Axioms

13 pages, 287 KB

Open AccessArticle

Bi-Univalent Function Classes Defined by Imaginary Error Function and Bernoulli Polynomials

by Ibtisam Aldawish, Sondekola Rudra Swamy, Basem Aref Frasin and Supriya Chandrashekharaiah

Axioms 2025, 14(10), 731; https://doi.org/10.3390/axioms14100731 (registering DOI) - 27 Sep 2025

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in

U =

[...] Read more.

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in

U =

{ς \in C : | ς | < 1}

, characterized by Bernoulli polynomials associated with imaginary error functions. For functions belonging to these subclasses, we establish bounds for their initial coefficients. For these classes, we also tackle the Fekete–Szegö problem. Several new results are also obtained as special cases by specifying certain parameter values in the general findings. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 4th Edition)

15 pages, 390 KB

Open AccessArticle

On Wijsman f_ρ-Statistical Convergence of Order α of Modulus Functions

by Gülcan Atıcı Turan and Mikail Et

Axioms 2025, 14(10), 730; https://doi.org/10.3390/axioms14100730 - 26 Sep 2025

Abstract

In the present paper, we introduce and investigate the concepts of Wijsman f_ρ-statistical convergence of order α and Wijsman strong f_ρ-convergence of order α. These notions are defined as natural generalizations of classical statistical convergence and Wijsman convergence, [...] Read more.

In the present paper, we introduce and investigate the concepts of Wijsman f_ρ-statistical convergence of order α and Wijsman strong f_ρ-convergence of order α. These notions are defined as natural generalizations of classical statistical convergence and Wijsman convergence, incorporating the tools of modulus functions and natural density through the function f. We provide a detailed analysis of their structural properties, including inclusion relations, basic characterizations, and illustrative examples. Furthermore, we establish the inclusion relations between Wijsman f_ρ-statistical convergence and Wijsman strong f_ρ-convergence of order α, showing conditions under which one implies the other. These notions are different in general, while coinciding under certain restrictions on the function f, the parameter α, and the sequence ρ. The results obtained not only extend some well-known findings in the literature but also open up new directions for further study in the theory of statistical convergence and its applications to analysis and approximation theory. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

42 pages, 509 KB

Open AccessArticle

Differential Galois Theory and Hopf Algebras for Lie Pseudogroups

by Jean-Francois Pommaret

Axioms 2025, 14(10), 729; https://doi.org/10.3390/axioms14100729 - 26 Sep 2025

Abstract

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups [...] Read more.

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that

d i m_{K} (L) = ∣ L / K ∣ < \infty

, the transcendence degree

t r d (L / K) < \infty

and

t r d (L / K) = \infty

with

d i f f t r d (L / K) < \infty

, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension

L = K (G)

; then, there is a Galois correspondence between the intermediate fields

K \subset K^{'} \subset L

and the subgroups

e \subset G^{'} \subset G

, provided that

K^{'}

is stable under a Lie algebra

Δ

of invariant derivations of

L / K

. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article

(This article belongs to the Special Issue Advances in Hopf Algebras, Tensor Categories and Related Topics)

24 pages, 1300 KB

Open AccessArticle

On the Construction and Analysis of a Fractional-Order Dirac Delta Distribution with Application

by Muhammad Muddassar, Adil Jhangeer, Nasir Siddiqui, Malik Sajjad Mehmood, Liaqat Khan and Tahira Jabeen

Axioms 2025, 14(10), 728; https://doi.org/10.3390/axioms14100728 - 26 Sep 2025

Abstract

We introduce the generalized fractional-order Dirac delta distribution

δ_{GFODDF}

, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. [...] Read more.

We introduce the generalized fractional-order Dirac delta distribution

δ_{GFODDF}

, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that

δ_{GFODDF}

reduces to the classical delta when

η = 1

, while providing additional flexibility for

0 < η < 1

. These results show that

δ_{GFODDF}

is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering. Full article

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

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22 pages, 2953 KB

Open AccessArticle

Integrating Proper Orthogonal Decomposition with the Crank–Nicolson Finite Element Method for Efficient Solutions of the Schrödinger Equation

by Xiaohui Chang, Hong Li, Jiahua Wang and Xuehui Ren

Axioms 2025, 14(10), 727; https://doi.org/10.3390/axioms14100727 - 25 Sep 2025

Abstract

This study proposes a dimensionality reduction method based on proper orthogonal decomposition (POD) for numerically solving the Schrödinger equation by optimizing the coefficient vectors of the Crank–Nicolson finite element (CNFE) solution. The fully discrete CNFE scheme is derived from the Schrödinger equation, with [...] Read more.

This study proposes a dimensionality reduction method based on proper orthogonal decomposition (POD) for numerically solving the Schrödinger equation by optimizing the coefficient vectors of the Crank–Nicolson finite element (CNFE) solution. The fully discrete CNFE scheme is derived from the Schrödinger equation, with the stability and convergence of the CNFE solution rigorously established. Utilizing the POD technology, POD basis functions are constructed and a reduced-dimension model is formulated. The uniqueness and unconditional stability are proved, and the error estimate of the reduced-dimension solution is derived. With a

100 \times 100

spatial grid, the reduced-dimension CNFE (RDCNFE) method employing POD technology reduces the degrees of freedom each time step from

101^{2}

to 6. Numerical results show that for each additional second of simulation time, the CPU runtime of the standard CNFE method increases by several seconds, while the RDCNFE method increase remains below one second. This demonstrates that the reduced-dimension method significantly enhances computational efficiency for the Schrödinger equation while preserving numerical accuracy. Full article

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20 pages, 769 KB

Open AccessArticle

Homotopy Analysis Method and Physics-Informed Neural Networks for Solving Volterra Integral Equations with Discontinuous Kernels

by Samad Noeiaghdam, Md Asadujjaman Miah and Sanda Micula

Axioms 2025, 14(10), 726; https://doi.org/10.3390/axioms14100726 - 25 Sep 2025

Abstract

This paper addresses first- and second-kind Volterra integral equations (VIEs) with discontinuous kernels. A hybrid method combining the Homotopy Analysis Method (HAM) and Physics-Informed Neural Networks (PINNs) is developed. The convergence of the HAM is analyzed. Benchmark examples confirm that the proposed HAM-PINNs [...] Read more.

This paper addresses first- and second-kind Volterra integral equations (VIEs) with discontinuous kernels. A hybrid method combining the Homotopy Analysis Method (HAM) and Physics-Informed Neural Networks (PINNs) is developed. The convergence of the HAM is analyzed. Benchmark examples confirm that the proposed HAM-PINNs approach achieves high accuracy and robustness, demonstrating its effectiveness for complex kernel structures. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

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26 pages, 2019 KB

Open AccessArticle

Statistical Convergence for Grünwald–Letnikov Fractional Differences: Stability, Approximation, and Diagnostics in Fuzzy Normed Spaces

by Hasan Öğünmez and Muhammed Recai Türkmen

Axioms 2025, 14(10), 725; https://doi.org/10.3390/axioms14100725 - 25 Sep 2025

Abstract

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove [...] Read more.

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set

{1, x, x^{2}}

propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Difference and Differential Equations)

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21 pages, 321 KB

Open AccessArticle

Trigonometric Sums via Lagrange Interpolation

by Marta Na Chen, Wenchang Chu and Xiaoyuan Wang

Axioms 2025, 14(10), 724; https://doi.org/10.3390/axioms14100724 - 25 Sep 2025

Abstract

By means of the Lagrange interpolation, we derive two trigonometric identities that are utilized to evaluate, in closed forms, eight classes of power sums of trigonometric functions over equally distributed angles around the unit circle. When the mth term is removed from [...] Read more.

By means of the Lagrange interpolation, we derive two trigonometric identities that are utilized to evaluate, in closed forms, eight classes of power sums of trigonometric functions over equally distributed angles around the unit circle. When the mth term is removed from the sums, four classes (among eight) are shown to admit also closed expressions, that are surprisingly independent of the integer parameter “m”. Full article

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

15 pages, 287 KB

Open AccessArticle

Weighted Integral Operators from the

H_{v}^{\infty}

Space to the

B_{μ}

Space on Cartan–Hartogs Domains

by Yinghao Huo and Jianbing Su

Axioms 2025, 14(10), 723; https://doi.org/10.3390/axioms14100723 - 24 Sep 2025

Viewed by 33

Abstract

In this paper, necessary and sufficient conditions are established for the boundedness and compactness of a weighted integral operator when acting between the weighted Bloch space

B_{μ}

and the weighted Hardy space

H_{v}^{\infty}

on the Cartan–Hartogs domain. [...] Read more.

In this paper, necessary and sufficient conditions are established for the boundedness and compactness of a weighted integral operator when acting between the weighted Bloch space

B_{μ}

and the weighted Hardy space

H_{v}^{\infty}

on the Cartan–Hartogs domain. Full article

28 pages, 463 KB

Open AccessArticle

A Novel p-Norm-Based Ranking Algorithm for Multiple-Attribute Decision Making Using Interval-Valued Intuitionistic Fuzzy Sets and Its Applications

by Sandeep Kumar, Saiful R. Mondal and Reshu Tyagi

Axioms 2025, 14(10), 722; https://doi.org/10.3390/axioms14100722 - 24 Sep 2025

Viewed by 7

Abstract

The main focus of this paper is to introduce an algorithm that enhances the outcomes of multiple-attribute decision making by harnessing the adaptability of interval-valued intuitionistic fuzzy (

I V I F

) sets (

I V I F S

s). This algorithm [...] Read more.

The main focus of this paper is to introduce an algorithm that enhances the outcomes of multiple-attribute decision making by harnessing the adaptability of interval-valued intuitionistic fuzzy (

I V I F

) sets (

I V I F S

s). This algorithm utilizes

I V I F

numbers (

I V I F N

s) to represent attribute values and attribute weights, enabling the decision maker to account for the intricate nuances and uncertainties that are inherent in the decision-making process. We introduce a novel generalized score function (

G S F

) designed to overcome the limitations of previous functions. This function incorporates two parameters, denoted as

γ_{1} and γ_{2} (γ_{1} + γ_{2} = 1)

with

γ_{1} \in (0, 0.5)

. The core concept of this algorithm centers around the computation of the p-distance for each alternative relative to the positive ideal alternative. The p-distance is derived from the p-norm associated with each alternative’s score matrix, providing the decision maker (

D M

) with a tool to rank the available alternatives. Various examples are given to demonstrate the practicality and effectiveness of the proposed algorithm. Additionally, we apply the algorithm to a real event-based multiple-attribute decision-making (

M A D M

) problem—the investment company problem—to identify the optimal alternatives through a comparative analysis. Full article

(This article belongs to the Special Issue Recent Advances in Fuzzy Theory Applications)

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17 pages, 524 KB

Open AccessArticle

Three-Way Approximations with Covering-Based Rough Set

by Mei Li and Renxia Wan

Axioms 2025, 14(10), 721; https://doi.org/10.3390/axioms14100721 - 24 Sep 2025

Viewed by 57

Abstract

In order to approximate an undefinable set of objects by using the extensions in OE-concept lattices, this study combines three-way concept analysis with covering-based rough set and introduces an innovative approach for managing uncertain information and decision-making. This approach employs the minimal neighborhood [...] Read more.

In order to approximate an undefinable set of objects by using the extensions in OE-concept lattices, this study combines three-way concept analysis with covering-based rough set and introduces an innovative approach for managing uncertain information and decision-making. This approach employs the minimal neighborhood of the maximal description, which is determined by meet-irreducible elements, to define the lower and upper of an undefinable set. On this basis, we formalize the concepts of lower and upper approximation OE-concepts and propose a three-way approximation optimization algorithm. Experimental results demonstrate the effectiveness and efficiency of our algorithm. Full article

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17 pages, 283 KB

Open AccessArticle

Notes on a New Class of Univalent Starlike Functions with Respect to a Boundary Point

by Kamaraj Dhurai, Amjad Saleh Alghamdi and Srikandan Sivasubramanian

Axioms 2025, 14(10), 720; https://doi.org/10.3390/axioms14100720 - 23 Sep 2025

Viewed by 92

Abstract

This article presents a newly defined subclass of univalent functions that are starlike with respect to a boundary point, closely related to the Robertson class and specifically associated with a vertical strip domain. Additionally, this study derives generalized coefficient estimates for these classes, [...] Read more.

This article presents a newly defined subclass of univalent functions that are starlike with respect to a boundary point, closely related to the Robertson class and specifically associated with a vertical strip domain. Additionally, this study derives generalized coefficient estimates for these classes, as well as for the Robertson class linked to the Nephroid domain and the Lemniscate of Bernoulli. Full article

(This article belongs to the Section Geometry and Topology)

25 pages, 570 KB

Open AccessArticle

Distribution-Free EWMA Scheme for Joint Monitoring of Location and Scale Based on Post-Sales Online Review Process

by Sirui An and Jiujun Zhang

Axioms 2025, 14(10), 719; https://doi.org/10.3390/axioms14100719 - 23 Sep 2025

Viewed by 80

Abstract

Nowadays, the online comment process of product after-sales has become a key part of product development. Quality problems, such as the failure of products or services, are more likely to exist or hide in negative comments. Therefore, this paper focuses on detecting abnormal [...] Read more.

Nowadays, the online comment process of product after-sales has become a key part of product development. Quality problems, such as the failure of products or services, are more likely to exist or hide in negative comments. Therefore, this paper focuses on detecting abnormal changes in both the time between review T and the emotional score S of negative comments. Due to the complexity of the online review process, the distribution assumption of S and T may be invalid. To solve this problem, this study propose a distribution-free monitoring scheme that combines the exponentially weighted moving average-based Lepage statistics of S and T using a max-type combining function. This scheme is designed for joint monitoring of location and scale parameters in Phase II of an unknown but continuous process. The scheme’s performance is evaluated via Monte Carlo simulation under in-control and out-of-control conditions, using statistical measures such as the mean, standard deviation, median, and selected percentiles of the run length distribution. Simulation results indicate that the scheme is effective in detecting shifts in both location and scale parameters. Furthermore, an application of the proposed scheme for monitoring online reviews is discussed to illustrate its implementation design. Full article

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17 pages, 3219 KB

Open AccessArticle

Stability of the Split-Step θ-Method for Stochastic Pantograph Systems with Markovian Switching and Jumps

by Guangjie Li, Zhipei Hu, Baishu Xu, Zilong Chen and Feiqi Deng

Axioms 2025, 14(10), 718; https://doi.org/10.3390/axioms14100718 - 23 Sep 2025

Viewed by 176

Abstract

This study focuses on analyzing the almost sure exponential stability of the split-step

θ

-method (

S S θ

-method) when applied to stochastic pantograph differential equations characterized by Markovian switching and jump processes. Initially, we establish the almost sure exponential stability of [...] Read more.

This study focuses on analyzing the almost sure exponential stability of the split-step

θ

-method (

S S θ

-method) when applied to stochastic pantograph differential equations characterized by Markovian switching and jump processes. Initially, we establish the almost sure exponential stability of the system’s trivial solution. Subsequently, under an additional sufficient condition, it is demonstrated that the discrete solutions generated by the

S S θ

-method also exhibit this stability property. Finally, a computational experiment is conducted to support the theoretical results. Full article

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16 pages, 311 KB

Open AccessArticle

Reformulation of Fixed Point Existence: From Banach to Kannan and Chatterjea Contractions

by Zouaoui Bekri, Nicola Fabiano, Mohammed Ahmed Alomair and Abdulaziz Khalid Alsharidi

Axioms 2025, 14(10), 717; https://doi.org/10.3390/axioms14100717 - 23 Sep 2025

Viewed by 109

Abstract

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via [...] Read more.

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via Green’s function and establish a sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if the derivative of j is bounded by K and

K \cdot {(η - ζ)}^{2} / 8 < 1 / 3

, then T is a Kannan contraction, ensuring a unique solution. This condition applies even when the Banach contraction principle fails. We also explore the plausibility of applying the Chatterjea contraction, though rigorous verification remains open. Examples illustrate the applicability of the results. This work highlights the utility of generalized contractions in differential equations. Full article

(This article belongs to the Special Issue Research in Fixed Point Theory and Its Applications)

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Axioms, Volume 14, Issue 10 (October 2025) – 15 articles

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