Axioms

16 pages, 713 KB

Open AccessFeature PaperArticle

Null Geodesics and Shadow Structure in Einstein–Weyl Gravity

by Joseph Sultana

Axioms 2026, 15(8), 610; https://doi.org/10.3390/axioms15080610 - 14 Aug 2026

We investigate null geodesics, photon spheres and black hole shadows for the static spherically symmetric non-Schwarzschild black hole solution of Einstein–Weyl gravity, a higher-derivative extension of General Relativity containing a quadratic Weyl-curvature term. Such higher-curvature theories are motivated by attempts to formulate a [...] Read more.

We investigate null geodesics, photon spheres and black hole shadows for the static spherically symmetric non-Schwarzschild black hole solution of Einstein–Weyl gravity, a higher-derivative extension of General Relativity containing a quadratic Weyl-curvature term. Such higher-curvature theories are motivated by attempts to formulate a quantum theory of gravity, where they improve the ultraviolet behaviour of the gravitational interaction, and also arise naturally as effective descriptions in approaches such as string theory. We employ the numerical black hole solution obtained by Lü et al. to compute the photon sphere, the shadow radius and the angular size of the shadow as observed by static observers. We show that, for black holes of equal mass, the photon sphere, shadow radius and angular size are consistently larger than those of the corresponding Schwarzschild black hole, with the deviations increasing monotonically with the higher-curvature coupling parameter

α

. Motivated by the Event Horizon Telescope observations of M87* and Sagittarius A*, we further compare the predicted shadow size with current observational uncertainties and derive phenomenological upper bounds on the dimensionless coupling

α / m^{2}

. These results demonstrate that black hole shadow observations provide a promising avenue for testing Einstein–Weyl gravity and constraining quantum-motivated higher-curvature corrections to General Relativity. Full article

(This article belongs to the Special Issue Mathematical Aspects of Black Holes in General Relativity and Beyond)

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32 pages, 3055 KB

Open AccessArticle

Dynamical Analysis of a Delayed Fractional-Order Food Chain Model with the Allee Effect and Prey Refuge

by Linjie Sun, Ruiqing Shi and Yunfeng Liu

Axioms 2026, 15(8), 609; https://doi.org/10.3390/axioms15080609 - 12 Aug 2026

Abstract

In this paper, a delayed fractional-order three-trophic-level food chain model with Allee effect and prey shelter is proposed and analyzed. The model adopts the Caputo fractional derivative to describe the memory effect, and introduces two discrete time delays which represent the gestation and [...] Read more.

In this paper, a delayed fractional-order three-trophic-level food chain model with Allee effect and prey shelter is proposed and analyzed. The model adopts the Caputo fractional derivative to describe the memory effect, and introduces two discrete time delays which represent the gestation and response delays of predators, respectively. We establish the positivity, boundedness, existence, uniqueness and continuity of solutions. By using Jacobian matrix analysis and fractional stability theory, we investigate the equilibrium points and their local stability. The corrected characteristic equation is derived, and sufficient conditions for a Hopf-type stability switch induced by the delays are obtained. In the integer-order case

α = 1

, the critical delay and transversality condition are verified numerically, supporting a classical Hopf-bifurcation conclusion subject to the usual nondegeneracy assumptions. Numerical simulations verify the theoretical results and illustrate the combined effects of time delays, fractional memory, prey shelter and Allee effect on system dynamics. The results show that time delays may destabilize the system and produce persistent oscillatory numerical behaviour, while stronger memory effects, shelter strength and Allee effects enhance system stability and suppress oscillatory behaviors. Full article

(This article belongs to the Special Issue Applied Mathematics and Mathematical Modeling, 2nd Edition)

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24 pages, 383 KB

Open AccessArticle

On Minimax Robust Estimation Problem for Stochastic Sequences with Harmonizable Symmetric α-Stable Increments

by Maksym Luz and Mikhail Moklyachuk

Axioms 2026, 15(8), 608; https://doi.org/10.3390/axioms15080608 - 12 Aug 2026

Abstract

We consider the problem of the optimal linear estimation of the functional

A_{N} ξ = \sum_{k = 0}^{N} a (k) ξ (k)

which depends on the unknown values

ξ (k)

, [...] Read more.

We consider the problem of the optimal linear estimation of the functional

A_{N} ξ = \sum_{k = 0}^{N} a (k) ξ (k)

which depends on the unknown values

ξ (k)

,

k = 0, 1, \dots, N

, of a stochastic sequence with harmonizable symmetric

α

-stable nth increments,

1 < α \leq 2

. The derived estimates are based on observations at points

m \in Z ∖ {0, 1, 2, \dots, N}

. Cases of observations without noise, with harmonizable symmetric

α

-stable noise and with noise with harmonizable symmetric

α

-stable increments are studied. Classical solutions as well as minimax robust ones are obtained. Full article

(This article belongs to the Special Issue Modern Trends and Application of Decision-Making Theory, Stability and Control, 2nd Edition)

30 pages, 364 KB

Open AccessArticle

Qualitative Axioms Within Probability Calculus and Their Role in Comparing Previsions of Random Quantities

by Pierpaolo Angelini

Axioms 2026, 15(8), 607; https://doi.org/10.3390/axioms15080607 - 11 Aug 2026

Viewed by 133

Abstract

In this paper, the notion of probability is not undefined, so we extend to multilinear indices qualitative axioms which are not in conflict with those characterizing the development of modern probability theory: this is the main objective achieved by the current paper. The [...] Read more.

In this paper, the notion of probability is not undefined, so we extend to multilinear indices qualitative axioms which are not in conflict with those characterizing the development of modern probability theory: this is the main objective achieved by the current paper. The logical foundation of probability calculus is here extended by studying the mathematical expectation of random variables having two or more marginal variables as their components. Random variables, studied along with their probability distributions of a nonparametric nature, are geometric entities of which fundamental invariance properties are made explicit. The research gap addressed by this paper is the following: since the Cartesian product of two or more sets, where each of them is the image of a marginal random variable, is not commutative, noncommutative geometric objects coinciding with tensors come into play to make previsions of entities treating high-dimensional data. Empirical data given by time series of a finite length are handled. Time series of a finite length are formally seen as frequency distributions. They are also random variables. Hence, frequency distributions and random variables are shown to be the two sides of the same coin. Full article

(This article belongs to the Special Issue Research on Applied Statistics and Stochastic Processes)

29 pages, 2050 KB

Open AccessArticle

Propagation-Constrained Stochastic Modeling and Robust Recursive Estimation of Air-to-Underwater ELF Signals Under Depth-Evolving Alpha-Stable Disturbances

by Yongxin Cui and Zheng Dou

Axioms 2026, 15(8), 606; https://doi.org/10.3390/axioms15080606 - 11 Aug 2026

Viewed by 45

Abstract

Air-to-underwater extremely low frequency (ELF) signal recovery requires a receiver that accounts for both cross-medium attenuation and impulsive interference. Seawater weakens the desired electromagnetic waveform while altering the tail behavior that remains observable within the receiver bandwidth. Conventional recursive robust filters normally choose [...] Read more.

Air-to-underwater extremely low frequency (ELF) signal recovery requires a receiver that accounts for both cross-medium attenuation and impulsive interference. Seawater weakens the desired electromagnetic waveform while altering the tail behavior that remains observable within the receiver bandwidth. Conventional recursive robust filters normally choose their linearity parameters and output scale from empirical or data-driven rules and therefore do not explicitly incorporate this depth-dependent physical–statistical coupling. This work formulates the propagation-guided scaled recursive weighted myriad (PG-SRWMy) filter as a mathematically structured framework that links a propagation-evolving stochastic process model with robust nonlinear recursive estimation for reference-normalized underwater ELF recovery. The propagation state determines a depth-evolving effective stable-like description through a characteristic exponent and a dispersion parameter, and these stochastic descriptors are transformed into branch-specific initial linearity parameters of the recursive myriad estimator. The same propagation model yields a positive and bounded normalization transform for the reconstruction scale, while first-order sensitivity relations characterize the local effect of environmental-state uncertainty. The original SRWMy sample-wise recursion is retained for data-driven adaptation. Numerical experiments show that PG-SRWMy accelerates bilinear-parameter stabilization, lowers waveform-reconstruction error, and improves end-to-end reference-normalized signal-to-noise ratio (SNR) gain across changes in receiver depth and surface-side impulsiveness. The results support propagation-aware initialization as a mathematically structured and statistically interpretable route to depth-adaptive underwater ELF reception. Full article

(This article belongs to the Special Issue Research on Applied Statistics and Stochastic Processes)

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14 pages, 780 KB

Open AccessArticle

Sharp Estimates for q-Convex Functions and the Associated Classical Family

by Kuppusami Sakthivel, Hari Mohan Srivastava and Srikandan Sivasubramanian

Axioms 2026, 15(8), 605; https://doi.org/10.3390/axioms15080605 - 11 Aug 2026

Viewed by 52

Abstract

In this article, we introduce and study two new subclasses of analytic univalent functions defined via the Ma–Minda function. Specifically, we consider the class

C_{ξ_{q}}

of q-convex functions involving a suitable Ma–Minda function

ξ_{q} (z),

together [...] Read more.

In this article, we introduce and study two new subclasses of analytic univalent functions defined via the Ma–Minda function. Specifically, we consider the class

C_{ξ_{q}}

of q-convex functions involving a suitable Ma–Minda function

ξ_{q} (z),

together with its classical counterpart

C_{ξ}

corresponding to

ξ (z),

where

0 < q < 1 .

We determine bounds for the first few Taylor–Maclaurin coefficients and deduce Fekete–Szegö and Kruskal inequality. Moreover, we obtain the associated Toeplitz determinants related to this class. We highlight several new consequences of our results that are of independent interest in geometric function theory. Full article

(This article belongs to the Section Mathematical Analysis)

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15 pages, 308 KB

Open AccessArticle

Fixed-Point Properties of the Bellman Operator in Discounted Stochastic Maintenance Optimization

by Jelena Vujaković, Nataša Kontrec and Biljana Panić

Axioms 2026, 15(8), 604; https://doi.org/10.3390/axioms15080604 - 11 Aug 2026

Viewed by 51

Abstract

This paper investigates an infinite-horizon discounted stochastic maintenance optimization problem within the framework of dynamic programming. The system degradation is modeled by a discrete-time stochastic process affected by maintenance actions and random disturbances, while the objective is to minimize the expected discounted maintenance [...] Read more.

This paper investigates an infinite-horizon discounted stochastic maintenance optimization problem within the framework of dynamic programming. The system degradation is modeled by a discrete-time stochastic process affected by maintenance actions and random disturbances, while the objective is to minimize the expected discounted maintenance and degradation costs. The analysis is carried out on the Banach space of continuous functions equipped with the supremum norm. It is proved that the associated Bellman operator is well defined, maps the function space into itself, and is a contraction with contraction modulus equal to the discount factor. Consequently, the existence and uniqueness of the optimal value function follow from the Banach Fixed Point Theorem, and the convergence of value iteration is established. In addition, rigorous a priori and a posteriori error estimates are derived, providing theoretical stopping criteria for numerical computation. The theoretical results are complemented by numerical experiments illustrating the optimal stationary maintenance policy, the stability of the computed solution under grid refinement, and the influence of the discount factor on both the optimal policy and the convergence rate of value iteration. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques, 2nd Edition)

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26 pages, 655 KB

Open AccessFeature PaperArticle

SIR Model with Dependent Infectivity and Death Rates

by Emma Breidenich, Joe Cooper, Qianzhao Huang, Camille Wagner, Sándor Kovács and Meir Shillor

Axioms 2026, 15(8), 603; https://doi.org/10.3390/axioms15080603 - 10 Aug 2026

Viewed by 95

Abstract

This work constructs, analyzes and simulates a new general SIR epidemiological model for the spread of a generic long-time disease, in which the coefficients of infectivity and death rate are system variables. Diseases, such as COVID-19, have demonstrated clearly that infectivity and death [...] Read more.

This work constructs, analyzes and simulates a new general SIR epidemiological model for the spread of a generic long-time disease, in which the coefficients of infectivity and death rate are system variables. Diseases, such as COVID-19, have demonstrated clearly that infectivity and death rates can change over time, even for the same variant of the virus, due to vaccination, improved treatments, better analysis, better medications, etc. This motivates us to construct the SIR-ID model for a generic disease in which the rate coefficients are state variables as a part of the systems’s evolution in time. The model consists of a coupled system of five differential equations, where the equations for the infectivity and death rate have general source functions. The analysis shows the existence, positivity and boundedness of the solutions. A discussion of the Endemic (EE) and Disease-Free (DFE) equilibria and their stability is provided. A bifurcation analysis of the DFE and EE is conducted, as well as a sensitivity analysis. Then, computer simulations depict two typical cases of dynamic behavior, one when the DFE is stable and attracting, and one in which the EE is stable and attracting. These also show the way the system approaches the steady states. Full article

(This article belongs to the Special Issue Advances in Mathematical Models and Applications)

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19 pages, 436 KB

Open AccessArticle

A Sobolev–Information Perspective on Derivative-Observation-Augmented PINNs for Parameter Identification of Second-Order Dynamical Systems

by Liwen Xu and Yixuan Lin

Axioms 2026, 15(8), 602; https://doi.org/10.3390/axioms15080602 - 9 Aug 2026

Viewed by 133

Abstract

Identifying parameters of dynamical systems from sparse measurements is a core task in structural health monitoring and vibration engineering. For second-order oscillators, standard physics-informed neural networks (PINNs) struggle because different parameter values can produce nearly identical displacement records, making the inverse problem ill-posed. [...] Read more.

Identifying parameters of dynamical systems from sparse measurements is a core task in structural health monitoring and vibration engineering. For second-order oscillators, standard physics-informed neural networks (PINNs) struggle because different parameter values can produce nearly identical displacement records, making the inverse problem ill-posed. We propose the derivative-observation-augmented PINN (D-PINN), which incorporates velocity measurements into the training loss to resolve this degeneracy. Three theoretical results support the method: a Sobolev-type inequality proves that constraining the velocity error automatically bounds the displacement error; a Fisher information analysis shows that velocity observations increase the information available for parameter estimation; and a residual-based estimate bounds the parameter error in terms of the solution accuracy and its derivatives. Experiments on linear, forced near-resonance, and Duffing oscillators (10 random seeds, 20,000 epochs) show that D-PINN reduces the damping coefficient relative error from 40% to 11.7% without any parameter prior. With a weak prior (

μ_{0} = 3.2

, a 20% deviation from the true value 4.0), the error drops further to 2.1%, a 19-fold improvement over standard PINN. We also analyze sensitivity to prior quality, derivative observation source, and measurement noise, and identify scenarios where derivative observations do not improve displacement fitting. Full article

(This article belongs to the Special Issue Application of Machine Learning and Optimization Methods in Engineering Mathematics, 2nd Edition)

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27 pages, 441 KB

Open AccessReview

Deep Learning for Solving Integral Equations: A Problem-Oriented Review with an Axiomatic Perspective

by Zhiyuan Ren, Ruilong Yu, Yi Zeng and Shijie Zhou

Axioms 2026, 15(8), 601; https://doi.org/10.3390/axioms15080601 - 9 Aug 2026

Viewed by 158

Abstract

This review surveys recent deep learning approaches for solving integral equations, categorizing them into three methodological families: physics-informed embedding, spectral/topological acceleration, and hybrid symbolic–numeric frameworks. The main findings are threefold. First, these methods achieve promising empirical accuracy in oscillatory, high-dimensional, and singular-kernel settings, [...] Read more.

This review surveys recent deep learning approaches for solving integral equations, categorizing them into three methodological families: physics-informed embedding, spectral/topological acceleration, and hybrid symbolic–numeric frameworks. The main findings are threefold. First, these methods achieve promising empirical accuracy in oscillatory, high-dimensional, and singular-kernel settings, yet their theoretical foundations remain largely incomplete. Second, from an axiomatic perspective, most approaches lack rigorous guarantees of convergence, stability, and spectral consistency; we formulate five testable propositions that a complete theory should satisfy. Third, we identify five specific unresolved theoretical questions and outline a focused research agenda toward a mathematically rigorous theory of neural operator approximation for integral equations. The novelty of this review lies in its dual computational–axiomatic evaluation and its provision of a structured, problem-oriented framework for future investigations. Full article

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29 pages, 847 KB

Open AccessArticle

Another Simple Proof of the Close Connection of SQS(10) and GQ(2,2)

by Stefano Innamorati

Axioms 2026, 15(8), 600; https://doi.org/10.3390/axioms15080600 - 9 Aug 2026

Viewed by 82

Abstract

Symmetry plays a key role in identifying the close connection between different finite incidence structures. In this paper, by studying the properties of points not belonging to an elliptic quadric of PG(3,3), a short demonstration is given of the close connection between the [...] Read more.

Symmetry plays a key role in identifying the close connection between different finite incidence structures. In this paper, by studying the properties of points not belonging to an elliptic quadric of PG(3,3), a short demonstration is given of the close connection between the Steiner Quadruple system SQS(10) and the Cremona–Richmond configuration. Full article

(This article belongs to the Special Issue Graph Invariants and Their Applications)

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22 pages, 863 KB

Open AccessFeature PaperArticle

GL (2, R)

in the Finite One-Dimensional Ising Model with Nonuniform Couplings

by Nicholay S. Tonchev and Daniel Dantchev

Axioms 2026, 15(8), 599; https://doi.org/10.3390/axioms15080599 - 8 Aug 2026

Viewed by 92

Abstract

Using the properties of

GL (2, R)

, the general linear group of invertible

2 \times 2

real matrices, we investigate random fields of spin variables on finite one-dimensional rings with a unit cell of

p \in N

sites. The [...] Read more.

Using the properties of

GL (2, R)

, the general linear group of invertible

2 \times 2

real matrices, we investigate random fields of spin variables on finite one-dimensional rings with a unit cell of

p \in N

sites. The interaction parameters are assumed to be periodic with period p. The cases

p = 1

and

p = 2

recover the one-dimensional and alternating Ising models, respectively. The couplings between adjacent spins may be ferromagnetic (positive) or antiferromagnetic (negative). Utilising the recurrence relations of the Chebyshev polynomials and a bijection between the number of spins and the polynomial index, we derive explicit formulae well suited to the finite-size analysis of the partition functions, free energy, and specific heat of both models. We show that, for the (

p = 2

) case, the specific heat exhibits a double-Schottky anomaly whenever the characteristic exchange energy scales are sufficiently separated. We prove that this double-peak structure originates from the coexistence of distinct energy scales induced by the periodic modulation of the coupling signs and characterise its dependence on the model parameters. We demonstrate that the universality hypothesis in critical Casimir force theory remarkably holds without requiring small fields or large interaction parameters, suggesting a form of “hyper-universal” behaviour valid for arbitrary model parameters. Full article

(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)

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26 pages, 1844 KB

Open AccessArticle

Trajectory Tracking Control of an Off-Axis Tractor-Trailer Wheeled Mobile System with Passive Steering

by Xiangrong Wen, Danhong Chen and Yusheng Zhou

Axioms 2026, 15(8), 598; https://doi.org/10.3390/axioms15080598 - 8 Aug 2026

Viewed by 94

Abstract

This paper investigates the system modeling and trajectory tracking control problem of an off-axis tractor-trailer wheeled mobile system and proposes a control strategy based on an integral sliding surface and a super-twisting algorithm. First, the motion relationship between the tractor and trailer is [...] Read more.

This paper investigates the system modeling and trajectory tracking control problem of an off-axis tractor-trailer wheeled mobile system and proposes a control strategy based on an integral sliding surface and a super-twisting algorithm. First, the motion relationship between the tractor and trailer is derived based on their geometric configuration, and the kinematic and dynamic models of the off-axis tractor-trailer wheeled mobile system with a passive steering angle are established. Then, a composite controller is designed by integrating the integral sliding surface with the super-twisting algorithm to achieve accurate tracking of the desired trajectory. During the controller design process, the desired trajectory is reconstructed as a curvature-consistent dynamic tracking target. Under this dynamic tracking target, the curvature deviation depends only on the yaw-rate error and is decoupled from longitudinal velocity disturbances, thereby improving trajectory tracking accuracy. Finally, numerical simulations are conducted to validate the effectiveness and generality of the proposed control strategy under three typical reference trajectories, including cycloidal, S-shaped, and circular trajectories. Comparisons with PID and feedback linearization control methods are also performed. The simulation results demonstrate that the off-axis tractor-trailer wheeled mobile system can closely follow the desired trajectories and exhibits satisfactory robustness. Full article

(This article belongs to the Topic Nonlinear Phenomena, Chaos, Control and Applications to Engineering and Science and Experimental Aspects of Complex Systems, 2nd Edition)

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47 pages, 4587 KB

Open AccessArticle

Classical and Bayesian Parameter Estimation for Generalised Exponential Competing Risks Models Under Improved Adaptive Type-II Progressive Censoring

by Hana N. Alqifari

Axioms 2026, 15(8), 597; https://doi.org/10.3390/axioms15080597 - 7 Aug 2026

Viewed by 124

Abstract

Competing-risks models play an important role in reliability and survival analysis because failures often arise from several latent causes acting simultaneously. In this paper, we study a two-cause independent competing-risks model in which the latent lifetimes follow the generalised exponential distribution under the [...] Read more.

Competing-risks models play an important role in reliability and survival analysis because failures often arise from several latent causes acting simultaneously. In this paper, we study a two-cause independent competing-risks model in which the latent lifetimes follow the generalised exponential distribution under the improved adaptive Type-II progressive censoring scheme. The proposed framework aims to estimate the model parameters together with the reliability and hazard-rate functions using both classical and Bayesian inference. The frequentist analysis develops maximum likelihood estimators and approximate confidence intervals based on asymptotic theory, whereas the Bayesian analysis employs independent Gamma priors, squared-error loss, and a Metropolis–Hastings algorithm to obtain posterior estimates and highest posterior density credible intervals. An extensive Monte Carlo simulation study is conducted to investigate their finite-sample performance under different censoring schemes, threshold settings, and sample sizes. Finally, two real competing-risks datasets are analysed to illustrate the practical applicability of the proposed methodology and to demonstrate that the generalised exponential competing-risks model provides a competitive alternative for reliability and survival data analysis. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimations, 3rd Edition)

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37 pages, 423 KB

Open AccessArticle

A Measure-Theoretic Formulation of Hybrid Systems Beyond Zeno Time

by Robert Vrabel

Axioms 2026, 15(8), 596; https://doi.org/10.3390/axioms15080596 - 7 Aug 2026

Viewed by 104

Abstract

Hybrid dynamical systems may exhibit Zeno behavior, where infinitely many discrete transitions occur in finite time, leading to a loss of well-posedness of trajectories beyond the accumulation point. This paper develops a measure-theoretic formulation of hybrid dynamics by representing discrete transitions through finite [...] Read more.

Hybrid dynamical systems may exhibit Zeno behavior, where infinitely many discrete transitions occur in finite time, leading to a loss of well-posedness of trajectories beyond the accumulation point. This paper develops a measure-theoretic formulation of hybrid dynamics by representing discrete transitions through finite vector-valued Radon measures and recasting the system as a measure differential inclusion. Within this framework, we establish a closure result for extended hybrid solutions in the space of functions of bounded variation and derive an existence result under suitable approximation assumptions. We also prove consistency with classical hybrid trajectories in the absence of Zeno behavior and characterize the state at the Zeno time as the left limit of the hybrid evolution, together with any additional vector atom deliberately assigned at the accumulation time. The proposed formulation provides a natural basis for continuation beyond the accumulation point and allows Lyapunov-based stability properties to be formulated directly at the level of the measure-driven dynamics. A central feature of the approach is that infinitely many discrete transitions with finite total variation of the jump increments are encoded by a finite vector-valued atomic measure whose atoms may accumulate at the Zeno time. An additional vector atom at the Zeno time may be introduced as a lumped effective jump; however, such an atom should be understood as a modeling choice, not as an automatic consequence of the jump sequence. The results are illustrated on an event-triggered control system exhibiting Zeno accumulation. Full article

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

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6 pages, 201 KB

Open AccessArticle

Legendre Polynomials and an Inequality for a Combinatorial Sum

by Horst Alzer and Hans W. Volkmer

Axioms 2026, 15(8), 595; https://doi.org/10.3390/axioms15080595 - 7 Aug 2026

Viewed by 153

Abstract

Let

P_{n}

be the Legendre polynomial of degree n. We use an estimate for the ultraspherical polynomials and a gamma function inequality to prove that [...] Read more.

Let

P_{n}

be the Legendre polynomial of degree n. We use an estimate for the ultraspherical polynomials and a gamma function inequality to prove that

| \int_{- 1}^{x} P_{n} (t) d t | < \sqrt{\frac{2}{π}} \frac{1}{n^{3 / 2}} (n \geq 1; - 1 \leq x \leq 1)

and we apply this result to obtain the combinatorial inequality

|\sum_{k = 0}^{n} (\binom{n}{k}) (\binom{(n + k - 1) / 2}{n}) \frac{x^{k + 1}}{k + 1} - \frac{1}{n + 1} (\binom{n / 2}{n})| < \sqrt{\frac{2}{π}} \frac{1}{2^{n} n^{3 / 2}} (n \geq 1; - 1 \leq x \leq 1) .

The factor

\sqrt{2 / π}

given in both inequalities is the best possible. Full article

(This article belongs to the Section Mathematical Analysis)

15 pages, 230 KB

Open AccessArticle

The Dynamic String-Averaging Method for Inverse Strongly-Monotone Operators with Summable Errors

by Alexander J. Zaslavski

Axioms 2026, 15(8), 594; https://doi.org/10.3390/axioms15080594 - 7 Aug 2026

Viewed by 130

Abstract

In the work by W. Takahashi and M. Toyoda (2003) it was introduced and studied an iterative process for solving a variational inequality problem which is induced by a inverse strongly-monotone mapping. They showed the weak convergence of the iteration process. Recently we [...] Read more.

In the work by W. Takahashi and M. Toyoda (2003) it was introduced and studied an iterative process for solving a variational inequality problem which is induced by a inverse strongly-monotone mapping. They showed the weak convergence of the iteration process. Recently we established that most of exact iterates of the same iterative process are approximate solutions of the variational inequality. In the present work we use the dynamic string-averaging algorithm for finding a common solution of a finite family of variational inequality problems, generated by inverse strongly-monotone mappings, and a finite family of fixed point problems. We study this algorithm in the presence of summable computational errors. It is shown that the cardinality of the set of iterates which are not approximate solutions is finite and does not exceed a certain constant which is calculated. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

23 pages, 1417 KB

Open AccessArticle

A Truncated-Kernel Mollification Method for the Cauchy Problem of the Modified Helmholtz Equation

by Huilin Xu, Fanli Xu and Baoxia Wang

Axioms 2026, 15(8), 593; https://doi.org/10.3390/axioms15080593 - 5 Aug 2026

Viewed by 115

Abstract

This paper addresses the Cauchy problem for the multi-dimensional modified Helmholtz equation, a classical and severely ill-posed problem. A truncated-kernel mollification method is proposed as an effective regularization approach. Both the a priori and a posteriori regularization parameter choice strategies are examined, and [...] Read more.

This paper addresses the Cauchy problem for the multi-dimensional modified Helmholtz equation, a classical and severely ill-posed problem. A truncated-kernel mollification method is proposed as an effective regularization approach. Both the a priori and a posteriori regularization parameter choice strategies are examined, and the associated error estimates and convergence rates of the regularized solutions are established. The practical viability and effectiveness of the method are further validated through numerical experiments. Full article

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

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34 pages, 1258 KB

Open AccessArticle

A Proportional-Arithmetic Framework for Fourier Analysis on the Positive Real Line

by Carlos M. Cruz-Rodas, Marlon M. López-Flores and William Campillay-Llanos

Axioms 2026, 15(8), 592; https://doi.org/10.3390/axioms15080592 - 5 Aug 2026

Viewed by 260

Abstract

This paper develops a Fourier framework internal to proportional arithmetic on the positive real line. We construct the corresponding complex scalar field, differential and integral operators, oscillatory kernel, Fourier transform, and proportional function spaces. A correspondence theorem proves that the representative of the [...] Read more.

This paper develops a Fourier framework internal to proportional arithmetic on the positive real line. We construct the corresponding complex scalar field, differential and integral operators, oscillatory kernel, Fourier transform, and proportional function spaces. A correspondence theorem proves that the representative of the proportional transform is the classical Fourier transform under the logarithmic identification. Consequently, inversion, Plancherel, convolution, Schwartz invariance, and Sobolev characterizations follow by transport. We establish the exact relation with Fourier analysis on the multiplicative group and with the Mellin transform on the imaginary axis. Model resolvent and heat equations illustrate the operational calculus, while a scale-localized profile shows how spectral modulus and phase encode log-scale width and preferred scale. The construction is therefore a systematic proportional-arithmetic realization of classical harmonic analysis, rather than an analytically independent Fourier theory. Full article

(This article belongs to the Section Mathematical Analysis)

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25 pages, 1683 KB

Open AccessArticle

Analytical Study of Impulsive Hilfer-Type Fractional p-Laplacian Problems Using Neural Networks and Finite-Difference Methods

by Rahman Ullah Khan, Ioannis K. Argyros, Taha Radwan and Yousif Altayeb

Axioms 2026, 15(8), 591; https://doi.org/10.3390/axioms15080591 - 5 Aug 2026

Viewed by 225

Abstract

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter

ϑ \in [0, 1]

is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting [...] Read more.

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter

ϑ \in [0, 1]

is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting cases of the formulation, not as separate cases. The variational functional is then built by adding the point-impulse contribution to the distributed potential and the use of an appropriate space of the Hilfer fractional derivative. Using variants of the fountain theorem, we prove the existence of two infinite sequences of weak solutions, one of which is of unbounded energy and another of which is of small energy and tends to zero from below. The weak residual based stability analysis is further developed, and local generalized Hyers–Ulam and Hyers–Ulam–Rassias stability estimates are obtained. Because of multiplicity of solutions, a uniqueness-based argument for stability, Ulam’s approach, is not possible and stability is instead achieved by providing residual-based arguments.The assumptions are verified through illustrative examples. Lastly, we examine the convergence behavior, residual decay, and effect of the Hilfer type parameter in conjunction with a Hilfer-type parameter neural surrogate with boundary constraints based on a discrete Hilfer scheme. The study, in general, proves a link between the solution multiplicity, residual stability, and the numerical realization in one impulsive fractional p-Laplacian framework. Full article

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

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35 pages, 579 KB

Open AccessArticle

A Unified Hybrid Estimation Strategy Using Multiple Auxiliary Transformations in Systematic Sampling with Simulation and Real-Life Applications

by Fatimah A. Almulhim, Hassan M. Aljohani and Umer Daraz

Axioms 2026, 15(8), 590; https://doi.org/10.3390/axioms15080590 - 5 Aug 2026

Viewed by 140

Abstract

Estimating the finite population mean under systematic sampling becomes challenging when auxiliary information is nonlinear, skewed, or structurally complex, as conventional linear estimators often lose efficiency. This study proposes a new class of weighted hybrid estimators that combine harmonic and geometric transformations of [...] Read more.

Estimating the finite population mean under systematic sampling becomes challenging when auxiliary information is nonlinear, skewed, or structurally complex, as conventional linear estimators often lose efficiency. This study proposes a new class of weighted hybrid estimators that combine harmonic and geometric transformations of the auxiliary variable. The proposed approach is designed to capture nonlinear relationships while handling skewed data and reducing sensitivity to extreme observations. Expressions for bias and mean squared error are derived, and optimal weights are obtained by minimizing the mean squared error. The theoretical results indicate that the proposed estimators are more efficient than traditional ratio, product, regression, and exponential-type estimators. A simulation study further confirms their improved performance across various population structures, correlation levels, and sampling fractions, with notable improvements in skewed and nonlinear settings. The proposed class provides a flexible and reliable alternative for practical applications in systematic sampling. Full article

(This article belongs to the Section Mathematical Analysis)

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3 pages, 151 KB

Open AccessEditorial

Theory and Application of Integral Inequalities, 2nd Edition

by Loredana Ciurdariu

Axioms 2026, 15(8), 589; https://doi.org/10.3390/axioms15080589 - 5 Aug 2026

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Abstract

In this Editorial, we present “Theory and Application of Integral Inequalities, 2nd Edition” a Special Issue of the Journal Axioms [...] Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities, 2nd Edition)

23 pages, 1525 KB

Open AccessArticle

Acceptance Sampling Plans for Exponential- and Weibull-Distributed Lifetimes Under the Group Sampling Framework

by Ching-Ho Yen, Kuen-Suan Chen, Mou-Yuan Liao, Chun-Min Yu and Ting Zhou

Axioms 2026, 15(8), 588; https://doi.org/10.3390/axioms15080588 - 4 Aug 2026

Viewed by 370

Abstract

Product lifetime is a critical quality characteristic of electronic products. Among various lifetime models, the Weibull distribution is one of the most flexible and widely used distributions in reliability analysis because it can describe different failure rate patterns through its shape parameter. In [...] Read more.

Product lifetime is a critical quality characteristic of electronic products. Among various lifetime models, the Weibull distribution is one of the most flexible and widely used distributions in reliability analysis because it can describe different failure rate patterns through its shape parameter. In this study, the Weibull distribution is treated as the primary lifetime model, while the exponential distribution is included as a simpler benchmark model and as a special case of the Weibull distribution with shape parameter m = 1. Based on the lifetime performance index, this research applies the group sampling concept to design two lifetime acceptance sampling plans under these lifetime distributions. The optimal sampling plan parameters of the lifetime acceptance sampling plans are determined by minimizing the number of groups while satisfying the two-point principle of the operating characteristic curve. For practical purposes, the parameters of the proposed plan are tabulated for some combinations of quality levels with commonly used producer risk and consumer risk. Moreover, a comparative analysis of the two lifetime testing methods is presented, and the results show that Testing Method II proposed in this study can implement sampling inspection more efficiently. Finally, an example is used to illustrate the proposed methodology. Full article

(This article belongs to the Special Issue Current Trends in the Mathematics of Fuzzy Sets and Logic)

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36 pages, 10718 KB

Open AccessArticle

Bayesian Inference via Markov Iterative Methods for Generalized Progressive Hybrid Unit Bilal Censoring and Its Applications to Thermodynamics and Meteorology

by Heba S. Mohammed, Ahmed Elshahhat, Osama E. Abo-Kasem and Asmaa Abdel-Hakim

Axioms 2026, 15(8), 587; https://doi.org/10.3390/axioms15080587 - 4 Aug 2026

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Abstract

The increasing availability of bounded lifetime observations in different disciplines has intensified the demand for flexible models capable of accommodating complex failure mechanisms. Motivated by this need, a comprehensive inferential framework is developed for the unit Bilal (UBilal) distribution using generalized progressive hybrid [...] Read more.

The increasing availability of bounded lifetime observations in different disciplines has intensified the demand for flexible models capable of accommodating complex failure mechanisms. Motivated by this need, a comprehensive inferential framework is developed for the unit Bilal (UBilal) distribution using generalized progressive hybrid censoring, which guarantees a minimum number of observed failures while controlling experimental duration. Classical inference is established through maximum likelihood estimation, which is accompanied by asymptotic confidence intervals based on both normal and log-transformed approximations. Moreover, a Bayesian framework using a Metropolis–Hastings Markov chain Monte Carlo algorithm is presented. The proposed methodology further provides inference for important reliability characteristics, including the reliability and hazard rate functions, through both frequentist and Bayesian paradigms proposed. An extensive Monte Carlo investigation is conducted under diverse censoring schemes, sample sizes, and prior specifications to evaluate estimation accuracy, interval performance, and the influence of censoring severity. The simulation results show that Bayesian methods always provide better estimates and more reliable interval estimates, especially when prior information is used. Using two real datasets from thermodynamics and meteorology, the numerical results demonstrate that the UBilal model provides an excellent fit and yields reliable inference under bounded observations. Overall, the proposed methodology presents an efficient Bayesian inferential framework for bounded lifetime data collected through the generalized progressive hybrid censoring and expands the applicability of the UBilal model to reliability and related fields. Full article

(This article belongs to the Special Issue Applications of Bayesian Methods in Statistical Analysis, Second Edition)

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55 pages, 65526 KB

Open AccessReview

Some Approaches to Solving the KP Equation: Different Representations and Various Types of Solutions

by Pierre Gaillard

Axioms 2026, 15(8), 586; https://doi.org/10.3390/axioms15080586 - 4 Aug 2026

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Abstract

We present different methods to construct solutions to the Kadomtsev–Petviashvili (KP) equation. In the first method, from the solutions to the NLS equation, we construct solutions to the KP equation in terms of Fredholm determinants. We deduce solutions written as quotients of Wronskians [...] Read more.

We present different methods to construct solutions to the Kadomtsev–Petviashvili (KP) equation. In the first method, from the solutions to the NLS equation, we construct solutions to the KP equation in terms of Fredholm determinants. We deduce solutions written as quotients of Wronskians of order

2 N

. When one of these parameters tends to zero, we obtain N-order rational solutions expressed as a quotient of two polynomials of degree

2 N (N + 1)

in x, y and t, depending on

2 N - 2

real parameters. We obtain, in this case, regular solutions to the KP equation. Using new results from the NLS equation, with solutions constructed in terms of quotients of determinants of order N depending on

2 N - 2

real parameters, we are able to highlight new forms of configurations, such as triangles and concentric rings. Another approach using the Darboux transformation is given to get multi-parametric solutions to the KP equation. In this approach, it is possible to construct an infinite hierarchy of solutions depending on the degree of summation and the degree of derivation. The third method involves choosing special polynomials and using the bilinear Hirota method to get other types of solutions to the KP equation. We also obtain an infinite hierarchy of solutions depending on the order of the determinants. The last method allows the construction of regular solutions to the KP equation. In this approach, we obtain another alternative to obtain regular solutions, as in the case of the first method, and we also observe the formation of configurations such as triangles or concentric rings. We study the configurations of these hierarchies of solutions to the KP equation as a function of their different parameters. Full article

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

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39 pages, 2878 KB

Open AccessArticle

Geometric Frequency Mixing in Helical Waveguides via a One-Dimensional Covariant Helmholtz Model: Gauge Reduction and Spectral Splitting

by Gülden Altay Suroğlu, Şeyma Firdevs Hızal and Hasan Bulut

Axioms 2026, 15(8), 585; https://doi.org/10.3390/axioms15080585 - 4 Aug 2026

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Abstract

This study develops a one-dimensional covariant Helmholtz model for a vector-valued wave field transported along a circular helical centerline and represented in the Frenet–Serret frame. For a helix with constant curvature

κ > 0

and torsion

τ \neq 0

, the geometric coupling [...] Read more.

This study develops a one-dimensional covariant Helmholtz model for a vector-valued wave field transported along a circular helical centerline and represented in the Frenet–Serret frame. For a helix with constant curvature

κ > 0

and torsion

τ \neq 0

, the geometric coupling is described by a constant skew-symmetric connection matrix

Ω \in so (3)

. The covariant Helmholtz operator is shown to admit an exact gauge reduction to the flat componentwise Helmholtz operator through

u (s) = e^{- Ω s} y (s)

. Thus, within the one-dimensional centerline formulation, the helix preserves the operator spectrum while redistributing the observed Frenet components through parallel transport. The closed-form solutions show that a monochromatic input with wavenumber k is decomposed into a carrier and two geometric sidebands governed by the Darboux rotation rate

λ = \sqrt{κ^{2} + τ^{2}}

. In the sub-geometric regime

k < λ

, the lower algebraic sideband is represented by the positive observable wavenumber

q_{-} = | k - λ |

, with associated scale

T_{beat} = L_{-} = 2 π / q_{-}

. The lossless energy analysis proves conservation of the total averaged energy and its redistribution among the carrier and observable sidebands. A representative helical acoustic-channel design is then examined as a conceptual realization of the centerline model. Monte Carlo perturbations and additive-noise tests show that the predicted sideband locations, lower-sideband scale, and energy partition remain stable under prescribed fabrication tolerances and spectrally identifiable under weak and moderate measurement noise. Full article

(This article belongs to the Section Mathematical Physics)

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25 pages, 402 KB

Open AccessArticle

Saddlepoint Inference for Nonlinear Statistics from Inverse Gaussian Models: Applications to Clinical, Engineering, and Environmental Data

by Abd El-Raheem M. Abd El-Raheem and Mona Hosny

Axioms 2026, 15(8), 584; https://doi.org/10.3390/axioms15080584 - 3 Aug 2026

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Abstract

This paper applies established saddlepoint approximation techniques to nonlinear statistics arising from inverse Gaussian models. In particular, we consider the product of independent inverse Gaussian random variables and the ratio of weighted linear combinations of inverse Gaussian random variables, for which exact distributions [...] Read more.

This paper applies established saddlepoint approximation techniques to nonlinear statistics arising from inverse Gaussian models. In particular, we consider the product of independent inverse Gaussian random variables and the ratio of weighted linear combinations of inverse Gaussian random variables, for which exact distributions are generally unavailable in closed form. For the product statistic, a logarithmic transformation converts the problem into one involving the cumulant generating function of a sum of log-transformed variables. This cumulant generating function is expressed in terms of fractional moments including modified Bessel functions of the second kind. For the ratio statistic, the event including the ratio is reformulated in terms of a linear statistic, which enables the use of saddlepoint density and Lugannani-Rice distribution approximations. The proposed formulation accommodates heterogeneous model parameters, overlapping numerator and denominator components, and flexible coefficient structures subject to positivity of the denominator. Simulation studies show that the proposed approximations provide accurate results across a range of sample sizes, skewness regimes, and parameter configurations. Furthermore, simulation results indicate that the saddlepoint approximation is more accurate than the normal approximation. Sensitivity analysis confirms that the proposed approximations are reasonably stable under moderate inverse Gaussian parameter misspecification. Three real data applications including clinical illness scores, engineering repair times, and environmental runoff measurements illustrate the practical usefulness of the approach. Overall, the results indicate that the saddlepoint approximation provides an accurate and computationally efficient tool for inference on nonlinear statistics from inverse Gaussian models when exact distributions are not available. Full article

(This article belongs to the Special Issue Recent Developments in Statistical Research)

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20 pages, 372 KB

Open AccessArticle

Construction of Multi-Rate QC-LDPC Codes Based on Permutation Method

by Hengzhou Xu, Jinru Wang, Mei Zhang, Mengmeng Xu and Qian Wang

Axioms 2026, 15(8), 583; https://doi.org/10.3390/axioms15080583 - 3 Aug 2026

Viewed by 218

Abstract

This paper proposes a systematic permutation-based construction method for multi-rate quasi-cyclic low-density parity-check (QC-LDPC) codes. We first present a graph-theoretic framework in which any regular QC-LDPC code can be normalized to a canonical base matrix that is uniquely determined by a permutation

π

[...] Read more.

This paper proposes a systematic permutation-based construction method for multi-rate quasi-cyclic low-density parity-check (QC-LDPC) codes. We first present a graph-theoretic framework in which any regular QC-LDPC code can be normalized to a canonical base matrix that is uniquely determined by a permutation

π

. This normalization reduces the complex code design to a single combinatorial optimization problem over the symmetric group. Based on this normalization, we analyze the cycle structure of the lifted Tanner graph and derive necessary and sufficient conditions for 4-cycle elimination in terms of the permutation difference function. We develop two complementary algorithms: a simulated annealing algorithm that searches for permutations that minimize a weighted sum of 4-cycles and 6-cycles, and a progressive column-ordering algorithm that ensures every prefix subgraph maintains high girth. This approach yields a nested family of rate-compatible codes. Simulation results show that the constructed codes outperform the 5G-LDPC codes. The nested base matrix structure facilitates seamless rate switching, which makes the proposed code family well suited for adaptive transmission systems in future wireless networks. Full article

(This article belongs to the Special Issue Combinatorics and Graph Theory with Applications in Computer Science)

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21 pages, 306 KB

Open AccessArticle

Taylor Recurrences and Coulomb-Corrected Asymptotics for the Schrödinger–Newton Ground State

by Mirko Tarulli, George Venkov and Petia Zorovska

Axioms 2026, 15(8), 582; https://doi.org/10.3390/axioms15080582 - 3 Aug 2026

Viewed by 192

Abstract

We study the positive, radial ground-state profile of the stationary Schrödinger–Newton system in the fixed-energy normalization

μ = 1

with

V (\infty) = 0

. Using regularity and radial symmetry of solutions, we derive convergent even Taylor expansions for the wave [...] Read more.

We study the positive, radial ground-state profile of the stationary Schrödinger–Newton system in the fixed-energy normalization

μ = 1

with

V (\infty) = 0

. Using regularity and radial symmetry of solutions, we derive convergent even Taylor expansions for the wave function and Newtonian potential near the origin, with explicit recurrence relations expressing all the coefficients in terms of the initial data

(a_{0}, b_{0}) = (y (0), V (0))

with

a_{0} > 0

and

b_{0} > 1

. Global existence and uniqueness of the positive radial ground state are taken from the known Schrödinger–Newton/Choquard theory, while the present work focuses on the local coefficient structure and the far-field expansion. In the far field, the Poisson equation yields the Coulomb tail

V (r) = \hat{M} / r + O (e^{- 2 r} r^{\hat{M} - 2})

with no algebraic corrections at any order, where

\hat{M} = \int_{0}^{\infty} r^{2} y^{2} d r

is the determined reduced mass. The decaying wave profile admits the Coulomb-corrected asymptotic expansion

y (r) = C e^{- r} r^{\hat{M} / 2 - 1} \sum_{m \geq 0} c_{m} r^{- m}

, obtained by reducing the radial equation to a Whittaker equation with an exponentially small perturbation controlled by asymptotic integration. The inverse-power series is divergent and interpreted in the Poincaré sense. The mass, energy and virial identities serve as compatibility conditions for the globally selected profile. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Numerical Modeling)

21 pages, 926 KB

Open AccessArticle

Admissible Convexity for Time-Varying Fractional Lyapunov Inequalities

by Jiale Chen, Osama F. Abdel Aal, Jairo Viola and Weigang Sun

Axioms 2026, 15(8), 581; https://doi.org/10.3390/axioms15080581 - 3 Aug 2026

Viewed by 235

Abstract

Fractional Lyapunov inequalities provide an effective tool for stability analysis of fractional-order systems, since classical chain and product rules are not directly applicable to fractional operators. This paper investigates admissible convexity conditions for time-varying Lyapunov functions in continuous Caputo and discrete nabla fractional [...] Read more.

Fractional Lyapunov inequalities provide an effective tool for stability analysis of fractional-order systems, since classical chain and product rules are not directly applicable to fractional operators. This paper investigates admissible convexity conditions for time-varying Lyapunov functions in continuous Caputo and discrete nabla fractional settings. It is shown that state convexity alone is insufficient: a historical ordering condition is needed to ensure the proper sign of the memory terms. The continuous inequality is characterized by a weighted integral of historical residuals, whereas the discrete nabla counterpart is characterized by a weighted sum over historical grid points. Reverse inequalities for concave functions are also derived under reversed ordering conditions. Product-form inequalities with nonnegative convex state factors and nonnegative nonincreasing time factors are recovered as natural admissible cases. Numerical examples and Lyapunov applications are presented to verify the results. Full article

(This article belongs to the Special Issue Advances in Discrete-Fractional Mathematics and Its Application)

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