Axioms doi: 10.3390/axioms15060421

Authors: Yinghui Liu Bo Zhang Jingwen Duan

In this paper, we characterize the conditions for a complex symmetric Toeplitz composition operator \(T_{\psi}C_{\phi}\) on the Bergman space \(A^2(\mathbb{D})\), where the symbol function \(\psi\) is harmonic or \(\psi(z)=A\overline{z}^nz^m+B\overline{z}^sz^t\) for \(A, B\in\mathbb{C}\backslash\{0\}\) and \(n-m=t-s\). We also present necessary and sufficient conditions for the commutativity of \(T_{\psi}C_{\phi}\) and \(C_{\xi,\theta}\), where \(C_{\xi,\theta}f(z)=e^{i\xi}\overline{f(e^{i\theta}\overline{z})}\) for \(\xi, \theta \in \mathbb{R}\).

Axioms doi: 10.3390/axioms15060422

Authors: Tassaddaq Hussain Enrique Villamor Sneh Gulati Mohammad Shakil Mohammad Ahsanullah Bhuiyan Mohammad Golam Kibria

In this study, we introduce a novel three-parameter discrete probability distribution, defined by the parameters α, β, and λ, called the Generalized Nabla Geometric Distribution (GNGD). The proposed model is constructed through the discretization of a generalized mixture of exponential distributions. Among its parameters, β plays a significant role in governing the tail behavior and determining the shape of the hazard rate function. The adaptability of the GNGD is further highlighted by its capability to model under-dispersed, equi-dispersed, and over-dispersed data. Several important theoretical properties of the proposed distribution are explored, including the probability generating function, moments, and reliability characteristics, all obtained in explicit closed forms. In addition, the distribution is characterized, and the asymptotic behavior of its minimum and maximum order statistics is investigated. Parameter estimation is performed using the maximum likelihood estimation (MLE) approach, and an extensive simulation study is conducted to evaluate the efficiency and consistency of the estimators. To illustrate its practical usefulness, the GNGD is fitted to two real engineering data sets arising from repairable and non-repairable systems. The findings demonstrate that the proposed distribution provides an excellent fit to both data sets, achieving the smallest information loss criteria and the largest p-values among the competing goodness-of-fit measures considered.

Axioms doi: 10.3390/axioms15060420

Authors: Ghaliah Alhamzi Georgia Irina Oros Mdi Begum Jeelani Kalika Prasad Shahid Ahmad Wani

In the present work, we introduce and study a new two-parameter generalization of Legendre-based Appell polynomials, defined through an explicit representation that unifies classical Legendre structures with the Appell polynomial framework. Starting from a generating function, we derive a three-term recurrence relation, a degree-lowering operator, an integro-partial degree-raising operator, and a corresponding integro-partial differential equation satisfied by the new family. A determinant representation is established via Cramer’s rule applied to the Cauchy-product expansion of the generating function. Several subfamilies of independent interest arise naturally as special cases, namely, Legendre-based Hermite–Frobenius–Euler polynomials, Legendre-based Miller–Lee polynomials, and both the probabilist’s and physicist’s variants of Legendre-based bi-variate Hermite polynomials. For each subfamily we record the corresponding recurrence relations, shift operators, differential equations, and determinant forms, and we illustrate the behavior of selected members through three-dimensional surface plots and real-root distribution diagrams. The framework presented here extends several constructions available in the recent literature and points to natural directions for future work, including connections with q-series, combinatorial identities, and symbolic-computation methods, which are outlined in the concluding section.

Axioms doi: 10.3390/axioms15060419

Authors: Hanan S. Gafel Luluah G. Albugami

This study analyzes and explores the influence of multiple physical mechanisms—namely the influences of heat and mass transfer, thermal radiation, and magnetic field effects on the peristaltic transport of a Rabinowitsch-type non-Newtonian fluid within an inclined channel. To accurately represent the intricate behavior of the fluid under these coupled physical phenomena, a nonlinear model was formulated that integrates thermal, magnetic, and radiative forces into its framework. The given coupled differential equations are transformed into ordinary differential equations (ODEs). Using assumptions of long-wavelength and low-Reynolds-number approximations, the governing equations were significantly simplified. The resulting set of equations was solved analytically using Mathematica, subject to appropriate boundary conditions for velocity, temperature, and concentration. Graphs for velocity, temperature and concentration are illustrated. Thermal radiation was incorporated into the energy equation via the Rosseland approximation, thereby enabling a more accurate characterization of heat transport within the system. Moreover, the rate of heat and mass transfer for different variables was also examined. These findings are essential for the progression of advanced fluid transport systems in biomedical engineering, chemical processing, and energy generation, improving the design and management of non-Newtonian fluid dynamics.

Axioms doi: 10.3390/axioms15060418

Authors: Mostafa Hassanlou Ebrahim Abbasi Maryam G. Alshehri

In this paper, we investigate the generalized integral-type operator acting between the fractional Cauchy transform space and two classical analytic function spaces: the weighted Bloch space and the weighted Dirichlet space. For the operator acting into the weighted Bloch space, we obtain two equivalent exact formulas for its operator norm. Furthermore, an estimate for its essential norm is provided, which leads to a necessary and sufficient condition for compactness. For the operator acting into the weighted Dirichlet space, we derive the exact operator norm and fully characterize its compactness.

Axioms doi: 10.3390/axioms15060417

Authors: Bryan Sanctuary

The double-slit experiment is commonly interpreted as evidence that a single electron must be described by a spatially extended wavefunction whose path amplitudes interfere. Here, we present an alternative geometric formulation within the Bivector Standard Model, in which the observed far-field interference pattern is reproduced while the phase is attributed to an internal bivector clock carried by the electron. In this approach, each electron remains localized and produces a single detection event, while the familiar fringe pattern emerges statistically from the accumulation of many impacts. Interference arises from the comparison of clock phases associated with geometrically distinct paths, rather than from the superposition of spatial waves. The resulting probability distribution recovers the standard two-slit interference factor, the single-slit diffraction envelope, and the usual fringe-spacing relation in the Fraunhofer regime. The de Broglie wavelength emerges as the spatial manifestation of this transported phase. This formulation provides a geometric account of the origin of phase in single-electron interference, consistent with standard results while offering a distinct physical interpretation.

Axioms doi: 10.3390/axioms15060416

Authors: Juan Luis González-Santander

We calculate several finite integrals involving trigonometric and hyperbolic functions by applying the Laplace convolution theorem and known inverse Laplace transform formulas. As a consequence, we obtain a new integral representation of the Kelvin function bei(z), and a new reduction formula for a particular generalized hypergeometric function. In addition, we present several new inverse Laplace transform formulas that do not appear to have been reported in the existing literature.

Axioms doi: 10.3390/axioms15060415

Authors: Danylo Vorvul Andrii Musienko Iryna Galchenko Mykola Myroniuk Andrii Sobchuk

Large language model (LLM)-driven computer use agents (CUAs) automate graphical user interface (GUI) tasks but often re-solve previously encountered subtasks, increasing token use and latency. We address this limitation with a directed graph-based persistent memory in which nodes represent observable GUI states and edges encode executable action sequences. We formalize the memory-augmented agent as S=⟨A,Σ,G,δ,π,Φ⟩, define task reachability and memory-coverage conditions inspired by functional stability theory, and derive token-cost efficiency bounds. In control-theoretic terms, the Manager–Worker architecture can be interpreted as a closed-loop system where memory provides experience-based feedback; this interpretation is used as an analogy rather than a full model-reference adaptive control proof. Experiments on OSWorld show that the proposed agent cuts both the LLM token consumption and execution time by about 50% versus a memoryless baseline while preserving comparable success rates (≈36.9% on 15-step and ≈46.9% on 50-step tasks). The demonstrated contribution is therefore operational efficiency through reusable graph memory, not a claim of improved task success or classical Lyapunov stability.

Axioms doi: 10.3390/axioms15060414

Authors: Olga Kostyukova Tatiana Tchemisova

A fundamental objective in the study of convex cones is the description and analysis of their extreme rays. In the case of the copositive cone, these rays are generated by extremal copositive matrices, which encode the boundary structure of the cone and are closely related to challenging instances of copositive and completely positive programming. In this work, we propose a constructive framework for generating copositive matrices from a given extremal copositive matrix of a smaller order and establish conditions under which copositivity and extremality are preserved. This approach highlights the interplay between the zero structure of a copositive matrix, its minimal zeros, and the facial geometry of the copositive cone. The results obtained allow one to generate new families of extremal copositive matrices in higher dimensions.

Axioms doi: 10.3390/axioms15060413

Authors: Giuseppe Filippone Mario Galici Gianmarco La Rosa Federica Piazza Marco Elio Tabacchi

This paper investigates the structure of fuzzy Lie subalgebras, with particular emphasis on isomorphisms and nilpotency. Building on two prior conference contributions, one of which established foundational results on fuzzy bases of Lie algebras, we develop here a more complete and unified treatment of these themes. We introduce a notion of isomorphism between fuzzy Lie subalgebras based on the transfer principle via t-cut sets, and we prove that isomorphic fuzzy Lie subalgebras necessarily share the same nilpotency measure. The central contribution of the paper is a fuzzy measure of nilpotency N(μ)∈[0,1], defined for any non-constant fuzzy Lie subalgebra μ of a Lie algebra g. This invariant equals 1 precisely when μ is fuzzy nilpotent, and decreases as the subalgebra departs from nilpotency. We show that nilpotency of the underlying Lie algebra implies N(μ)=1, but that the converse fails in general, as witnessed by an explicit counterexample.

Axioms doi: 10.3390/axioms15060412

Authors: Tong Li Junhan Wang Pan Shi

In this paper, we consider the one-dimensional liquid crystal flow with a vacuum free boundary and a degenerate viscosity coefficient. The global existence and long-time dynamics of strong solutions are established under a smallness condition on the initial energy at the basic level. The main challenges come from the degeneracy near the moving boundary and the strong nonlinear coupling between the velocity and the director field. To overcome these, we obtain uniform-in-time and space point-wise bounds of the deformation variable, and we construct uniform-in-time weighted energy estimates via singular multiplier techniques. Unlike previous works, the density is allowed to vanish and the viscosity coefficient is taken to be density-dependent rather than constant.

Axioms doi: 10.3390/axioms15060411

Authors: Wen-Xiu Ma

This paper aims to introduce a real four-component integrable extension of the complex Kaup–Newell soliton hierarchy. Following a general idea for extending the standard Kaup–Newell spectral matrix, we propose a specific matrix eigenvalue problem involving four real potentials and construct the corresponding integrable Hamiltonian hierarchy via the zero-curvature formulation. A recursion operator and a bi-Hamiltonian structure are presented to demonstrate the Liouville integrability of the resulting hierarchy. As an illustrative example, we derive an integrable system of four real derivative nonlinear Schrödinger equations, each containing two linear dispersion terms and generalizing the standard complex derivative nonlinear Schrödinger equations.

Axioms doi: 10.3390/axioms15060410

Authors: Alexander Prokopenya Mukhtar Minglibayev Balnur Assan

We consider the translational–rotational motion of a small axisymmetric body in the gravitational field of two stars of variable masses moving under the influence of their mutual gravitational attraction. The two stars are assumed to lose their masses isotropically with different rates and their total mass decreases according to the joint Meshcherskii law. The relative motion of the stars is described by the corresponding exact solution to Gyldén’s equation and is considered to be given. The small axisymmetric body may change its mass, size and shape while its initial dynamic structure is retained. The problem is analyzed in the framework of Lagrange’s formalism, and equations of translational–rotational motion are derived in terms of the osculating elements of aperiodic motion on quasi-conic sections and the Andoyer variables. As equations of motion of the small body are not integrable, the perturbation theory is applied with the perturbing function expanded into power series in terms of eccentricity and inclination, which are assumed to be small. Averaging these equations over the mean longitudes of the two bodies and two Andoyer angles in the absence of commensurability of frequencies, we obtain the differential equations describing the long-term evolution of the orbital elements and Andoyer variables which may be investigated numerically for different laws of the system parameters’ variation. All the relevant symbolic calculations are performed with the computer algebra system Wolfram Mathematica.

Axioms doi: 10.3390/axioms15060409

Authors: Yertay Kazez Zhanars A. Abdiramanov Nauryzbay Adil Abdumauvlen S. Berdyshev

We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals modelling viscoelastic memory damping. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (βNM=1/4) combined with an implicit–explicit linearisation in which the linear spatial operator is treated implicitly and the nonlinear terms are treated explicitly through a second-order extrapolation. This linearisation eliminates the need for Newton–Raphson iterations at each time step. To overcome the dense memory bottleneck arising from two distinct fractional orders α≠β, the convolution memory kernels are compressed by independent sum-of-exponentials approximations obtained from a double-exponential quadrature of the kernel’s integral representation, which significantly reduces the computational complexity of the history term. A rigorous stability estimate and a global convergence bound are established using a discrete Grönwall inequality. Numerical experiments confirm the theoretical temporal and spatial convergence rates and demonstrate the practical speed-up afforded by the sum-of-exponentials acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory.

Axioms doi: 10.3390/axioms15060408

Authors: Sarah Alotibi Mohamed Ben Ayed

In this paper, we study energy bounded solutions uε converging weakly to 0 of the subcritical problem −Δu+gu=hun+2n−2−ε,u>0inΩ,u=0on∂Ω, where Ω is a C2 bounded domain in Rn with n≥4, g is a C1 positive function on Ω¯, h is a C3 positive function on Ω¯, and ε is a small positive parameter. Assuming that the normal derivative of h is negative on the boundary, we prove that uε must blow up in the interior of the domain. Moreover, we determine the precise location of the blow-up points and the corresponding blow-up rates. Conversely, for sufficiently small ε, we construct blowing-up solutions that converge weakly to zero, which allows us to obtain a multiplicity result for the problem. In contrast, when the normal derivative of h is positive at a boundary point b, we show that it is possible to construct solutions converging to zero and blowing up precisely at b.

Axioms doi: 10.3390/axioms15060407

Authors: Xiangqing Zhao Jifeng Bao

This paper establishes the global existence and uniqueness of solutions to the initial-boundary value problem for a fifth-order KdV equation posed on a finite interval [0, d]. Overcoming the challenge of global solvability imposed by non-conservative boundary conditions, we introduce a nonlinear boundary feedback mechanism inspired by control theory to enforce energy dissipation. The proof hinges on deriving rigorous a priori estimates that capture both the Kato smoothing effect and boundary trace regularity, complemented by a tailored nonlinear estimate to handle the feedback term. Consequently, local solutions are extended to global ones. Furthermore, comprehensive numerical experiments validate the proposed approach and yield strong empirical evidence of exponential energy decay, a property crucial for control applications.

Axioms doi: 10.3390/axioms15060406

Authors: Amani S. Alghamdi Rana A. Alzahrani

Detecting structural changes or change points in statistical models is a fundamental component of data analysis, as it plays a crucial role in understanding the dynamic behavior of real-world data, particularly in financial markets and cryptocurrency returns. In this study, we developed a procedure based on the likelihood ratio test (LRT) to identify change points in the parameters of the exponentiated exponential logistic (EEL) distribution. Furthermore, the binary segmentation technique was employed to efficiently detect multiple change points and determine their locations. The proposed methodology was derived under the null and alternative hypotheses, and its statistical properties were examined in depth. To evaluate its performance, extensive Monte Carlo simulations were conducted to assess the empirical power of the test under various sample sizes and significance levels. Furthermore, the method was applied to real cryptocurrency return data to demonstrate its practical ability to detect change points and illustrate its effectiveness at identifying structural breaks. The empirical results indicate that the LRT-based procedure exhibits strong capability in detecting significant distributional changes in cryptocurrency data over time, confirming its effectiveness as an analytical tool in statistical and financial studies.

Axioms doi: 10.3390/axioms15060405

Authors: Gültekin Atalik

This study proposes a novel methodology for constructing intuitionistic X¯˜−R˜ and X¯˜−S˜ control charts by incorporating a ranking method developed explicitly for triangular intuitionistic fuzzy numbers. Unlike traditional fuzzy control charts, the proposed model eliminates the need for defuzzification in the computation process. Instead, all decisions are derived directly through a fuzzy ranking approach, thereby preserving the integrity of the original fuzzy information and preventing information loss. This characteristic constitutes a key contribution of the study. To validate the practical applicability and effectiveness of the proposed methodology, real-world data collected from an engineering facility were utilized. The results were thoroughly analyzed through a real-world manufacturing case study. The application demonstrated that all samples, except for an assignable cause in one specific sub-range, were statistically in control. Most importantly, the proposed approach monitored the process variations without stripping away the inherent subjective hesitation of the decision-makers, thereby demonstrating its potential capability in handling structural uncertainty compared to traditional crisp methods.

Axioms doi: 10.3390/axioms15060404

Authors: Minghua Shi Jianbing Su Kang Wang

This paper investigates weighted composition–differentiation operators acting between Bers-type spaces defined on a generalized Hua domain of the first kind. By establishing a key norm inequality for functions in these spaces, we obtain necessary and sufficient conditions for the boundedness and compactness of such operators.

Axioms doi: 10.3390/axioms15060403

Authors: Zhen Wang Yi Wu Kai Zhou

A new concept of weight-array-based stochastic domination in the Cesàro sense is developed. Under Gut’s condition, we establish the weak law of large numbers and Lr-convergence for weighted sums of multidimensional arrays of pairwise negatively quadrant dependent random variables. These results improve and extend corresponding known results in the literature.

Axioms doi: 10.3390/axioms15060402

Authors: Cheng Zeng Daohua Wang Zixuan Zhao

In this paper, we define the local Hausdorff dimension on fractal networks inspired by the idea of α-potential at a point x∈Rd. We prove some basic properties of the local Hausdorff dimension and then obtain the local Hausdorff dimension of some networks. We also discuss a more generalized dimension definition, namely the subset Hausdorff dimension.

Axioms doi: 10.3390/axioms15060401

Authors: Saddaf Noreen Muhammad Imran Muhey U. Din Zhang Wei Adil Murtaza

The article derives sufficient conditions under which the normalized Rabotnov function becomes q-close-to-convex relative to specific starlike functions on the open unit disk. To enhance the impact of our results, we include some consequences derived from the main theorems, along with graphical illustrations. The starlikeness of the Rabotnov function with respect to different aspects also falls within the scope of this study.

Axioms doi: 10.3390/axioms15060400

Authors: Yasser Almoteri Ahmed Ghezal

This paper investigates the closed-form solvability and dynamical behavior of a class of nonlinear triangular difference systems with overlapping indices, emphasizing the role of coefficient symmetry and asymmetry in determining the qualitative behavior of the system. A unified analytical framework is developed by transforming the original nonlinear system into equivalent linear or multiplicative difference equations, thereby enabling the derivation of explicit general solutions for various parameter configurations. The results show that the structure of the coefficients plays a fundamental role in determining stability, periodicity, and long-term dynamics. In particular, symmetric configurations tend to produce regular and more structured periodic behavior, whereas asymmetric configurations lead to more irregular oscillatory patterns and increased sensitivity to initial conditions. These theoretical findings are supported by numerical simulations and graphical illustrations, which demonstrate how variations in coefficient values and signs influence the evolution of the system. Finally, an application to discrete survival dynamics is presented, illustrating the capability of the proposed model to describe interacting survival processes under both symmetric and asymmetric parameter regimes, thereby highlighting its potential relevance in the study of applied discrete dynamical systems.

Axioms doi: 10.3390/axioms15060399

Authors: Şuara Onbaşıoğlu Altuhovs Banu Pazar Varol

In this paper, we propose a new class of intuitionistic fuzzy b-metric spaces in the sense of Romaguera and investigate their fixed-point properties. Within this framework, we define and analyze Hicks-type contraction mappings. In addition, the concept of K-stationary intuitionistic fuzzy b-metrics is introduced and examined through illustrative examples. Our findings generalize classical results in fuzzy b-metric spaces and extend fixed-point theorems to the intuitionistic fuzzy setting. This study enriches fixed-point theory in intuitionistic fuzzy environments and provides a basis for further theoretical investigations and applications.

Axioms doi: 10.3390/axioms15060398

Authors: Sharief Deshmukh Amira Ishan

We find restrictions on a trans-Sasakian structure F,u,γ,α,β on a 3-dimensional Riemannian manifold M3,g so that M3,g is homothetic to a Sasakian manifold. In that, first we show that if the vector u of the trans-Sasakian structure F,u,γ,α,β on a 3-dimensional Riemannian manifold M3,g is an affine conformal vector with affine potential α≠0 and the condition uα=−β2 holds, necessarily implies M3,g is homothetic to a Sasakian manifold. Similarly, it is shown that if the vector u of the trans-Sasakian structure F,u,γ,α,β on a 3-dimensional Riemannian manifold M3,g is a projective vector and the sectional curvatures of the plane sections containing u are positive constant, then M3,g is homothetic to a Sasakian manifold. Finally, we find certain generic conditions on a 3-dimensional Riemannian manifold M3,g possessing a trans-Sasakian structure F,u,γ,α,β so that M3,g is homothetic to a Sasakian manifold.

Axioms doi: 10.3390/axioms15060397

Authors: Fatimah A. Almulhim Mohammed B. Alamari Ali Laksaci

In this paper, we introduce a novel kernel-based estimator for the regression operator of a scalar response variable R given a functional covariate F taking values in a semi-metric space. The estimator is constructed through the minimization of the least absolute relative error (LARE) criterion, which provides an invariant scale and more balanced measure of predictive performance than conventional squared error methods. By focusing on relative deviations, the LARE approach effectively reduces the influence of extreme response values and enhances robustness in the presence of heteroscedasticity. From a theoretical point of view, we investigate the asymptotic behavior of the proposed estimator under strong mixing conditions for functional time series data. We show that, despite the temporal dependence structure, the estimator remains consistent and achieves convergence rates comparable to those obtained under independence. In the computational part, we show that the proposed method is computationally efficient and straightforward to implement. Its empirical performance is evaluated through simulation studies conducted under different dependence scenarios. In addition, the applicability of the method is illustrated through the analysis of a real data set.

Axioms doi: 10.3390/axioms15060396

Authors: Awn Alqahtani Eltiyeb Ali Khalid I. A. Ahmed

In this work, we establish the relationships between weak e-reversibility and weak e-semicommutativity, introducing weak e-reflexive rings as a natural generalization of reflexive and e-reflexive rings. We demonstrate that, under specific conditions, weak e-reflexivity and weak e-reversibility coincide in Baer rings, and we provide characterizations via corner subrings and semicentral idempotents. Furthermore, we introduce e-nilpotent reflexive rings and examine their structural stability and connections within the class of generalized reflexive rings. A comprehensive analysis is provided regarding the behavior of these properties under polynomial and Dorroh extensions, as well as within matrix rings and quotient structures.

Axioms doi: 10.3390/axioms15060394

Authors: Youjiang Lin Jiaming Lan Jinghong Zhou

In this paper, we studied the limiting regimes α→n− and α→0+ in the affine Hardy–Littlewood–Sobolev (HLS) inequalities. Specifically, we established affine logarithmic HLS inequalities and affine Beckner-type logarithmic Sobolev inequalities for pairs of functions. The results extended classical logarithmic inequalities to the affine setting and recovered known sharp inequalities in the limiting cases.

Axioms doi: 10.3390/axioms15060395

Authors: Alina Sîntămărian Ovidiu Furdui

The purpose of the paper is to give some sequences that converge quickly to a generalization of Ioachimescu’s constant, i.e., the limit of the sequence 1as+1(a+1)s+⋯+1(a+n−1)s−11−s(a+n−1)1−s−a1−sn∈N, where a∈(0,+∞) and s∈(0,1).

Axioms doi: 10.3390/axioms15060393

Authors: Doaa Filali Fatemah A. Alghamdi Mohamed E. Elnair Faizan Ahmad Khan

In this paper we study superderivations and Jordan superderivations of incidence superalgebras of locally finite Z2-graded posets. We prove that every superderivation can be expressed as a sum of an inner superderivation, an additive superderivation, and a ring superderivation; if the coefficient ring has inner derivations only, then every superderivation is inner. We also show that every Jordan superderivation decomposes uniquely as a sum of a superderivation and a proper Jordan superderivation that vanishes on the even part. These results extend classical results on incidence algebras to the supersetting.

Axioms doi: 10.3390/axioms15060392

Authors: Abdul Haseeb Sudhakar Kumar Chaubey Mohammad Nazrul Islam Khan

In this article, some remarkable results on general relativistic spacetimes with non-constant scalar curvature τ, admitting almost Ricci-Bourguignon solitons and gradient almost Ricci-Bourguignon solitons, have been established. Finally, a non-trivial example of a general relativistic spacetime is constructed by using partial differential equations to validate some of our findings.

Axioms doi: 10.3390/axioms15060391

Authors: Ghaliah Alhamzi Sajad A. Sheikh Prakash Jadhav Veena Beleyur Mdi Begum Jeelani

Hardy-type averaging operators arise in real analysis, rearrangement theory, weighted inequalities, and Volterra integral equations. This paper develops a distribution-function transport on (0,∞) equipped with an atomless Borel measure μ, showing that the cumulative map Φ(x)=μ((0,x]) implements a measure isomorphism onto Lebesgue measure under transparent support and continuity hypotheses. Under this transport, the Hardy averaging operator relative to μ is conjugate to the classical Hardy operator on (0,∞) with Lebesgue measure. The main contribution is the systematic transport principle: classical constants, extremizing sequences, weighted criteria, endpoint estimates, and resolvent information are transferred exactly to the μ-scale. We establish sharp Lp(μ) bounds, sharp power-weight extensions in Lp(Φγdμ) for −1<γ<p−1, a transported one-weight Hardy class beyond powers, endpoint weak and strong estimates, spectral interpretation of the Volterra threshold, and numerical illustrations for the transported constants and a Volterra feedback equation.

Axioms doi: 10.3390/axioms15060390

Authors: Xin Zhao Pengcheng Li

In this paper, we investigate the relationships and singularities of three associated curves, T-dual curves, N-dual curves and evolutes, for mixed-type curves in the Minkowski plane. While T-dual curves and evolutes have been studied in previous works, the N-dual curve has remained unexplored in the mixed-type setting. To fill this gap, this paper makes three main contributions. Firstly, we provide a rigorous definition of the N-dual curve, explicitly resolving the technical difficulties that arise at lightlike points where the normal line is not well-defined. Secondly, we analyze its singularities and classify its point types. Thirdly, based on these results, we establish new geometric relations among the T-dual curve, N-dual curve, and evolute. In particular, we prove that at lightlike points, the T-dual and N-dual curves coincide when the fixed point lies on the tangent line, and that the T-dual curve of the evolute coincides with the N-dual curve of the original curve under suitable conditions. These results reveal a coherent geometric framework linking the three objects. All theoretical findings are supported and validated by a variety of examples throughout the paper.

Axioms doi: 10.3390/axioms15060389

Authors: Jan Jekl Josef Rebenda

Recently, a new generalization of hyperbolic functions called para-hyperbolic functions has been introduced. However, properties of the para-hyperbolic functions have not been investigated yet. In this paper, we derive the correct explicit formulas of the para-hyperbolic sine and cosine, study elementary properties of these functions, and explore which of the inequalities that hold for trigonometric and hyperbolic functions find their counterparts for para-hyperbolic functions. Namely, we prove a Wilker type inequality, Cusa-Huygens and Lazarević type inequality, Wu-Debnah modification of Wilker type inequality and Shafer type inequality for para-hyperbolic functions. Our results provide the first thorough exploration of para-hyperbolic functions which establishes groundwork for further discoveries and applications.

Axioms doi: 10.3390/axioms15060388

Authors: Vladica S. Stojanović Nikola Mitrović Kristina Tomović Hassan S. Bakouch Shuhrah Alghamdi

This manuscript introduces a new class of count time series models, referred to as the minification integer-valued Split-BREAK (MIN–SB) process. The proposed framework extends the Split-BREAK modeling philosophy to the integer-valued setting and provides a flexible mechanism for capturing rare events, zero inflation, and structural regime changes frequently observed in safety-related data. The main stochastic properties of the MIN–SB process are derived, including stationarity conditions, explicit moment structure, and correlation dynamics. A key theoretical result reveals an implicit hidden Markov structure underlying the observable process, providing a structural explanation for zero clustering observed in rare-event count processes. Parameter estimation is developed using a simulated method of moments (SMM) approach based on zero-related statistics, and the asymptotic properties of the resulting estimators are established. A Monte Carlo simulation study demonstrates favorable finite-sample performance of the proposed estimation procedure. The practical usefulness of the model is illustrated through an empirical application to time series of injuries and fatalities caused by fire accidents in Serbia. The results show that the MIN–SB specification provides a flexible and accurate framework for modeling zero-inflated count processes arising in fire safety dynamics.

Axioms doi: 10.3390/axioms15060387

Authors: Taoufik Moulahi Najmeddine Attia

This paper introduces a novel class of condensing mappings through the concept of α-admissibility. We prove several extensions of Darbo’s fixed point theorem within this framework, including a generalized contractive condition and a coupled fixed point result. Our main results extend the classical condensing condition to a flexible inequality involving α-admissibility and an auxiliary function. An example is given to demonstrate the applicability of our theorems where traditional approaches fail. Also, by leveraging the properties of α-admissible condensing operators, we proceed to establish a coupled fixed point theorem that relaxes traditional compactness conditions. At the end of the article, we apply our main theorem to prove the existence of solutions for a nonlinear integral equation.

Axioms doi: 10.3390/axioms15050386

Authors: Álvaro Antón-Sancho

Let X be a compact connected Riemann surface of genus g≥2 and let E be a holomorphic vector bundle of rank n over X. The compactness and connectedness of X imply that the characteristic polynomial of any holomorphic endomorphism φ∈H0(X,End(E)) has constant coefficients, a fact we call the Principle of Spectral Constancy. As a consequence, the eigenvalues of φ are globally constant over X, the primary decomposition of E with respect to φ consists of globally defined holomorphic subbundles, and the Jordan decomposition φ=φs+φn into semisimple and nilpotent parts is globally well defined as a decomposition of sections of End(E). This paper provides a systematic analysis of Jordan normal forms for endomorphisms of holomorphic vector bundles over X, relating the Jordan type of φ to the stability properties of E. In particular, it is proved that endomorphisms of stable bundles are necessarily scalar, that the Jordan decomposition of an endomorphism of a polystable bundle is determined componentwise by the classical Jordan normal forms of matrices in the associated endomorphism algebra, and that finite-order endomorphisms are always semisimple. These results are applied to the study of fixed points of automorphisms of the moduli space BX(SL(n,C)) of rank n and trivial determinant polystable vector bundles over X. Specifically, a new result establishes that the commutative subalgebra of H0(X,End(E)) generated by the endomorphism associated with a fixed-point condition is semisimple, so nilpotent endomorphisms of E are precisely those incompatible with the fixed-point structure.

Axioms doi: 10.3390/axioms15050385

Authors: Jibin Yang Maozai Tian Fuguo Liu

Standard Bayesian autoregressive changepoint models can become unstable near sample boundaries. As a candidate changepoint approaches either edge of the series, the local residual degrees of freedom shrink, producing a Gamma-function singularity in the marginal likelihood that can strongly bias the posterior toward spurious edge detections. To address this issue, we introduce a regularization framework driven by local degrees of freedom. By incorporating a centripetal prior of the form π(k)∝(ν1ν2)λ—where ν1=k−2p−1 and ν2=n−k−p−1—the proposed method is designed to counteract this boundary effect. Theoretical analysis shows that a regularization intensity of λ≥1 is sufficient to offset this boundary effect asymptotically. Simulation results confirm that this approach substantially mitigates the U-shaped error profile typical of unregularized estimators, yielding a more favorable accuracy–robustness trade-off relative to the standard frequentist baselines considered in our study. Finally, empirical applications to several 2022 natural gas benchmarks, including TTF, SHPGX LNG, JKM, NBP, and NYMEX Henry Hub, demonstrate the framework’s ability to distinguish persistent structural transitions from transient market turbulence. These results suggest that degree-of-freedom-based centripetal prior regularization can improve the stability of Bayesian changepoint inference in nonstationary time series.

Axioms doi: 10.3390/axioms15050384

Authors: Haipeng Yu Manjie Zhou Zhongyang Yu Mengmeng Xu Hengzhou Xu

Tanner quasi-cyclic low-density parity-check (QC-LDPC) codes form an important family of structured LDPC codes with favorable girth properties. This paper studies the girth of Tanner (2, L)-regular QC-LDPC codes (referred to as Tanner QC-LDPC cycle codes) for arbitrary integers L>2 and develops a novel algebraic number theoretic method to determine the girth for all sufficiently large primes p with p≡1(mod2L). We first analyze the case L=3 and prove that the girth is 12 for every prime p≡1(mod6) through exhaustive resultant computations. We then extend the method to arbitrary L and obtain a clear classification: when L is even, the girth is exactly 8 for all admissible primes; when L is odd, the girth attains the maximum value 12 for all sufficiently large admissible primes. The proof transforms cycle existence conditions into polynomial equations and applies resultant theory. This approach converts the infinite task of checking all primes into a finite set of algebraic checks. Numerical simulations show that the Tanner (2, 5)-regular non-binary code over GF(64) achieves a coding gain of approximately 0.2 dB over the 5G LDPC code of equivalent binary length.

Axioms doi: 10.3390/axioms15050383

Authors: Yusra Taj Sarfraz Nawaz Malik Alina Alb Lupaş

In this paper, certain subclasses of meromorphic harmonic functions which are formulated using a q-differential operator are meticulously analyzed. Initially, two new subclasses WHq(k;E,F) and Wηq(k;E,F) associated with the Janowski function with relevance to the idea of weak subordination are defined. These classes are further studied through their various analytical and geometric properties. Some of these explored properties include the necessary and sufficient coefficient condition, the radii of starlikeness, characterizations of extreme points, distortion estimation, closeness under convolution, and convex combination features. Additionally, the asymptotic behavior of the coefficients is also examined, and to express the findings, the Big-O, little-o, and asymptotic equivalency notations are used. These findings significantly represent the interaction between the growth, dominant terms, and limiting behavior of functions within these subclasses.

Axioms doi: 10.3390/axioms15050382

Authors: Simon Brezovnik Janez Žerovnik

Roman-type domination parameters form an important class of graph invariants that model protection and resource allocation problems on networks. Among them, [k]-Roman domination provides a unified framework that generalizes Roman, double Roman, and higher-order variants. In this paper we investigate the [k]-Roman domination number of cylindrical grids Cm☐Pn and derive several new constructive upper bounds. Our approach combines three complementary techniques: linear periodic constructions, uniform ceiling-type labelings, and packing-based refinements. We first analyze the case C9☐Pn, where these three families of bounds can be compared explicitly and their relative efficiency is shown to depend on the parameter k. We then extend the linear constructions to cylindrical grids whose circumference is a multiple of one of the values r∈{3,4,5,…,9}, obtaining a unified family of upper bounds for Crt☐Pn. Motivated by the asymptotic behavior of these estimates, we further derive general upper bounds depending only on the residue class of m modulo 5, which apply to all cylindrical grids. As a consequence, we obtain explicit estimates for the double Roman domination number γ[2R](Cm☐Pn) and compare the resulting multiple-based constructions with the residue-class bounds. This comparison shows that the residue-class construction becomes asymptotically superior for all sufficiently large admissible circumferences, while several exceptional small cases remain better covered by tailored constructions.

Axioms doi: 10.3390/axioms15050381

Authors: Byungdo Park

With this Editorial, we present a Special Issue of Axioms entitled “Trends in Differential Geometry and Algebraic Topology [...]

Axioms doi: 10.3390/axioms15050380

Authors: Zafar Duman Abbasov Ghadah Albeladi Mohamed Gamal Youssri Hassan Youssri

This research paper investigates the solution of diffusion equations characterized by Third-Kind (Robin) boundary conditions within n-dimensional complex domains. The analysis is conducted in the L2 Hilbert space, which facilitates the substantiation of both the existence and uniqueness of solutions through variational methods. Analytical solutions are derived for multidimensional domains by employing the Fourier method and spectral analysis techniques. Complementing this theoretical framework, a high-accuracy numerical approach based on the Associated Legendre Polynomials Collocation Spectral Method (ALP-CSM) with Chebyshev–Gauss–Lobatto nodes is developed. Rigorous convergence analysis confirms spectral accuracy, with numerical examples in one, two, and three dimensions demonstrating error decay from O(10−3) to machine precision O(10−15). The mathematical impact of Third-Kind boundary conditions on the diffusion rate and the steady state of the system is demonstrated. The obtained results provide a robust tool for modeling physical processes, particularly in systems involving heat exchange on the surfaces of complex-structured domains, offering both theoretical insight and computational efficiency.

Axioms doi: 10.3390/axioms15050379

Authors: Jean-Renaud Pycke

One of the main tasks in the field of directional statistics is to build goodness-of-fit tests for uniformity on circles, spheres or more abstract manifolds. We discuss a goodness-of-fit test for uniformity on a distance-regular graph. Our main tool is the pseudo-inverse of the combinatorial Laplacian, for which we give explicit expressions in terms of the intersection array of the graph. Such a test can be used as a test for uniformity of data from the circle or the sphere, grouped on the tiles of a regular tessellation associated with some finite group of isometries. We describe the cases of Platonic graphs and provide examples based on real circular and spherical data, vectorial or axial.

Axioms doi: 10.3390/axioms15050378

Authors: Jonathan Washburn Milan Zlatanović Philip Beltracchi

In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural n-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination S=α·t, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an (n−1)-dimensional null distribution. On the other hand, when the same function is expressed in the original x-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in x-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura–Saito and Bregman divergences, and give a Fisher–Rao realization of the logarithmic Hessian metric.

Axioms doi: 10.3390/axioms15050377

Authors: Yuan-Yuan Chen Wei-Gang Jian Hai-Ping Zhong

This paper extends Loomis’s classical spectral criterion for almost periodic functions to bounded sequences in discrete settings. We first establish a discrete Kadets-type theorem for bounded sequences by regarding almost periodic functions as continuous with respect to the Bohr metric. We then introduce three spectra for bounded sequences via the Carleman transform and prove their equivalence. Using the Beurling–Gelfand theorem, we derive a discrete Loomis-type theorem, providing a spectral criterion for almost periodicity of bounded sequences. Our results extend the continuous theory to the discrete case and offer new tools for analyzing almost periodicity in sequence spaces.

Axioms doi: 10.3390/axioms15050376

Authors: Lei Deng Chong Li

Traditional DEA models can neither effectively handle fuzzy random variables nor achieve a complete ranking of decision-making units (DMUs). Based on the conventional fuzzy stochastic DEA model, this study introduces an exponential distribution extension. By incorporating fuzzy random variables, it significantly simplifies the deterministic transformation of chance-constrained models. Moreover, most existing DEA ranking methods only consider the relative efficiencies among DMUs while ignoring their internal structural characteristics. To address this issue, we develop a deterministic model for the exponentially extended fuzzy stochastic DEA and design a weight formula that reflects the internal input–output structure of each DMU. This approach makes the complete ranking of DMUs more reasonable and better aligned with practical situations. Finally, the rationality and effectiveness of the proposed model are verified through a comparative analysis of rankings obtained from different DEA models. The results indicate that the input–output structure within a decision-making unit plays a significant role in its efficiency ranking.

Axioms doi: 10.3390/axioms15050375

Authors: Yuchen Li Jiang Zhou

We introduce the abstract weighted Morrey spaces Mp,uκ(X,M,μ), where μ is a general measure; investigate the properties of their predual spaces; and prove the boundedness of the Hardy–Littlewood maximal operator. Furthermore, we obtain an extrapolation theorem on Mp,uκ(X,M,μ), and consequently establish the norm inequalities for bounded oscillation (BO) operators on Mp,uκ(X,M,μ). As an application, we verify that BO operators include the maximal operators, Littlewood–Paley square operators, and Carleson operators on classical weighted Morrey spaces, as well as Calderón–Zygmund operators on weighted Morrey spaces of homogeneous type.

Axioms doi: 10.3390/axioms15050374

Authors: Ghadah Alomani Raouf Fakhfakh

This paper presents two analytic mappings defined on probability measures that extend and unify the (a,b)- and (s,t)-deformations arising in free probability for s, b>0 and a, t∈R. These unified operators, denoted U(a,b,s,t)′ and U(a,b,s,t)″, are characterized by a functional equation involving the Cauchy–Stieltjes transform, providing a transform-based formulation of measure deformation. They reduce to the (a,b)-transformation when s=t=1 and to the (s,t)-transformation when a=b=1. Working in the framework of Cauchy–Stieltjes kernel families, we study the induced effect of these transformations on the associated variance functions and obtain explicit transformation formulas. These results yield a stability theorem showing that the free Meixner class is stable under both operators. In addition, we derive two properties of the semicircle law via the restricted deformations U(a,b,1/b,t)′ and U(a,b,1/b,t)″, thereby emphasizing the structural role of symmetry in measure transformations and in the preservation of canonical measures.

Axioms doi: 10.3390/axioms15050373

Authors: Sahar H. Nazra Sunil Kumar Yadav Sameh Shenawy Carlo Mantica

This manuscript investigates conformal η-Ricci–Yamabe solitons of type (κ,l) on Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field. Necessary conditions are obtained under which the soliton exhibits expanding, steady, or shrinking behavior. The analysis is further extended to several physically relevant fluid models, including dark fluid, dust fluid, stiff matter, and radiational fluid, and the corresponding geometric constraints are derived. In addition, structural results are established for Kählerian Norden space-times with a vanishing space–matter tensor and with a divergence-free matter tensor, highlighting their influence on the curvature geometry. The study also addresses several intrinsic curvature conditions of the space-time, such as conformal flatness, Ricci semi-symmetry, Ricci recurrence, and pseudo-Ricci symmetry, leading to a collection of geometric and physical characterizations. The results obtained provide a unified geometric framework linking Ricci–Yamabe soliton structures, fluid dynamics, and curvature properties within the setting of Kählerian Norden geometry.

Axioms doi: 10.3390/axioms15050372

Authors: Siyi Xie Chengzhou Wei Muhammad Zainul Abidin

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on Rn with fractional diffusion (−Δ)α, where α∈(1/2,1], and with positive viscosity and resistivity coefficients μ,ν>0. The problem is treated at the scale-invariant regularity sc=np+1−2α. For small divergence-free initial data in the critical Triebel–Lizorkin–Lorentz space F˙p,rsc,q, we construct a unique global mild solution. The main contribution is the use of the single-norm time–frequency space mm′F˙p,rsc,q, built on Meyer wavelets and the parabolic gauge t22αj. This space keeps the critical spatial size, the short-time behavior, and the high-frequency decay in one norm. By using a Gevrey-weighted Duhamel formulation, we prove boundedness of the corresponding fractional heat propagators and establish the bilinear paraproduct estimate required for the fixed-point argument. Consequently, e(t(−Δ)α)γ(u,b)∈mm′F˙p,rsc,q2n for some γ>0 depending on the parameters. This gives a Gevrey-type spatial smoothing effect, which is stronger than ordinary analyticity in the adopted scale. The restriction α>12 enters through the factor 2j(1−2α), which supplies the high-frequency gain needed to close the critical bilinear estimates; in this sense it is sharp for the present method. The classical viscous–resistive case is recovered when α=1.

Axioms doi: 10.3390/axioms15050371

Authors: Steven Duplij

This paper introduces and systematically develops the theory of polyadic group rings, a higher arity generalization of classical group rings. We construct the fundamental operations of these structures, defining the m-ary addition and n-ary multiplication for a polyadic group ring built from an (mr,nr)-ring and an ng-ary group. A central result is the derivation of the “quantization” conditions that interrelate these arities, governed by the arity freedom principle, which also extends to operations with higher polyadic powers. We establish key algebraic properties, including conditions for total associativity and the existence of a zero element and identity. The concepts of the polyadic augmentation map and augmentation ideal are generalized, providing a bridge to the classical theory. The framework is illustrated with explicit examples, solidifying the theoretical constructions.

Axioms doi: 10.3390/axioms15050370

Authors: Renato Petek Brigita Ferčec Matej Mencinger

We study the center problem for polynomial maps y=f(x)=−x−∑n=1∞anxn+1, arising from homogeneous algebraic curves x+y+∑i=0nαn−i,ixn−iyi=x+y+Hn(x,y)=0. While explicit conditions were previously known only for low even degrees n=2,4,6,8,10, their general structure remained conjectural. In this paper we resolve the case n=12 and prove that for all even degrees n=2k, the center condition is completely characterized by two families of algebraic relations: mirror symmetry conditions and alternating-sum conditions. The proof combines algebraic methods with a direct structural argument. In particular, the necessity part is established without relying on explicit formulas for focus quantities, instead, we make use of the involutive property of the associated map and analyze the symmetric difference Hn(x,f(x))−Hn(f(x),x), which leads to a simple and rigorous characterization of the center condition. This provides a complete and conceptually transparent solution of the homogeneous center problem for polynomial maps.

Axioms doi: 10.3390/axioms15050369

Authors: Yufei Wang Yang Yang

In this paper, we consider the existence of normalized solutions for the fractional Kirchhoff system with Hardy critical nonlinearities. The problem combines the challenges of nonlocal operators, singular critical effects, and a variational framework with fixed mass conditions. By employing the minimax theorem, analyzing the Palais–Smale sequence and restricting to radial functions, we overcome the loss of strong convergence at critical energy levels. Under L2-supercritical case, we prove the existence of positive radial normalized solutions, along with the positivity of the corresponding Lagrange multipliers.

Axioms doi: 10.3390/axioms15050368

Authors: Wojciech Ptas Krzysztof Siminski

The three-way decision paradigm is a new and auspicious paradigm approach to classification. It introduces a non-commitment region, allowing classifiers to abstain from (defer) uncertain predictions. This is a key mechanism in cascade classification systems, where samples assigned to the non-commitment region in one classifier are passed to the next one. Fixed thresholds are often used to determine the non-commitment region, but they require fine-tuning and provide limited insight into the reasons for deferment. We propose an automatic mechanism for determining the non-commitment region using auxiliary metaclassifiers. We reframe deferment as a learnable decision problem rather than a thresholding problem. Each metaclassifier predicts whether its accompanying classifier is likely to make a correct prediction for a given sample and decides whether to return a final answer or defer the decision to the next cascade stage. In this approach, deferment is based on a broader context than a single confidence threshold, and it is tailored to the characteristics of each classifier and dataset. With neuro-fuzzy systems used as metaclassifiers, deferment decisions can be expressed as human-readable rules.

Axioms doi: 10.3390/axioms15050367

Authors: Samia M. Said Emad K. Jaradat Sayed M. Abo-Dahab Sarhan Y. Atwa

An investigation is presented into the coupled thermo-mechanical behavior of a fiber-reinforced composite solid, with specific consideration given to the influences of gravity and thermal preconditions under inclined loading. The theoretical foundation of this work is based on the generalized dual-phase Green–Naghdi theory for constitutive modeling. Normal mode analysis has been utilized in the fundamental equations of coupled thermoelasticity. Ultimately, the derived equations are expressed as a vector-matrix differential equation, which is subsequently solved using the eigenvalue method. The outcomes of this analysis are then interpreted through numerical simulations, the details of which are presented graphically and discussed comprehensively to draw pertinent conclusions. A detailed parametric study was conducted to elucidate the individual and synergistic effects of these parameters on the material’s behavior. The findings confirm the model’s efficacy in capturing complex thermo-mechanical couplings, providing a robust framework for the design and optimization of composite structures in different environments. Analysis of the results indicates that the gravity field, reference temperature, and inclined load all exert a notable influence on the physical field variables. Furthermore, the numerical calculations performed in MATLAB R2013a demonstrate close consistency with the theoretical solution, verifying the model’s accuracy.

Axioms doi: 10.3390/axioms15050366

Authors: Bin Yan Juan Zhao

Cluster Validity Indices (CVIs) act as a pivotal tool in machine learning for assisting in the determination of the optimal number of clusters. Nevertheless, traditional CVIs often exhibit subpar performance when confronted with the complex characteristics prevalent in real-world data, such as inter-cluster overlap, outliers and uneven density distribution. To address this challenge, this paper proposes a multiplicative, adaptive and robust Cluster Validity Index, designated as the Robust Adaptive (RA) index. This index takes the kernel density function of sample points as the fundamental tool and reconstructs its two core components: in the measurement of intra-cluster compactness, the concept of density quantiles is incorporated, which markedly enhances its robustness against outliers; in the measurement of inter-cluster separability, a density-based Jeffrey divergence method is developed to effectively characterize inter-cluster differences in overlapping datasets. To mitigate the impact of bandwidth selection on kernel density estimation, this study adopts strategies including Scott’s and Silverman’s heuristic algorithms, thus enabling adaptive learning of the inherent distribution characteristics of data. For experimental validation, a comprehensive set of experiments was conducted on both synthetic and real-world datasets. The results show that, in comparison with the classical indices (CH, DB, SIL, I) that demonstrate prominent performance on overlapping datasets, the RA index delivers superior performance in scenarios involving mild to moderate overlap, uneven density distribution and the presence of outliers. Among nine synthetic datasets, the RA index correctly identified the optimal number of clusters in eight cases, achieving a high success rate of 88.89% and outperforming all the comparative indices. On eight real-world datasets with diverse scales, dimensionalities and inherent structural features, the RA index was also verified to be the most robust and effective metric among the five participating indices for comparison. Meanwhile, its failure on complex datasets such as S-set4 and Iris, which contain both severe inter-cluster overlap and outliers, also indicates that density-based CVIs have inherent limitations when faced with data structures characterized by high overlap and faint cluster boundaries. This finding points to a clear direction for future research: constructing novel CVIs from the perspective of sparse matrices may serve as a feasible breakthrough path to address such limitations.

Axioms doi: 10.3390/axioms15050365

Authors: Gülden Altay Suroğlu Şeyma Firdevs Hızal Hasan Bulut

This study investigates the robustness of Frenet–Serret curvature (κ) and torsion (τ) estimates derived from noisy discretely-sampled three-dimensional space curves, with emphasis on the comparative performance of cubic spline and cubic Hermite interpolation methods. Accurate estimation of these geometric invariants is essential for reliable analysis of curves arising in signal processing and shape reconstruction; yet, the higher-order derivatives required for their computation exhibit pronounced sensitivity to measurement noise. We examine curves constructed through a Hilbert transform-based parameterization of the form r(t)=X(t),A(t)sinϕ(t),g(t), where discrete samples are contaminated with additive white Gaussian noise at varying signal-to-noise ratios. Reconstruction is performed using cubic spline interpolation, which ensures global C2 continuity, as well as cubic Hermite spline interpolation, which provides C1 continuity with local tangent control. Frenet frame computations are then applied via regularized finite difference schemes. To characterize noise amplification theoretically, we derive the Curvature Stability Index (CSI) and Torsion Stability Index (TSI) as first-order variance bounds under the delta method. While these indices formalize the derivative-order dependence of noise sensitivity, Monte Carlo simulations reveal that empirical variance exceeds theoretical predictions by factors of 104 to 106, indicating dominance of nonlinear error propagation. Nevertheless, the indices establish that torsion instability arises fundamentally from third-order derivative structure rather than ground-truth magnitude. Numerical experiments across three geometric regimes constant-invariant helices, variable-curvature helices, and planar curves with identically zero torsion demonstrate that the ratio of the torsion root mean square error to curvature root mean square error consistently ranges from 6.5 to 9.8. This disparity persists even in the degenerate planar case, where τ≡0 analytically, confirming that torsion sensitivity is an intrinsic property of the Frenet–Serret formulation. Across all configurations, cubic spline reconstruction yields lower Monte Carlo mean RMSE and reduced empirical variance compared to Hermite spline, providing superior stability for derivative-based invariant estimation.

Axioms doi: 10.3390/axioms15050364

Authors: Flavio Prebianca Bruna G. Pedro Gabriel B. Corrêa Anderson Hoff Cesar Manchein Holokx A. Albuquerque

We investigate the nonlinear dynamics of a linearly coupled van der Pol–Duffing system using numerical continuation, time-domain simulations, stochastic analysis, and analog circuit experiments. The model exhibits a rich variety of dynamical regimes, including periodic oscillations, period-doubling cascades, and chaotic attractors arising from the interplay between self-excitation and nonlinear stiffness. Numerical continuation is employed to reconstruct the bifurcation structure, enabling the identification of equilibrium branches, periodic solutions, and their stability in parameter space. The time-domain numerical results reveal the mechanisms governing transitions between regular and chaotic dynamics. To assess robustness under realistic conditions, intrinsic stochastic perturbations are introduced, showing that increasing noise intensity progressively erodes fine periodic structures, while larger dynamical domains remain comparatively robust. Experimental results obtained from an analog circuit implementation confirm the main dynamical regimes predicted numerically. Overall, the combined computational and experimental approach provides a systematic characterization of the system’s bifurcation structure and its robustness to noise. The results support the concept of chaos-based sensing and are consistent with previous findings in chaotic bioimpedance detection, indicating that maximum sensitivity occurs near regions of high bifurcation complexity, where small parameter variations induce significant qualitative changes in the system dynamics.

Axioms doi: 10.3390/axioms15050363

Authors: Xuefeng Yao Yusi Yang Hongwei Jiao

This paper proposes an outer space branch-and-search method for a class of linear fractional programming problems over a polytope. First, the original problem is reformulated in an equivalent problem by applying the equivalent transformation. Second, by using a new linearization technique, a linear programming relaxation problem of the equivalent problem is constructed. Third, lower bounds are obtained by solving a sequence of linear programming relaxation problems. Fourth, the convergence of the proposed algorithm is proved, and its worst-case computational complexity is estimated. Finally, numerical experimental results are reported to demonstrate the effectiveness of the algorithm. Additionally, an investment decision-making problem was solved to validate the applicability of the method proposed in this paper.

Axioms doi: 10.3390/axioms15050362

Authors: He Li Jinyu Zou Shumin Zhang

Connectivity is a classic measure that evaluates the fault tolerance of multiprocessor systems when processor failures occur. To better evaluate the reliability of multiprocessor systems, researchers have proposed many indicators based on connectivity with additional constraints, for example the number of components formed by removing an edge subset or a vertex subset. If the subgraph obtained from G by deleting an edge subset contains at least g components, the minimum size among all such edge subsets is denoted by the g-component edge connectivity of G. Regarding the g-component edge connectivity of many well-known networks, numerous results exist. However, we are particularly interested in general graphs. In this paper, we first establish basic properties of g-component edge connectivity and determine its exact value for complete graphs, paths, cycles and complete multipartite graphs. We then characterize graphs achieving a given g-component edge connectivity. Finally, we study three parameters related to g-component edge connectivity inspired by classical extremal problems.

Axioms doi: 10.3390/axioms15050361

Authors: Samuel L. Krushkal

The paper provides a new approach to estimating the coefficients of arbitrary holomorphic functions, which still remains an important problem of complex analysis. This approach is intrinsically connected with the features of univalent functions and with Teichmüller spaces.

Axioms doi: 10.3390/axioms15050360

Authors: Dmitrii Karp Elena Prilepkina

We extend the classical Euler-type integral representations for the Appell functions F1, F2, and F3, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, these representations provide analytic continuation of the corresponding Kampé de Fériet functions. We further focus on the following two applications. First, we obtain sufficient conditions for complete monotonicity on the positive quadrant for three families of the Kampé de Fériet functions. These conditions can be expressed directly in terms of parameters and imply, among other things, joint log-convexity and related inequalities for partial derivatives of the Kampé de Fériet functions. Second, we show how known reduction and transformation formulas for the Appell and the generalized hypergeometric functions can be lifted to Kampé de Fériet functions by concatenating parameter arrays via the integral representations. This yields several reduction formulas, including extensions of some classical and new product identities. Further combining integration against the Meijer–Nørlund G-function with Slater’s double series transformation we obtain several exotic identities for infinite sums of the generalized hypergeometric functions.

Axioms doi: 10.3390/axioms15050359

Authors: Farrukh Jamal Mohamed A. Abd Elgawad Muhammad Imran Shahid Mohammad

This paper introduces a novel continuous probability distribution on the unit interval called the unit Jamal distribution and explores its properties. The proposed distribution performs well in modeling bathtub-shaped data, effectively capturing its characteristic hazard rate behavior. Key mathematical characteristics such as moments, the moment generating function, order statistics, entropy, and the quantile function are thoroughly derived. Parameter estimation is conducted using maximum likelihood and Bayesian estimation methods. A simulation study is conducted to evaluate the accuracy of parameter estimates and to examine the distribution’s behavior. Additionally, the applicability of the proposed distribution is demonstrated through analysis of two real-world datasets, allowing for a comparison of its performance against existing models.

Axioms doi: 10.3390/axioms15050358

Authors: Muhammad Zakria Javed Muhammad Uzair Awan Loredana Ciurdariu Eugenia Grecu Hala Mostafa

This study presents some novel sharp estimates of the Euler–Maclaurin inequality using a new higher-order derivative Maclaurin identity. By utilizing the properties of convexity and classical inequalities, we exploit various novel tight boundaries of the Euler–Maclaurin inequality. They offer alternatives to measuring the sharp bounds of the mean integral of the higher-order differentiable mappings. In order to prove the importance and precision of the key findings, we apply graphical and numerical techniques. Another important section evaluates the behavior and validity of inequalities using a neural network model. The method is not only utilized to authenticate the results but also brings out the practical advancements of the study within a computational framework. The method and results of the article provide an insight and develop a solid connection between inequalities, higher-order derivative convex mappings, numerical analysis, approximation theory, and artificial neural networking.

Axioms doi: 10.3390/axioms15050357

Authors: Song-Kyoo Kim

This paper focuses on optimizing machine learning-based time-series forecasting models by constructing compact data. Compact Data Learning for Time Series (CDL-TS) is a novel framework aimed at minimizing forecasting model errors. By utilizing reduced sampling and robust comparison procedures, CDL-TS addresses the challenges of forecasting models on extensive real-time data systems. By strategically minimizing data size while maintaining accuracy, CDL-TS presents an innovative framework that facilitates robust predictions and enhances operational efficiency. The combined bivariate performance measure-based optimization effectively balances sampling frequency with the mean square error (MSE) to improve the trade operation performance in stock markets. Through a series of empirical applications, particularly involving the M/M/1 queuing system and various stock trade optimizations with global high-tech companies, the CDL-TS framework has proven its effectiveness by significantly minimizing both forecasting errors and operational costs. This accomplishment highlights the robust capabilities of CDL-TS in enhancing predictive accuracy while facilitating operation cost savings across different domains, including complex systems like queues and real-time financial markets.

Axioms doi: 10.3390/axioms15050356

Authors: Wilbert Nkomo Anis Ben Ghorbal Broderick Oluyede Fastel Chipepa

This work presents a new family of distributions (FoDs) called the exponentiated half logistic-generalized-Topp-Leone-G (EHL-GEN-TL-G) family. This family can be expressed as an infinite linear combination of exponentiated-G densities, which facilitates the derivation of its important statistical properties. The shapes of the density and hazard rate functions were investigated for special cases. The model parameters were estimated using six different methods, with the maximum likelihood technique emerging as the best approach. The consistency of the parameter estimates was then validated through Monte Carlo simulations. The exponentiated half logistic-generalized-Topp-Leone-Weibull (EHL-GEN-TL-W) distribution, a sub-model of the EHL-GEN-TL-G family, was applied to three sets of engineering failure time data. The results indicated that, based on in-sample goodness-of-fit criteria, the EHL-GEN-TL-W model provided the best fit among the several established models considered. Additionally, the EHL-GEN-TL-W regression model was developed, and its practical utility in modeling failure data was demonstrated.

Axioms doi: 10.3390/axioms15050355

Authors: Antonio E. Bargellini Daniele Ritelli

An endogenous extension of the classical logistic equation, in which the carrying capacity κ(t) varies in a periodic manner, is developed within this study. Existing theoretical results characterize the behavior of models of this nature. Nevertheless, to the best of our knowledge, the extant literature does not seem to include contributions with a comparable pragmatic computational purpose. By leveraging the Bernoulli structure of the model alongside a Fourier representation of the periodic forcing, closed-form expressions for the associated periodic solution are derived, and a unified Fourier-series framework is constructed to reconstruct trajectories under general periodic inputs. This approach refines and extends earlier findings on periodic logistic dynamics, resulting in an explicit characterization of the oscillatory regime induced by κ(t), which influences the system’s nonlinear feedback and long-term behavior.

Axioms doi: 10.3390/axioms15050354

Authors: F. A. H. Alomari M. Z. Youssef A. A. Alkinani S. S. Ezz-Eldien

Singular behavior near the initial times, arising from the presence of fractional-order derivatives in fractional differential equations, often leads to the deterioration of solutions obtained using spectral methods when applied based on classical orthogonal polynomials. Consequently, instead of relying on smooth polynomials as the basis for spectral approaches, significant attention has been devoted to developing appropriate nonsmooth functions that can form the basis for various spectral methods to address the limitation imposed by the fractional-order derivatives. In this study, we develop a numerical framework for solving one- and two-dimensional time-fractional coupled FitzHugh–Nagumo (FHN) models. We construct a novel, time-nonsmooth but spatially smooth function, called the orthogonal shifted Chebyshev function, in both one- and two-dimensional dimensions, that serves as the basis of the spectral collocation approach. Furthermore, we derive novel operational matrices of second-order and fractional-order derivatives, based on the new basis function in the spatial and time directions, respectively. These matrices are then used in conjunction with the spectral collocation technique in both spatial and time directions to reduce the problem to a system of algebraic equations. The numerical results demonstrate the accuracy of the presented numerical scheme and confirm the superiority of the new basis over the classical shifted Chebyshev polynomials when applied to time-fractional models.

Axioms doi: 10.3390/axioms15050353

Authors: Bang-Yen Chen Foued Aloui Majid Ali Choudhary Ibrahim Al-Dayel

The notion of 2-conformal vector fields on pseudo-Riemannian manifolds, which arose naturally in the study of hyperbolic solitons, is introduced by Fasihi-Ramandi, De, and Shamkhali. In this paper, we study invariant 2-conformal vector fields on four-dimensional non-reductive pseudo-Riemannian homogeneous manifolds G/H. Consequently, the complete classification of such vector fields is achieved, together with the necessary and sufficient conditions for their existence. The results are then applied to Lorentzian and neutral signatures, where 2-conformal vector fields provide an effective criterion for detecting 2-conformal equivalences in geometries with limited algebraic symmetries.

Axioms doi: 10.3390/axioms15050352

Authors: Jorge Pereira Fatelo Nelson Martins-Ferreira

Mobi spaces were introduced by the authors as a possible algebraic axiomatization of spaces in which any two points are connected by a geodesic path. Previous work has focused primarily on the algebraic properties of these structures; here, we return to the original geometric motivation. We present new characterizations of mobi spaces inspired by the important class of smooth manifolds that arise as open subsets of Euclidean (n)-space endowed with a Riemannian metric. We show that such manifolds satisfy the axioms of a mobi space, thereby providing a broad family of natural geometric examples.

Axioms doi: 10.3390/axioms15050351

Authors: Zongxu Li Zhuo Fang Mingyuan Cao Yueting Yang Ruobing Mei Siqi Liu

We propose a novel subspace derivative-free conjugate gradient method for solving large-scale nonlinear monotone equations with convex constraints. At each iteration, the search direction is constructed by minimizing a quadratic model within a subspace spanned by the current negative function value vector and the two most recent search directions. The algorithm incorporates a hyperplane projection technique to generate feasible iterative points. Under reasonable assumptions, we establish the global convergence and R-linear convergence rate of the proposed method. Extensive numerical experiments on benchmark problems demonstrate that the new algorithm significantly outperforms state-of-the-art derivative-free methods in terms of number of iterations, function evaluations, and CPU time. The results confirm the efficiency and robustness of the proposed approach for solving large-scale monotone systems.

Axioms doi: 10.3390/axioms15050350

Authors: Ahmad Al-Omari Mohammad H. M. Rashid

We investigate the local well-posedness of semilinear fractional integro-differential equations in Banach spaces with the Hadamard fractional derivative. The equation is DtβHu(t)=Au(t)+φt,u(t),∫1tK(t,s)ρ(s,u(s))ds,u(1)=u0, where A generates a compact C0 semigroup. Using Schauder’s fixed point theorem, we prove local existence under linear growth conditions. Uniqueness is obtained via Banach’s contraction principle under Lipschitz assumptions. The main contribution is a detailed theorem for non-Lipschitz nonlinearities satisfying Carathéodory conditions and Osgood-type growth, where we prove the existence and additional regularity of mild solutions. An illustrative example with Lipschitz nonlinearities is provided.

Axioms doi: 10.3390/axioms15050349

Authors: Guomengchao Bian Feifei Chen Senlin Wu

Z.K. Li established the first explicit decoupling estimate for the parabola in the range 4<p<6. Such explicit bounds are essential in number theory applications. Following the approach of J. Bourgain and C. Demeter, we derive an explicit bound for the implicit constant in an alternate form of decoupling in J. Bourgain and C. Demeter’s work. This is the first part of our three-part project to extend Z.K. Li’s result on the explicit bound for the decoupling constant to the n-dimensional case.

Axioms doi: 10.3390/axioms15050348

Authors: Eleftherios Makariadis Stefanos Makariadis Avrilia Konguetsof Basil K. Papadopoulos

This work introduces Fuzzy Implication Modeling Theory, a connective-centered framework that supplies an intermediate structural layer between the class and family levels in the subset of fuzzy implications generated by compositions of the three basic fuzzy connectives. The theory distinguishes Fuzzy Implication Models from Fuzzy Implication Model Instances and formalizes their relationship through three propositions addressing generation, traceability and non-uniqueness. Two model-level quantitative descriptors—a Behavior Index and an Intra-Model Variability Index—are proposed to enable comparative analysis across models, and a new Fuzzy Implication Model is introduced and proven to satisfy the defining conditions of a fuzzy implication. A MATLAB-based companion tool with a dedicated model-design module operationalizes the theory. Taken together, the framework provides a structured lens for organizing, analyzing and selecting fuzzy implication operators, and opens paths for both further theoretical development and applied studies.

Axioms doi: 10.3390/axioms15050347

Authors: Brahim Chnioune Mohammed Rahmani Abderrahmane Nitaj Mhammed Ziane

A new RSA variant based on the cubic Pell curve operates with a modulus N=pq where the encryption and the decryption exponents e and d are linked by the congruence ed≡1 (mod(p−1)2(q−1)2). At Africacrypt 2025, Rahmani and Nitaj demonstrated that this scheme is susceptible to lattice-based attacks when the secret exponent d is small. In this work, we present a refined attack on the same scheme when the prime factors p and q share a sufficient portion of their least significant bits (LSBs). Our new method extends the former results, and yields improved bounds on the decryption exponents. Leveraging a combination of Coppersmith-type techniques and lattice methods, our approach is capable of recovering the RSA prime factors in polynomial time.

Axioms doi: 10.3390/axioms15050345

Authors: Antanas Laurinčikas Darius Šiaučiūnas

Let P be a system of generalized prime numbers, and NP the corresponding system of generalized integers. Assuming that ∑m⩽xm∈NP1−ax≪xβ with a>0 and 0⩽β<1, we consider the Beurling zeta-function ζP(s), s=σ+it. Beurling zeta-functions constitute a wide class of non-standard zeta-functions which pose interesting mathematical problems. Numerous authors are searching for restrictions on the systems P and NP that the corresponding Beurling zeta-functions have some properties similar to those of classical zeta-functions. One of such properties is the functional independence which was initiated by O. Hölder and D. Hilbert, and, in the most general form, by S.M. Voronin. This is a motivation to obtain the functional independence in the Voronin sense for a certain class of Beurling zeta-functions. Under a certain additional condition involving the generalized von Mangoldt function, we obtain the functional independence of the function ζP(s). We prove that the function ζP(s) does not satisfy the equation ∑k=0rskFkζP(s),ζP′(s),…,ζP(n−1)(s)=0 with continuous functions Fk, k=0,…,r. The proof is based on the universality property of ζP(s) on approximation of analytic functions by shifts ζP(s+iτ), τ∈R.

Axioms doi: 10.3390/axioms15050346

Authors: Ze-Ping Wang Li-Hua Qin Xue-Yi Chen

We first study the f-biharmonicity of totally umbilical hypersurfaces in a Riemannian manifold of dimension n≥3 and prove that any totally umbilical proper f-biharmonic hypersurface without boundaries in a nonpositively curved manifold must be noncompact. Since biharmonic submanifolds are special cases of f-biharmonic submanifolds, our results on f-biharmonic hypersurfaces in nonpositively curved conformally flat spaces provide a natural extension of the generalized Chen’s conjecture. We then investigate the f-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Next, we study f-biharmonic surfaces in a conformally flat 3-space, and for those with nonzero constant mean curvature (CMC), we provide a complete classification of them in 3-space forms. Finally, we investigate the f-biharmonicity of hypersurfaces in a conformally flat space with negative sectional curvature. Our results generalize some previous conclusions on biharmonic hypersurfaces.

Axioms doi: 10.3390/axioms15050344

Authors: Osman Yürekli

This paper revisits the use of the Laplace transform in the study of Bessel and modified Bessel differential equations. Rather than presenting the method as a new alternative to the classical Frobenius approach, the paper is organized as a pedagogical review of how standard operational rules for the Laplace transform lead to transform-domain differential equations for the regular solutions of these variable-coefficient problems. For Bessel’s equation, the transformed equation yields the classical Laplace transform of Jν(x); for the modified Bessel equation, the same procedure yields the corresponding transform of Iν(x). These formulas are then used to recover standard recurrence and derivative identities and to derive several illustrative transform evaluations. The paper also explains why the singular companion solutions Yν(x) and Kν(x) are not obtained directly from the regular Laplace-transform framework and must instead be introduced through classical connection formulas. Particular attention is given to placing the calculations in the context of the earlier literature, especially the classical treatise of Watson and later work on Laplace and Lipschitz–Hankel integrals. In this form, the paper is intended as a self-contained review and tutorial on a useful operational approach to Bessel-type equations.

Axioms doi: 10.3390/axioms15050343

Authors: Gregory Natanson

The paper links Stevenson’s formally complex hypergeometric polynomials to the real-by-definition Romanovski-Routh (R-Routh) polynomials. The spectral problem for Stevenson’s second-order normal ordinary differential equation (NODE) was formulated in such a way that it could be re-used for the two SLPs associated with the ‘trigonometric Rosen-Morse’ (t-RM) potential on the finite interval and the implicit Milson potential on the line (both solvable by the R-Routh polynomials). Namely, the sought-for eigenfunction was required to represent the principal Frobenius solutions at both minus- and plus-infinity. We refer to these boundary conditions as the ‘dual-PFS’ problem. The exact solvability of the former SLP with the trigonometric Liouville potential was then proven by taking into account that the Romanovski-Routh polynomial of degree n must have exactly n real zeros as well as that the discrete energy spectrum in question had no upper bound. As the direct consequence of this proof, we then found that the mentioned d-PFS problem Stevenson’s NODE and therefore the second SLP associated with the Milson potential on the line were exactly solvable via the quasi-rational solutions (q-RSs) composed of the R-Routh polynomials with degree-dependent indexes.

Axioms doi: 10.3390/axioms15050342

Authors: Svetozar Margenov

This survey presents an overview of recent developments in the analysis and numerical treatment of spectral fractional diffusion equations. Particular attention is devoted to efficient strategies for solving spectral fractional diffusion problems, including approaches based on rational approximation that enable efficient numerical realization of fractional powers of elliptic operators. Building on these approximations, we discuss adaptive finite element discretization techniques for polygonal domains, where singularities and geometric irregularities require carefully designed mesh refinement strategies. The survey also highlights the role of fractional diffusion operators in the preconditioning of coupled and multiphysics problems, where they can significantly improve the robustness and convergence of iterative solvers. Furthermore, we review recent results on maximum principles and monotonicity preservation for spectral fractional diffusion–reaction equations, which are essential for ensuring physically meaningful numerical solutions. Finally, we discuss current efforts aimed at improving robustness and computational efficiency through reduced and multilevel iteration methods. These approaches provide scalable algorithms for large-scale problems while maintaining accuracy and stability. The survey concludes by outlining several open problems and promising directions for future research in the numerical analysis of fractional diffusion models.

Axioms doi: 10.3390/axioms15050341

Authors: Fatemah Mofarreh Ahmed M. Elshenhab

This survey provides a systematic review of delayed matrix functions and their role in deriving explicit solutions for linear discrete delay systems. Tracing the evolution from foundational single-delay first-order systems to sophisticated multi-delay configurations, we cover delayed matrix exponentials, sines, and cosines for both commutative and non-commutative coefficient matrices, as well as generalizations for two-sided delay terms. We synthesize closed-form solution representations for a wide spectrum of initial value problems and highlight applications across stability analysis, controllability, iterative learning control, and finite-time stability. The paper concludes with a critical discussion identifying open problems-including extensions to higher-order differences, Volterra-type systems, and non-commutative multi-delay scenarios-serving as a unified reference connecting algebraic construction to analytical utility in linear discrete dynamical systems with memory. The review is reported in accordance with the PRISMA 2020 guidelines; the completed checklist and flow diagram are provided.

Axioms doi: 10.3390/axioms15050340

Authors: Zeeshan Yousaf Timothy Ganesan Muhammad Zaeem Ul Haq Bhatti

Complex metrics play a fundamental role in applied mathematical analysis and in the study of physical phenomena. In this work, two complex Minkowski spacetime metrics are examined in detail. The investigation centers on the spin matrices that generate these spacetimes, their corresponding algebraic properties, and the quaternionic structures that emerge from them. The key findings of this study include the identification of a novel set of spin matrices and the characterization of their symmetry, spectral, commutator, and anticommutator properties. Furthermore, a new generalized Lorentz algebra is derived, and a quaternionic mapping of the proposed spin matrices is performed, leading to the construction of new orthonormal quaternionic basis vectors.

Axioms doi: 10.3390/axioms15050339

Authors: Md Aquib

In this paper, we investigate Chen’s δ-invariant for partially slant (PS) submanifolds of complex space forms. A PS-submanifold admits an orthogonal decomposition of the tangent bundle into a proper slant distribution and an arbitrary ambiguous distribution. Using the Gauss equation together with algebraic optimization techniques, we derive a Chen-type inequality relating the δ-invariant to the squared mean curvature, the holomorphic sectional curvature of the ambient space, and the slant angle of the slant distribution. Unlike the classical Chen inequality for slant submanifolds, the obtained estimate contains an additional term reflecting the contribution of the ambiguous distribution. Several corollaries are derived, including dimension-dependent bounds and special cases corresponding to hemi-slant and semi-slant submanifolds. The equality case is also characterized in terms of the structure of the shape operators. These results provide a natural extension of Chen-type inequalities to the broader framework of partially slant geometry in Kähler manifolds.

Axioms doi: 10.3390/axioms15050338

Authors: Venelin Todorov Ivan Dimov

We investigate biased and unbiased Monte Carlo algorithms for solving Fredholm integral equations of the second kind and for estimating linear functionals of their solutions. Fredholm integral equations provide a common mathematical framework in uncertainty quantification, Bayesian inference, physics, finance, engineering modeling, telecommunication systems, signal processing, and other applied problems where system responses depend on distributed, uncertain, or noise-affected inputs. The comparison covers Crude Monte Carlo and Markov Chain Monte Carlo baselines, modified Sobol quasi–Monte Carlo schemes (MSS variants), the classical Unbiased Stochastic Algorithm (USA), and a new variance-controlled unbiased estimator, the Novel Unbiased Stochastic Algorithm (NUSA). NUSA preserves unbiasedness via a randomized-trajectory representation while improving stability through two mechanisms: adaptive absorption control, governed by a parameter Pd that regulates the effective trajectory length, and kernel-weight normalization based on an auxiliary proposal density to curb heavy-tailed weight products. Extensive experiments in one- and multi-dimensional settings (including regular and discontinuous kernels and weak/strong coupling regimes) show that NUSA consistently reduces dispersion and achieves smaller errors than USA under identical sampling budgets. In representative tests, NUSA attains relative errors below 10−3 and improves average accuracy by approximately 30–50% compared with USA, while maintaining near-linear runtime scaling in N and competitive scaling with dimension. Although NUSA is moderately more expensive per run than USA, the variance reduction yields a superior accuracy–cost trade-off, especially near strong-coupling regimes and in higher dimensions where standard unbiased estimators become variance-limited.

Axioms doi: 10.3390/axioms15050337

Authors: Bowen Liu Jiashu Wang Boris Melnikov

The score set of a tournament is defined as the set of its distinct out-degrees. In 1978, Reid proposed the conjecture that for any set of nonnegative integers D, there exists a tournament T with a score set D. In 1989, Yao presented an arithmetic proof of the conjecture, but a general polynomial-time construction algorithm has not been discovered. This paper proposes a necessary and sufficient condition and a separate necessary condition, based on the existing Landau’s theorem for the problem of reconstructing score sequences from score sets of tournament graphs. The necessary condition introduces a structured set that enables the use of group-theoretic techniques, offering not only a framework for solving the reconstruction problem but also a new perspective for approaching similar problems. In particular, the same theoretical approach can be extended to reconstruct valid score sets given constraints on the frequency of distinct scores in tournaments. Based on these conditions, we have developed three algorithms that demonstrate the practical utility of our framework: a polynomial-time algorithm and a scalable algorithm for reconstructing score sequences, and a polynomial-time network-building method that finds all possible score sequences for a given score set. Moreover, the polynomial-time algorithm for reconstructing the score sequence of a tournament for a given score set can be used to verify Reid’s conjecture. These algorithms have practical applications in sports analysis, ranking prediction, and machine learning tasks such as learning-to-rank models and data imputation, where the reconstruction of partial rankings or sequences is essential for recommendation systems and anomaly detection.

Axioms doi: 10.3390/axioms15050336

Authors: Yong Chan Kim Young-Hee Kim

In this paper, we introduce topological information systems between objects with a distance and attributes with a distance based on a complete co-residuated lattice. We can obtain two distances and Alexandrov topologies for objects and attributes from an information system. Moreover, we investigate the relations between residuated frames and residuated connections on Alexandrov topologies induced by distances instead of fuzzy equalities. We define a fuzzy concept lattice based on Alexandrov topologies and show that it is a complete lattice. We study its properties as a topological viewpoint and provide examples. Moreover, we introduce R-R embedding maps and R-R frame embedding maps.

Axioms doi: 10.3390/axioms15050335

Authors: Myroslav Sheremeta Oksana Holovata Oksana Mulyava

In terms of order, lower order, type, lower type and two-member power asymptotic for the function A(z)=∑n=1∞anf(λnz), analytic in a finite disk, where f is an entire transcendental function and (λn) is sequence of positive numbers increasing to +∞, the relationship between the growth of M(r,A)=∑n=1∞|an|Mf(rλn), where Mf(r)=max{|f(z)|:|z|=r}, and the behavior of the coefficients an, is studied.

Axioms doi: 10.3390/axioms15050334

Authors: Andriy Bandura Mykhailo Pivkach Tetyana Salo Oleh Skaskiv Andriy Kuryliak

This article investigates the lower bounds of analytic functions defined by absolutely convergent Dirichlet series in the left half-plane and establishes conditions under which such functions exhibit the pits property. The study extends classical results for entire Dirichlet series and lacunary power series by refining assumptions on the sequence of exponents and resolving gaps in earlier proofs found in the literature. Central to the analysis is the introduction of a modified function k(σ), whose behavior determines the size of exceptional sets surrounding the zeros of the series. Under suitable growth conditions on the exponents, the authors prove that outside small disks centered at the zeros, the modulus of the Dirichlet series admits explicit lower bounds involving its maximum term. Several auxiliary lemmas provide sharp estimates for the maximum modulus, the distribution of zeros, and the behavior of truncated Dirichlet polynomials. The main theorem demonstrates that the pits property holds uniformly in vertical strips approaching the imaginary axis. The paper concludes with a discussion of how the imposed step-size condition on exponents might be weakened, formulating a conjecture regarding a more general condensation index.

Axioms doi: 10.3390/axioms15050333

Authors: Laure Gouba Carine Ornela Megne Nono

Ordinary differential equations are fundamental tools for modeling dynamic systems in science, engineering, and applied mathematics. Solving these equations accurately and efficiently is crucial, particularly in cases where analytical solutions are challenging or impossible to obtain. This paper presents a method for solving inhomogeneous linear ordinary differential equations using an artificial neural network. The network is composed of a single input layer with one neuron, one hidden layer with three neurons, and a single output layer with one neuron. A multiple regression model is employed to determine the weights from the input layer to the hidden layer, while radial basis functions are used to compute the weights from the hidden layer to the output layer. The bias values are chosen within the range of −1 to 1 to optimize learning behavior. A trial solution is constructed as a sum of two parts. One part satisfies the initial condition, and the other part is the output of the network to approximate the function. The neural network is trained to minimize the mean squared error of the residuals obtained by doing the substitution of the trial solution into the given ordinary differential equation. The methodology is tested on first-order and second-order ordinary differential equations to evaluate its accuracy, stability and how its capability can be generalized. The results show that the method can approximate the exact solutions of these ordinary differential equations with reasonable accuracy.

Axioms doi: 10.3390/axioms15050332

Authors: Derek H. Smith Roberto Montemanni

The possible existence of a regular Moore graph of diameter 2 and degree 57 with the maximum number 3250 of vertices has been an open question for over 65 years. One approach to a construction focuses on the set of permutations that describe the 1-factors that give the adjacencies between leaf vertices of pairs of branches of a tree. Most of these permutations are derangements, that is they are permutations with no fixed points. As many products of 2, 3, or 4 of these derangements must also be derangements, it is tempting to use a group of derangements, that is a group of permutations in which every non-identity element is a derangement. The first case to consider is when the group of derangements is a cyclic group of permutations. In this paper it is proved that a construction using only a cyclic group of permutations is impossible. This leaves only the possibility of using some other group of derangements, or a set of derangements that do not form a group. The prospects for extending the work to these cases is considered at the end of the paper.

Axioms doi: 10.3390/axioms15050331

Authors: Alexander J. Zaslavski

In 2003 W. Takahashi and M. Toyodaestablished the weak convergence of an iteration process to solve a variational inequality problem induced by an inverse strongly-monotone mapping. Recently we proved that for the same iterative process, most of its exact iterates are approximate solutions of the variational inequality. It was also shown that the iteration process for solving a variational inequality problem for an inverse strongly-monotone mapping generates approximate solutions in the presence of computational errors. In this work we employ the Cimmino algorithm in order to generalize these results for common approximate solutions of a finite family of variational inequalities with inverse strongly-monotone mappings and of a finite family of fixed point problems in the presence of computational errors.

Axioms doi: 10.3390/axioms15050330

Authors: Josip Pečarić Jinyan Miao

Some equivalent statements and basic properties for the generalized convexity assumption p(x)f(x)+q(x)f′(x)+f″(x)≥0 are proved. Then based on these conclusions, various Jensen type inequalities under the generalized convexity are established. The idea is to transform such p(x),q(x)-convex functions to some simpler p(x), 0-convex or 0, q(x)-convex functions, or even convex functions. Ky Fan and Wang-Wang type inequalities are also generalized as applications.

Axioms doi: 10.3390/axioms15050329

Authors: Omar Mossa Alsalhi

This paper studies the algebraic properties of little Hankel operators on Hardy–Sobolev spaces Hs2, focusing on a notion of commutativity defined via adjoint products. For symbols φ and ψ, whose co-analytic parts are trigonometric polynomials, we consider the condition Hφ(s)*Hψ(s)=Hψ(s)*Hφ(s)onHs2. It is shown that this adjoint-product commutativity holds if and only if the co-analytic parts of the symbols are real scalar multiples of one another. As a consequence, the commutant of a nonzero Hankel operator on Hs2, within the class of Hankel operators whose co-analytic symbols are trigonometric polynomials, is one-dimensional over R. The proof relies on a direct coefficient analysis exploiting the finite Hankel structure induced by polynomial symbols. The result applies uniformly to all Sobolev exponents s≥0, including the classical Hardy space s=0 and the Dirichlet case s=1/2.

Axioms doi: 10.3390/axioms15050328

Authors: Sobia Ashraf Muhammad Yaseen Khidir Shaib Mohamed Alawia Adam Muntasir Suhail

This study presents an extended cubic B-spline Galerkin scheme for the numerical solution of multi-term time-fractional differential equations. The proposed formulation employs extended cubic B-splines together with the Caputo fractional derivative to model the time-fractional operators. Gauss quadrature is used to accurately evaluate the resulting integral. A stability analysis of the scheme is provided and its accuracy is assessed through L2 and L∞ error norms over different spatial nodes and mesh refinements. The numerical results demonstrate excellent agreement with the exact solutions, as illustrated in the tables and figures. These findings confirm the robustness, efficiency and reliability of the proposed method for solving multi-term time-fractional differential equations.

Axioms doi: 10.3390/axioms15050327

Authors: Tsvetelin Zaevski Nikolay Kyurkchiev Anton Iliev Vesselin Kyurkchiev Asen Rahnev

Many authors consider modified SIQR (susceptible, infected, isolated (quarantined), and recovered individuals) models for childhood diseases. In this paper, we examine a modified differential system with N free parameters that may be of interest to epidemiology experts. We pay special attention to the Melnikov function, which corresponds to the proposed new model. We create the Melnikov equation (M(t)=0) and analyze all of its roots using a specially designed software program. This gives the researcher the chance to accurately comprehend and articulate the classical Melnikov criterion for the potential appearance of chaos in the dynamical system. Additionally, we present a few specific modules for examining the new model’s dynamics. Additionally covered is a potential use of the Melnikov function that corresponds to the differential model under consideration, with particular potential in the modeling and synthesis of antenna diagrams. Last but not least, we consider the proposed generalization from a stochastic point of view.

Axioms doi: 10.3390/axioms15050326

Authors: Burçin Oktay

Let G⊂C be a bounded simply connected domain with rectifiable Jordan boundary. Denote by ϕ the conformal map of the exterior of G onto the exterior of the unit disk. For R>1, let ΓR be the level curve defined by ϕ(z)=R, and let GR denote its interior, so that G⊂GR. Suppose that f is analytic in GR. In this paper, we investigate the maximal convergence properties of the Fourier series of f with respect to the Bergman orthogonal polynomials of G. By employing the strong asymptotics of Bergman polynomials outside the domain G of orthogonality, determined by the boundary properties of G, we obtain estimates for the maximal convergence rate of the partial sums of the Fourier series of f in the uniform norm on G¯. These estimates are expressed in terms of the best polynomial approximation of f in the domain GR where f is analytic.

Axioms doi: 10.3390/axioms15050325

Authors: Zehra Özdemir Esra Parlak Johan Gielis

The Gielis superformula is a powerful parametric tool that generates an infinite variety of natural and organic curves and surfaces through a compact set of parameters. However, classical differential geometry has lacked a unified framework for analyzing their curvature, torsion, and intrinsic geometric properties. This study addresses this gap by developing a novel superelliptic geometric framework that integrates the superformula with the differential geometry of curves and surfaces. We define the superelliptic inner and cross products, the star derivative, and the superelliptic Frenet frame to extend Euclidean and Riemannian interpretations of curvature and torsion to a more flexible parametric structure. The framework provides a uniform geometric characterization of all Gielis curves and surfaces in an intrinsic sense with respect to the proposed superelliptic metric, rather than relying on their classical Euclidean parametric representations; singular cases (e.g., n1<2), which correspond to non-smooth or corner-like behavior in the Euclidean setting due to degeneracies in the radial function r(t), are regularized within this framework, since the induced metric maps such Gielis-type curves to intrinsically circular geometries with constant superelliptic curvature. This unifies the entire family under a common, robust foundation while preserving orthonormality and differentiability. This superelliptic approach offers a consistent and computationally tractable model that bridges mathematical abstraction with real-world morphology, with the superformula serving as a representative example of the framework’s broad generality for diverse geometric structures. The proposed theoretical framework is further supported by computational visualization, and all figures and numerical illustrations presented in this study were generated using MATLAB R2024a, ensuring a consistent implementation of the proposed superelliptic model.

Axioms doi: 10.3390/axioms15050324

Authors: Ning Zhang Jingyang Chen Hao Chen Jingzhen Liu

Accurate mortality prediction is essential to actuarial practice as it is directly linked to insurance pricing, reserving, and the management of longevity risk. This study proposes a deep neural network (DNN) model for the mortality rates of multiple populations; it is composed of long short-term memory (LSTM) and convolutional neural network (CNN) components. As mortality trends evolve over long time horizons, and as capturing the complex dependencies among mortality rates across countries or regions with a linear model is challenging, the LSTM and CNN were applied to mortality modeling. The former can automatically learn long-term dependencies of sequential data, whereas the latter can extract local features from grid or sequential data. Formulated as a nonlinear generalization of the Lee–Carter decomposition, the model maps the log-mortality matrix logM to future logm(x,t) end-to-end and generates multi-step forecasts through dynamic recursive prediction. Then, the DNN and baseline models were used to fit mortality data of 21 countries from the Human Mortality Database (HMD), which were divided into training and test sets with the year 2000 as the split point. Extensive numerical experiments from the perspectives of accuracy, stability, and reliability of long-term forecasting revealed that DNN models yield better predictive performance, particularly the LSTM-CNN model. It combines the LSTM, CNN, and fully connected network (FCN) layers and thus exploits each deep neural network to fit nonlinear age, period, and cohort effects as well as their interactive terms to achieve better predictive performance. However, the CNN still outperformed other models for certain groups. In addition, the conclusions hold for remaining life expectancy.

Axioms doi: 10.3390/axioms15050323

Authors: Rafael Cubarsi

The harmonic structure of a chord C composed of rational frequency ratios is studied from its tonal graph. The graph describes harmoncity/periodicity properties of the chord allowing to characterize the chord from several parameters, which are easily visualized by the chord spheroid. Relevant harmonic structures are analyzed, such as self-symmetric graphs associated with harmonically symmetric chords. Recurrence relationships for the 1cm(C) in terms of the gcd’s of the powerset of C, and for the gcd(C) in terms of the lcm’s, are derived to unveil how the harmonic quotient lcm(C)/gcd(C) depends on the common undertones and overtones. In particular, for any number of tones, the harmonic quotient can be univocally expressed from the chord’s common undertones. Therefore, although undertones and overtones have not been explicitly taken into account in the current model, the harmonic quotient actually incorporates information about them.