Bäcklund pairs. Our results extend existing theories by unifying several known curve pair classifications under a single geometric framework in Lorentzian space. Full article

(This article belongs to the Section Geometry and Topology)

16 pages, 616 KiB

Open AccessArticle

Bayesian Quantile Regression for Partial Functional Linear Spatial Autoregressive Model

by Dengke Xu, Shiqi Ke, Jun Dong and Ruiqin Tian

Axioms 2025, 14(6), 467; https://doi.org/10.3390/axioms14060467 - 16 Jun 2025

Viewed by 199

Abstract

When performing Bayesian modeling on functional data, the assumption of normality is often made on the model error and thus the results may be sensitive to outliers and/or heavy tailed data. An important and good choice for solving such problems is quantile regression. [...] Read more.

When performing Bayesian modeling on functional data, the assumption of normality is often made on the model error and thus the results may be sensitive to outliers and/or heavy tailed data. An important and good choice for solving such problems is quantile regression. Therefore, this paper introduces the quantile regression into the partial functional linear spatial autoregressive model (PFLSAM) based on the asymmetric Laplace distribution for the errors. Then, the idea of the functional principal component analysis, and the hybrid MCMC algorithm combining Gibbs sampling and the Metropolis–Hastings algorithm are developed to generate posterior samples from the full posterior distributions to obtain Bayesian estimation of unknown parameters and functional coefficients in the model. Finally, some simulation studies show that the proposed Bayesian estimation method is feasible and effective. Full article

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13 pages, 483 KiB

Open AccessArticle

The Mean and the Variance as Dual Concepts in a Fundamental Duality

by David Ellerman

Axioms 2025, 14(6), 466; https://doi.org/10.3390/axioms14060466 - 16 Jun 2025

Viewed by 225

Abstract

A basic duality arises throughout the mathematical and natural sciences. Traditionally, logic is thought to be based on the Boolean logic of subsets, but the development of category theory in the mid-twentieth century shows the duality between subsets and partitions (or equivalence relations). [...] Read more.

A basic duality arises throughout the mathematical and natural sciences. Traditionally, logic is thought to be based on the Boolean logic of subsets, but the development of category theory in the mid-twentieth century shows the duality between subsets and partitions (or equivalence relations). Hence, there is an equally fundamental dual logic of partitions. At a more basic or granular level, the elements of a subset are dual to the distinctions (pairs of elements in different blocks) of a partition. The quantitative version of subset logic is probability theory (as developed by Boole), and the quantitative version of the logic of partitions is information theory re-founded on the notion of logical entropy. The subset side of the duality uses a one-sample (or one-element) approach, e.g., the mean of a random variable; the partition side uses a two-sample (or pair-of-elements) approach. This paper gives a new derivation of the variance (and covariance) based on the two-sample approach, which positions the variance on the partition and information theory side of the duality and thus dual to the mean. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics)

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26 pages, 2724 KiB

Open AccessReview

From Different Systems to a Single Common Model: A Review of Dynamical Systems Leading to Lorenz Equations

by Juan Carlos Chimal-Eguía, Florencio Guzmán-Aguilar, Víctor Manuel Silva-García, Héctor Báez-Medina and Manuel Alejandro Cardona-López

Axioms 2025, 14(6), 465; https://doi.org/10.3390/axioms14060465 - 13 Jun 2025

Viewed by 384

Abstract

This paper presents an analytical exploration of how diverse dynamical systems, arising from different scientific domains, can be reformulated (under specific approximations and assumptions) into a common set of equations formally equivalent to the Lorenz system originally derived to model atmospheric convection. Unlike [...] Read more.

This paper presents an analytical exploration of how diverse dynamical systems, arising from different scientific domains, can be reformulated (under specific approximations and assumptions) into a common set of equations formally equivalent to the Lorenz system originally derived to model atmospheric convection. Unlike previous studies that focus on analyzing or applying the Lorenz equations, our objective is to show how these equations emerge from distinct models, emphasizing the underlying structural and dynamical similarities. The mathematical steps involved in these reformulations are included. The systems examined include Lorenz’s original atmospheric convection model, the chaotic water wheel, the Maxwell–Bloch equations for lasers, mechanical gyrostat, solar dynamo model, mesoscale reaction dynamics, an interest rate economic model, and a socioeconomic control system. This work includes a discussion of the unifying features that lead to similar qualitative behaviors across seemingly unrelated systems. By highlighting the Lorenz system as a paradigmatic limit of a broad class of nonlinear models, we underscore its relevance as a unifying framework in the study of complex dynamics. Full article

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38 pages, 4814 KiB

Open AccessArticle

Influence Analysis in the Lognormal Regression Model with Fitted and Quantile Residuals

by Muhammad Habib, Muhammad Amin and Sadiah M. A. Aljeddani

Axioms 2025, 14(6), 464; https://doi.org/10.3390/axioms14060464 - 13 Jun 2025

Viewed by 322

Abstract

Influence analysis is a critical diagnostic tool in regression modeling to ensure reliable parameter estimates. This study evaluates the effectiveness of diagnostic methods for detecting influential observations in the lognormal regression model using fitted and quantile residuals. We assess Cook’s distance, modified Cook’s [...] Read more.

Influence analysis is a critical diagnostic tool in regression modeling to ensure reliable parameter estimates. This study evaluates the effectiveness of diagnostic methods for detecting influential observations in the lognormal regression model using fitted and quantile residuals. We assess Cook’s distance, modified Cook’s distance, covariance ratio, and the Hadi method through a Monte Carlo simulation with varying sample sizes, dispersion parameters, perturbation values, and numbers of explanatory variables, and a real-world application to an atmospheric environmental dataset. Simulation results demonstrate that Cook’s distance and the Hadi method achieve a good performance under all scenarios, with quantile residuals generally outperforming fitted residuals. The sensitivity analysis confirms their robustness, with minimal variation in detection rates. The covariance ratio performs well but shows slight variability in high-dispersion cases, while modified Cook’s distance consistently underperforms, particularly with quantile residuals. The real-world application confirms these findings, with Cook’s distance and the Hadi method effectively identifying influential points affecting ozone concentration estimates. These results highlight the superiority of Cook’s distance and the Hadi method for lognormal regression model diagnostics, with quantile residuals enhancing detection accuracy. Full article

(This article belongs to the Special Issue Advances in the Theory and Applications of Statistical Distributions)

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30 pages, 999 KiB

Open AccessArticle

Codimension-Two Bifurcation Analysis and Global Dynamics of a Discrete Epidemic Model

by Raja Ramiz Ahmed Khan, Abdul Qadeer Khan, Turki D. Alharbi and Jawharah G. AL-Juaid

Axioms 2025, 14(6), 463; https://doi.org/10.3390/axioms14060463 - 13 Jun 2025

Viewed by 230

Abstract

In this paper, we study the global dynamics, boundedness, existence of invariant intervals, and identification of codimension-two bifurcation sets with detailed bifurcation analysis at the epidemic fixed point of a discrete epidemic model. More precisely, under definite parametric conditions, it is proved that [...] Read more.

In this paper, we study the global dynamics, boundedness, existence of invariant intervals, and identification of codimension-two bifurcation sets with detailed bifurcation analysis at the epidemic fixed point of a discrete epidemic model. More precisely, under definite parametric conditions, it is proved that every positive solution of the discrete epidemic model is bounded, and furthermore, we have also constructed the invariant interval. By the linear stability theory, we have derived the sufficient condition, as well as the necessary and sufficient condition(s) under which fixed points obey certain local dynamical characteristics. We also gave the global analysis at fixed points and proved that both disease-free and epidemic fixed points become globally stable under certain conditions and parameters. Next, in order to study the two-parameter bifurcations of the discrete epidemic model at the epidemic fixed point, we first identified the two-parameter bifurcation sets, and then a detailed two-parameter bifurcation analysis is given by the bifurcation theory and affine transformations. Furthermore, we have given the biological interpretations of the theoretical findings. Finally, numerical simulation validated the theoretical results. Full article

(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)

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15 pages, 294 KiB

Open AccessArticle

Analysis of Operators with Complex Gaussian Kernels over

ℰ^{'}

(ℝ): Abelian Theorems

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

Axioms 2025, 14(6), 462; https://doi.org/10.3390/axioms14060462 - 12 Jun 2025

Viewed by 332

Abstract

This paper investigates Abelian theorems for operators with complex Gaussian kernels over distributions of compact support. Furthermore, our investigation encompasses an examination of the Gauss–Weierstrass semigroup, the linear canonical transform, and the Ornstein–Uhlenbeck semigroup as particular cases within the scope of our study. [...] Read more.

This paper investigates Abelian theorems for operators with complex Gaussian kernels over distributions of compact support. Furthermore, our investigation encompasses an examination of the Gauss–Weierstrass semigroup, the linear canonical transform, and the Ornstein–Uhlenbeck semigroup as particular cases within the scope of our study. Full article

18 pages, 695 KiB

Open AccessArticle

Modified Bimodal Exponential Distribution with Applications

by Jimmy Reyes, Barry C. Arnold, Yolanda M. Gómez, Osvaldo Venegas and Héctor W. Gómez

Axioms 2025, 14(6), 461; https://doi.org/10.3390/axioms14060461 - 12 Jun 2025

Viewed by 252

Abstract

In this paper, we introduce a new distribution for modeling bimodal data supported on non-negative real numbers and particularly suited with an excess of very small values. This family of distributions is derived by multiplying the exponential distribution by a fourth-degree polynomial, resulting [...] Read more.

In this paper, we introduce a new distribution for modeling bimodal data supported on non-negative real numbers and particularly suited with an excess of very small values. This family of distributions is derived by multiplying the exponential distribution by a fourth-degree polynomial, resulting in a model that better fits the shape of the second mode of the empirical distribution of the data. We study the general density of this new family of distributions, along with its properties, moments, and skewness and kurtosis coefficients. A simulation study is performed to estimate parameters by the maximum likelihood method. Additionally, we present two applications to real-world datasets, demonstrating that the new distribution provides a better fit than the bimodal exponential distribution. Full article

(This article belongs to the Special Issue Advances in Statistical Simulation and Computing)

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23 pages, 544 KiB

Open AccessArticle

Estimation of Parameters and Reliability Based on Unified Hybrid Censoring Schemes with an Application to COVID-19 Mortality Datasets

by Mustafa M. Hasaballah, Mahmoud M. Abdelwahab and Khamis A. Al-Karawi

Axioms 2025, 14(6), 460; https://doi.org/10.3390/axioms14060460 - 12 Jun 2025

Viewed by 903

Abstract

This article presents maximum likelihood and Bayesian estimates for the parameters, reliability function, and hazard function of the Gumbel Type-II distribution using a unified hybrid censored sample. Bayesian estimates are derived under three loss functions: squared error, LINEX, and generalized entropy. The parameters [...] Read more.

This article presents maximum likelihood and Bayesian estimates for the parameters, reliability function, and hazard function of the Gumbel Type-II distribution using a unified hybrid censored sample. Bayesian estimates are derived under three loss functions: squared error, LINEX, and generalized entropy. The parameters are assumed to follow independent gamma prior distributions. Since closed-form solutions are not available, the MCMC approximation method is used to obtain the Bayesian estimates. The highest posterior density credible intervals for the model parameters are computed using importance sampling. Additionally, approximate confidence intervals are constructed based on the normal approximation to the maximum likelihood estimates. To derive asymptotic confidence intervals for the reliability and hazard functions, their variances are estimated using the delta method. A numerical study compares the proposed estimators in terms of their average values and mean squared error using Monte Carlo simulations. Finally, a real dataset is analyzed to illustrate the proposed estimation methods. Full article

(This article belongs to the Special Issue Recent Stochastic and Statistical Approaches for Modeling Complex Systems and Dependent Variables)

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13 pages, 243 KiB

Open AccessArticle

Complex Riemannian Spacetime and Singularity-Free Black Holes and Cosmology

by John W. Moffat

Axioms 2025, 14(6), 459; https://doi.org/10.3390/axioms14060459 - 12 Jun 2025

Viewed by 453

Abstract

An approach is presented to address singularities in general relativity using a complex Riemannian spacetime extension. We demonstrate how this method can be applied to both black hole and cosmological singularities, specifically focusing on the Schwarzschild and Kerr black holes and the Friedmann–Lemaître–Robertson–Walker [...] Read more.

An approach is presented to address singularities in general relativity using a complex Riemannian spacetime extension. We demonstrate how this method can be applied to both black hole and cosmological singularities, specifically focusing on the Schwarzschild and Kerr black holes and the Friedmann–Lemaître–Robertson–Walker (FLRW) Big Bang cosmology. By extending the relevant coordinates into the complex plane and carefully choosing integration contours, we show that it is possible to regularize these singularities, resulting in physically meaningful, singularity-free solutions when projected back onto real spacetime. The removal of the singularity at the Big Bang allows for a bounce cosmology. The approach offers a potential bridge between classical general relativity and quantum gravity effects, suggesting a way to resolve longstanding issues in gravitational physics without requiring a full theory of quantum gravity. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

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26 pages, 478 KiB

Open AccessArticle

Treatment Effect Estimation in Survival Analysis Using Copula-Based Deep Learning Models for Causal Inference

by Jong-Min Kim

Axioms 2025, 14(6), 458; https://doi.org/10.3390/axioms14060458 - 10 Jun 2025

Viewed by 470

Abstract

This paper presents the use of Copula-based deep learning with Horvitz–Thompson (HT) weights and inverse probability of treatment weighting (IPTW) for estimating propensity scores in causal inference problems. This study compares the performance of the statistical methods—Copula-based deep learning with HT and IPTW [...] Read more.

This paper presents the use of Copula-based deep learning with Horvitz–Thompson (HT) weights and inverse probability of treatment weighting (IPTW) for estimating propensity scores in causal inference problems. This study compares the performance of the statistical methods—Copula-based deep learning with HT and IPTW weights, propensity score matching (PSM), and logistic regression—in estimating the treatment effect (ATE) using both simulated and real-world data. Our results show that the Copula-based recurrent neural network (RNN) with the method of HT weights provides the most precise and robust treatment effect estimate, with narrow confidence intervals indicating high confidence in the results. The PSM model yields the largest treatment effect estimate, but with greater uncertainty, suggesting sensitivity to data imbalances. In contrast, logistic regression and causal forests produce a substantially smaller estimate, potentially underestimating the treatment effect, particularly in structured datasets such as COMPAS scores. Overall, copula-based methods (HT and IPTW) tend to produce higher and more precise estimates, making them effective choices for treatment effect estimation in complex settings. Our findings emphasize the importance of method selection based on both the magnitude and precision of the treatment effect for accurate analysis. Full article

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

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14 pages, 680 KiB

Open AccessArticle

Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability

by Sun Xie, Haixing Zhao, Jun Yin and Jinyu Zou

Axioms 2025, 14(6), 457; https://doi.org/10.3390/axioms14060457 - 10 Jun 2025

Viewed by 187

Abstract

A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability q. In this [...] Read more.

A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability q. In this paper, we focus on exploring the characteristics of more reliable three-terminal graphs when the edge failure probability approaches 1. Three reliability comparison criteria are proposed to characterize the locally most reliable three-terminal graph progressively when the number of edges m is in the range of

[5, 4 n - 10]

and

[(\binom{n}{2}) - n + 4, (\binom{n}{2})]

. At the same time, the locally optimal structures in the range of the edge number m with

(4 n - 10, (\binom{n}{2}) - n + 4)

are restricted to six specific classes of graphs. Furthermore, based on these criteria, a method is introduced to search local optimal structures and offer a theoretical foundation for constructing optimal networks and repairing faulty ones. Full article

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26 pages, 331 KiB

Open AccessArticle

A Stochastic Nash Equilibrium Problem for Crisis Rescue

by Cunlin Li and Yiyan Li

Axioms 2025, 14(6), 456; https://doi.org/10.3390/axioms14060456 - 10 Jun 2025

Viewed by 194

Abstract

This paper proposes a two-stage stochastic non-cooperative game model to solve relief supplies procurement and distribution optimization of multiple rescue organizations in crisis rescue. Rescue organizations with limited budgets minimize rescue costs through relief supply procurement, storage, and transportation in an uncertain environment. [...] Read more.

This paper proposes a two-stage stochastic non-cooperative game model to solve relief supplies procurement and distribution optimization of multiple rescue organizations in crisis rescue. Rescue organizations with limited budgets minimize rescue costs through relief supply procurement, storage, and transportation in an uncertain environment. Under a mild assumption, we establish the existence and uniqueness of the equilibrium point and derive the optimality conditions by using the duality theory, characterizing the saddle point in the Lagrange framework. The problem is further reformulated as a constraint system governed by Lagrange multipliers, and its optimality is characterized by the Karush–Kuhn–Tucker condition. The economic interpretation of the multipliers as shadow prices is elucidated. Numerical experiments verify the effectiveness of the model in cost optimization in crisis rescue scenarios. Full article

25 pages, 1700 KiB

Open AccessArticle

Pearson and Deviance Residual-Based Control Charts for the Inverse Gaussian Ridge Regression Process: Simulation and an Application to Air Quality Monitoring

by Muhammad Amin, Samra Rani and Sadiah M. A. Aljeddani

Axioms 2025, 14(6), 455; https://doi.org/10.3390/axioms14060455 - 9 Jun 2025

Viewed by 325

Abstract

In manufacturing and service industries, monitoring processes with correlated input variables and inverse Gaussian (IG)-distributed quality characteristics is challenging due to the limitations of maximum likelihood estimator (MLE)-based control charts. When input variables exhibit multicollinearity, traditional MLE-based inverse Gaussian regression model (IGRM) control [...] Read more.

In manufacturing and service industries, monitoring processes with correlated input variables and inverse Gaussian (IG)-distributed quality characteristics is challenging due to the limitations of maximum likelihood estimator (MLE)-based control charts. When input variables exhibit multicollinearity, traditional MLE-based inverse Gaussian regression model (IGRM) control charts become unreliable. This study introduces novel Shewhart control charts using Pearson and deviance residuals based on the inverse Gaussian ridge regression (IGRR) model to address this issue. The proposed IGRR-based charts effectively handle multicollinearity, offering a robust alternative for process monitoring. Their performance is evaluated through Monte Carlo simulations using average run length (ARL) as the main criteria, demonstrating that Pearson residual-based IGRR charts outperform deviance residual-based charts and MLE-based methods, particularly under high multicollinearity. A real-world application to a Pakistan air quality dataset confirms their superior sensitivity in detecting pollution spikes, enabling timely environmental negotiations. These findings establish Pearson residual-based IGRR control charts as a practical and reliable tool for monitoring complex processes with correlated variables. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications, 2nd Edition)

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11 pages, 220 KiB

Open AccessArticle

Remarks on an Identity of Anastase and Díaz-Barrero

by Horst Alzer and Robert Frontczak

Axioms 2025, 14(6), 454; https://doi.org/10.3390/axioms14060454 - 9 Jun 2025

Viewed by 261

Abstract

We extend an algebraic identity of Anastase and Díaz-Barrero (2022) and apply our results to deduce various formulas for sums and series involving (among others) Fibonacci and Lucas numbers, Bernoulli polynomials, and the Riemann zeta function. Full article

15 pages, 314 KiB

Open AccessArticle

Notes on the Free Additive Convolution

by Shokrya S. Alshqaq, Raouf Fakhfakh and Fatimah Alshahrani

Axioms 2025, 14(6), 453; https://doi.org/10.3390/axioms14060453 - 9 Jun 2025

Viewed by 304

Abstract

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free [...] Read more.

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free Gamma laws in the setting of Cauchy-Stieltjes Kernel (CSK) families. Our investigation reveals the essential links between analytic function theory and free probability, highlighting the usefulness of CSK families in developing the theoretical and computational aspects of free additive convolution. Full article

(This article belongs to the Section Mathematical Analysis)

10 pages, 445 KiB

Open AccessArticle

About Some Unsolved Problems in the Stability Theory of Stochastic Differential and Difference Equations

by Leonid Shaikhet

Axioms 2025, 14(6), 452; https://doi.org/10.3390/axioms14060452 - 9 Jun 2025

Viewed by 461

Abstract

This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In [...] Read more.

This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In contrast with the known stability condition, in which it is assumed that stochastic perturbations fade on the infinity quickly enough, a new situation is studied, in which stochastic perturbations can either fade on the infinity slowly or not fade at all. Some unsolved problem in this connection is brought to readers’ attention. Additionally, some unsolved problems of stabilization for one stochastic delay differential equation and one stochastic difference equation are also proposed. Full article

(This article belongs to the Section Mathematical Analysis)

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14 pages, 286 KiB

Open AccessArticle

A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions

by Aleksandar Petojević, Hari M. Srivastava and Sonja Orlić

Axioms 2025, 14(6), 451; https://doi.org/10.3390/axioms14060451 - 7 Jun 2025

Viewed by 419

Abstract

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers

τ_{s} (n)

(s \in N \cup {- 1})

, where, as usual,

N

is the set of positive [...] Read more.

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers

τ_{s} (n)

(s \in N \cup {- 1})

, where, as usual,

N

is the set of positive integers. The authors consider the associated partition functions and derive connections of the Eisenstein series with the numbers

τ_{s} (n)

. Several related corollaries and consequences of each of the presented results are also given. The paper concludes by presenting an open problem that is related to one of these results. Full article

(This article belongs to the Section Mathematical Analysis)

16 pages, 793 KiB

Open AccessArticle

Note on Iterations of Nonlinear Rational Functions

by Michal Fečkan, Amira Khelifa, Yacine Halim and Ibraheem M. Alsulami

Axioms 2025, 14(6), 450; https://doi.org/10.3390/axioms14060450 - 7 Jun 2025

Viewed by 259

Abstract

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By [...] Read more.

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system’s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area. Full article

(This article belongs to the Section Mathematical Analysis)

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35 pages, 382 KiB

Open AccessArticle

Generalized Pauli Fibonacci Polynomial Quaternions

by Bahadır Yılmaz, Nazmiye Gönül Bilgin and Yüksel Soykan

Axioms 2025, 14(6), 449; https://doi.org/10.3390/axioms14060449 - 6 Jun 2025

Viewed by 311

Abstract

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, [...] Read more.

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature. Full article

(This article belongs to the Special Issue Advances in Applied Algebra and Related Topics)

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Axioms, Volume 14, Issue 6 (June 2025) – 86 articles

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