Axioms, Volume 12, Issue 10 (October 2023) – 101 articles

In this paper we investigate the solutions of the so-called $\overline{\alpha }$-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and $(p,q)$-harmonic functions. Full article

In the explainable artificial intelligence (XAI) field, an algorithm or a tool can help people understand how a model makes a decision. And this can help to select important features to reduce computational costs to realize high-performance computing. But existing methods are usually used to visualize important features or highlight active neurons, and few of them show the importance of relationships between features. In recent years, some methods based on a white-box approach have taken relationships between features into account, but most of them can only work on some specific models. Although methods based on a black-box approach can solve the above problems, most of them can only be applied to tabular data or text data instead of image data. To solve these problems, we propose a local interpretable model-agnostic explanation approach based on feature relationships. This approach combines the relationships between features into the interpretation process and then visualizes the interpretation results. Finally, this paper conducts a lot of experiments to evaluate the correctness of relationships between features and evaluates this XAI method in terms of accuracy, fidelity, and consistency. Full article

The aim of this paper is to present and study nonlinear bivariate $C^1$ quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results. Full article

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained. Full article

In the literature, it is usually assumed that Leibniz described proof for the Fundamental Theorem of Calculus (FTC) in 1693. However, did he really prove it? If the answer is no from today’s perspective, are there works in which Leibniz introduced arguments that can be understood as formulations and justifications of the FTC? In order to answer this question, we used a historiographic methodology with expert triangulation. From the study of Leibniz’s manuscripts describing the inverse problem of tangents and its relationship with the quadrature problem, we found evidence of a geometrical argument from which the FTC can be inferred. We present this argument using technological resources and modern notation. This result can be used to teach the FTC due to the existence of dynamic and geometrical software, which makes it suitable for the classroom. Moreover, it provides another interpretation of the FTC complementary to the interpretation using Riemann sums. Full article

The space, ${Ƒ}_{cc}\left(\mathbb{R}\right)$, of all fuzzy intervals in $\mathbb{R}$ cannot form a vector space. However, the space ${Ƒ}_{cc}\left(\mathbb{R}\right)$ maintains a vector structure by treating the addition of fuzzy intervals as a vector addition and treating the scalar multiplication of fuzzy intervals as a scalar multiplication of vectors. The only difficulty in taking care of ${Ƒ}_{cc}\left(\mathbb{R}\right)$ is missing the additive inverse element. This means that each fuzzy interval that is subtracted from itself cannot be a zero element in ${Ƒ}_{cc}\left(\mathbb{R}\right)$. Although ${Ƒ}_{cc}\left(\mathbb{R}\right)$ cannot form a vector space, we still can endow a norm on the space ${Ƒ}_{cc}\left(\mathbb{R}\right)$ by following its vector structure. Under this setting, many different types of open sets can be proposed by using the different types of open balls. The purpose of this paper is to study the topologies generated by these different types of open sets. Full article

To quantitatively identify internal wire breakage damage in mining wire ropes, a wire rope internal wire breakage signal identification method is proposed. First, the whale optimization algorithm is used to find the optimal value of the variational mode decomposition parameter [$\mathrm{K},\mathsf{\alpha }]$ to obtain the optimal combination of the parameters, which reduces the signal noise with a signal-to-noise ratio of 29.29 dB. Second, the minimum envelope entropy of the noise reduction signal is extracted and combined with the time-domain features (maximum and minimum) and frequency-domain features (frequency–amplitude average, average frequency, average power) to form a fusion feature set. Finally, we use a particle swarm optimization–least squares support vector machine model to identify the internal wire breakage of wire ropes. The experimental results show that the method can effectively identify the internal wire rope breakage damage, and the average recognition rate is as high as 99.32%, so the algorithm can greatly reduce the system noise and effectively identify the internal damage signal of the wire rope, which is superior to a certain extent. Full article

In this paper, we investigate a delayed matrix exponential and utilize it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. To begin with, we present a novel definition of $\alpha$-exponential stability for these systems. Subsequently, we put forward several adequate conditions to ensure the $\alpha$-exponential stability of solutions for such delay systems. Moreover, by constructing a Grammian matrix that accounts for delays, we provide a criterion to determine the relative controllability of a linear problem. Additionally, we extend our analysis to nonlinear problems. Lastly, we offer several examples to verify the effectiveness of our theoretical findings. Full article

Several phenomena from natural sciences can be described by partial differential equations of Sobolev-type. On the other hand, it was shown that in many cases, the use of fractional derivatives provides a more realistic model than the use of standard derivatives. The goal of this paper is to study the nonexistence of weak solutions to a time-fractional differential inequality of Sobolev-type. Namely, we give sufficient conditions for the nonexistence or equivalently necessary conditions for the existence. Our method makes use of the nonlinear capacity method, which consists in making an appropriate choice of test functions in the weak formulation of the problem. This technique has been employed in previous papers for some classes of time-fractional differential inequalities of Sobolev-type posed on the whole space ${\mathbb{R}}^N$. The originality of this work is that the considered problem is posed on an annulus domain, which leads to some difficulties concerning the choice of adequate test functions. Full article

This paper considers nonparametric regression estimation with errors in the variables. It is a standard assumption that the characteristic function of the covariate error does not vanish on the real line. This assumption is rather strong. In this paper, we assume the covariate error distribution is a convolution of uniform distributions, the characteristic function of which contains zeros on the real line. Our regression estimator is constructed via the Laplace transform. We prove its strong consistency and show its convergence rate. It turns out that zeros in the characteristic function have no effect on the convergence rate of our estimator. Full article

The existence of at least three weak solutions to a discrete fractional boundary value problem containing a p-Laplacian operator and subject to perturbations is proved using variational methods. Some applications of the main results are presented. The results obtained generalize some recent results on both discrete fractional boundary value problems and p-Laplacian boundary value problems. Examples illustrating the results are given. Full article

This paper calculates numerical solutions of an extended three-coupled Korteweg–de Vries system by the q-homotopy analysis transformation method (q-HATM), which is a hybrid of the Laplace transform and the q-homotopy analysis method. Multiple investigations inspecting planetary oceans, optical cables, and cosmic plasma have employed the KdV model, significantly contributing to its development. The uniqueness, convergence, and maximum absolute truncation error of this algorithm are demonstrated. A numerical simulation has been performed to validate the accuracy and validity of the proposed approach. With high accuracy and few algorithmic processes, this algorithm supplies a series solution in the form of a recursive relation. Full article

In this paper, we have parameterized a timelike ($\mathcal{T}like$) circular surface ($\mathcal{CI}\mathit{surface}$) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a $\mathcal{T}like$ roller coaster surface ($\mathcal{RCO}\mathit{surface}$) to be a flat or minimal surface is examined in detail. Further, we illustrate the approach’s outcomes with a number of pertinent examples. Full article

In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is also constructed based on the idea. The numerical methods that can preserve the stability and asymptotical stability of the exact solutions are provided. Two numerical examples are provided to demonstrate the theoretical results. Full article

Let R be a commutative ring with identity, and $Z\left(R\right)$ be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by $W\Gamma \left(R\right)$ is an undirected (simple) graph with vertex set $Z{\left(R\right)}^{*}$, and two distinct vertices x and y are adjacent, if and only if there exist $r\in ann\left(x\right)$ and $s\in ann\left(y\right)$, such that $rs=0$. Importantly, it is worth noting that $W\Gamma \left(R\right)$ contains the zero-divisor graph $\Gamma \left(R\right)$ as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring ${\mathbb{Z}}_p\times {\mathbb{Z}}_t\times {\mathbb{Z}}_s$, denoted by $\mathrm{W}\Gamma ({\mathbb{Z}}_p\times {\mathbb{Z}}_t\times {\mathbb{Z}}_s)$, where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of $\mathrm{W}\Gamma ({\mathbb{Z}}_p\times {\mathbb{Z}}_t\times {\mathbb{Z}}_s)$. Full article

The purpose of the present study is to propose an interval-valued type 2 fuzzy set (IT2FS)-based analytic hierarchy process (AHP) framework to unfold the critical challenging factors influencing the sustenance and growth of the Indian tea industry. The current work follows an expert opinion-based group decision-making approach. The challenging factors have been identified through a literature review and finalized after a pilot study based on the opinions of professionals, consumers, and experts. Finally, the critical challenging factors and sub-factors have been figured out through analysis of the responses of the experts. To offset the subjective bias, an IT2FS-based granular analysis has been carried out. The findings reveal that market diversification and productivity are the central issues. Additionally, it is important to give attention to improving the quality of the products, increasing the use of modern technology and organic farming, and developing a variety of products. The result shows a considerable level of consistency in the group decision-making (CR < 0.1) for all pairwise comparisons. The present work shall be of use to formulate appropriate strategies and policy decisions. It shows a robust application of IT2FS-AHP for complex decision-making in real life. Full article

Agricultural product prices are subject to various uncertainties, including unpredictable weather conditions, pest infestations, and market fluctuations, which can significantly impact agricultural yields and productivity. Accurately assessing and understanding price is crucial for farmers, policymakers, and stakeholders in the agricultural sector to make informed decisions and implement appropriate risk management strategies. This study used the ordered weighted average (OWA) operator and its extensions as mathematical aggregation techniques incorporating ordered weights to capture and evaluate the factors influencing price variation. By generating different vectors related to different inputs to the traditional formulation, it is possible to aggregate information to calculate and provide a new view of the outcomes. The results of this research can help enhance risk management practices in agriculture and support decision-making processes to mitigate the adverse effects of price. Full article

In the pursuance of engineering excellence and sustainable practices, the optimization of material selection processes plays a crucial role. Using Fermatean fuzzy aggregation Operators (AOs), this study introduces an innovative method for improving material selection procedures. Combining the advantages of Fermatean fuzzy set (FrFS) and AOs, the proposed method enables a comprehensive evaluation of materials based on multiple criteria. The authors propose two operators: the “Fermatean fuzzy hybrid weighted arithmetic geometric aggregation (FrFHWAGA) operator” and the “Fermatean fuzzy hybrid ordered weighted arithmetic geometric aggregation (FrFHOWAGA) operator”. This method facilitates informed decision making in a number of industries by taking into account factors such as cost, durability, environmental impact, and availability. This research enables engineers, designers, and decision makers to optimize material selection, resulting in more efficient, cost-effective, and sustainable solutions across multiple domains. Full article

Currently, two-component integrable nonlinear equations from the hierarchies of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrödinger equation are being actively investigated. In this paper, we propose a new hierarchy of two-component integrable nonlinear equations, which have an important difference from the already known equations. To construct the hierarchical equations, we use the monodromy matrix method, as first proposed by B.A. Dubrovin. The method we use consists of solving the following sequence of problems. First, using the Lax operator, we find the monodromy matrix, which is a polynomial in the spectral parameter. More precisely, we find a sequence of monodromy matrices dependent on the degree of this polynomial. Each Lax operator has its own sequence of monodromy matrices. Then, using the terms from the decomposition of the monodromy matrix, we construct a sequence of second operators from a Lax pair. A hierarchy of evolutionary integrable nonlinear equations follows from the conditions of compatibility of the sequence of Lax pairs. Also, knowledge of the monodromy matrix allows us to find stationary equations that are analogs of the Novikov equations for the Korteweg–de Vries equation. In addition, the characteristic equation of the monodromy matrix corresponds to the spectral curve equation of the relevant multiphase solution for the integrable nonlinear equation. Since the coefficients of the spectral curve equation are integrals of the hierarchical equations, they can be utilized to find the simplest solutions of the constructed integrable nonlinear equations. In this paper, we demonstrate the operation of this method, starting with the assignment of the Lax operator and ending with the construction of the simplest solutions. Full article

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes. Full article

To significantly reduce the amplitude of longitudinal vibrations of the beam in the vicinity of its natural frequencies, a fundamentally new method of damping vibrations is used. For this purpose, the beam surfaces are covered with layers of polarized piezoceramics with a strong piezoelectric effect. We will use two types of electrical conditions on the electrodes of the piezoelectric layers: short-circuited electrodes and disconnected electrodes. On short-circuited electrodes, the electric potential is zero. As a result of the piezoelectric effect, an electric charge appears on the disconnected electrodes when the beam is deformed. The electroelastic state of a beam with different electrical conditions is described by different boundary value problems. A new approach to damping vibrations in the vicinity of natural frequencies is based on the following rule for controlling the dynamic characteristics of a structure: when the beam vibration frequency approaches its natural vibration frequency, we change the electrical conditions on the electrodes of the piezoelectric layers, thereby changing the spectrum of its natural frequencies. Let, for example, the vibration frequency of a beam with short-circuited electrodes approach its natural frequency. In this case, the amplitudes of the sought quantities grow without limit. The natural frequency spectrum of a beam with disconnected electrodes will differ from the spectrum of a beam with short-circuited electrodes. As a result, the amplitudes of the sought quantities will decrease. It is shown that the efficiency of vibration damping can be significantly increased by choosing the direction of the preliminary polarization of the piezoelectric material and the location of its electrodes. Numerical examples are given that demonstrate the effectiveness of the proposed method. The advantage of the method lies in its simplicity and the low cost of the piezoelectric material, which serves as a non-inertial damper. Full article

The need to find the global minimum in multivariable functions is a critical problem in many fields of science and technology. Effectively solving this problem requires the creation of initial solution estimates, which are subsequently used by the optimization algorithm to search for the best solution in the solution space. In the context of this article, a novel approach to generating the initial solution distribution is presented, which is applied to a genetic optimization algorithm. Using the k-means clustering algorithm, a distribution based on data similarity is created. This helps in generating initial estimates that may be more tailored to the problem. Additionally, the proposed method employs a rejection sampling algorithm to discard samples that do not yield better solution estimates in the optimization process. This allows the algorithm to focus on potentially optimal solutions, thus improving its performance. Finally, the article presents experimental results from the application of this approach to various optimization problems, providing the scientific community with a new method for addressing this significant problem. Full article

It is difficult to describe the hesitation and uncertainty of experts by single-valued information, and the differences in the importance of attributes are often ignored during the decision-making process. This paper introduces the probability and interval values into Fermatean hesitant fuzzy set (FHFS) and creatively proposes the probabilistic interval-valued Fermatean hesitant fuzzy set (PIVFHFS) to deal with information loss. This new fuzzy set allows decision makers to use interval-valued information with probability to express their quantitative evaluation, which broadens the range of information expression, effectively reflects the important degree of different membership degrees, and can describe uncertain information more completely and accurately. Under the probabilistic interval-valued Fermatean hesitant fuzzy environment, several new aggregation operators based on Hamacher operation are proposed, including the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher weighted averaging (PIVFHFHWA) operator and geometric (PIVFHFHWG) operator, and their basic properties and particular forms are studied. Then, considering the general correlation between different attributes, this paper defines the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher Choquet integral averaging (PIVFHFHCIA) operator and geometric (PIVFHFHCIG) operator and discusses related properties. Finally, a multi-attribute decision-making (MADM) method is presented and applied to the decision-making problem of reducing carbon emissions of manufacturers in the supply chain. The stability and feasibility of this method are demonstrated by sensitivity analysis and comparative analysis. The proposed new operators can not only consider the correlation between various factors but also express the preference information of decision makers more effectively by using probability, thus avoiding information loss in decision-making progress to some extent. Full article

The quantum-inspired genetic algorithm (QGA), which combines quantum mechanics concepts and GA to enhance search capability, has been popular and provides an efficient search mechanism. This paper proposes a modified QGA, called dynamic QGA (DQGA). The proposed algorithm utilizes a lengthening chromosome strategy for a balanced and smooth transition between exploration and exploitation phases to avoid local optima and premature convergence. Apart from that, a novel adaptive look-up table for rotation gates is presented to boost the algorithm’s optimization abilities. To evaluate the effectiveness of these ideas, DQGA is tested by various mathematical benchmark functions as well as real-world constrained engineering problems against several well-known and state-of-the-art algorithms. The obtained results indicate the merits of the proposed algorithm and its superiority for solving multimodal benchmark functions and real-world constrained engineering problems. Full article

Many phenomena can be described by random variables that follow asymmetrical distributions. In the context of regression, when the response variable Y follows such a distribution, it is preferable to estimate the response variable for predictor values using the conditional median. Quantile regression models can be employed for this purpose. However, traditional models do not incorporate a distributional assumption for the response variable. To introduce a distributional assumption while preserving model flexibility, we propose new varying-coefficients quantile regression models based on the family of log-symmetric distributions. We achieve this by reparametrizing the distribution of the response variable using quantiles. Parameter estimation is performed using a maximum likelihood penalized method, and a back-fitting algorithm is developed. Additionally, we propose diagnostic techniques to identify potentially influential local observations and leverage points. Finally, we apply and illustrate the methodology using real pollution data from Padre Las Casas city, one of the most polluted cities in Latin America and the Caribbean according to the World Air Quality Index Ranking. Full article

Efficient truck flow is essential for the efficient operation of a factory and the distribution of its products. This study demonstrated methods to improve truck loading times and overall efficiency at a major sugar manufacturing facility in the Middle East. The objective was to reduce truck waiting times at loading units and increase capacity. The data were collected through questionnaires, observations, and interviews with stakeholders. A simulation software was employed to analyze truck activity at loading stations at the factory. Multi-criteria decision making (MCDM) tools, AHP and TOPSIS, addressed five primary criteria and nine sub-criteria to assist in identifying, evaluating, and ranking feasible solutions. The study suggested different utilization of the various factory loading platforms at different times of the day. The findings from this study emphasize the importance of simulation-based approaches supplemented with decision-making processes to improve efficiency in sugar manufacturing facilities that may have broader applications in the factories of other industries. The study highlights remarkable improvements in operational efficiency, as seen in Alternative ₄ substantial 27.9% enhancement, resulting in cost savings and time efficiency. By implementing these findings, factories can enhance their truck flow management system, reduce waiting times, increase capacity utilization, optimize resource allocation, and improve overall efficiency. Full article

In this paper, we investigate the existence of positive solutions to a system of fractional differential equations that include the $(r_1,r_2,r_3)$-Laplacian operator, three-point boundary conditions, and various fractional derivatives. We use a combination of techniques, including cone expansion and compression of the functional type, and the Leggett–Williams fixed point theorem, to prove the existence of positive solutions. Finally, we provide two examples to illustrate our main results. Full article

Recently, there has been a lot of interest in comparative life testing for items under jointly censored schemes for products from multiple production lines. The inverse Weibull distribution (IWD) is commonly used in life testing and reliability theory. In this paper, we address the problem of statistical inference from comparative inverse Weibull distributions under joint samples. An adaptive type-II hybrid progressive censoring scheme (HPCS) is used to save the balance between the ideal test time and the number of observed failures. Under the adaptive type-II HPCS, unknown parameters of the inverse Weibull populations are estimated using both maximum likelihood and Bayesian approaches. Asymptotic confidence intervals are established using the observed Fisher information matrix and bootstrap confidence intervals. We suggest using Markov chain Monte Carlo (MCMC) techniques to compute credible intervals under independent gamma priors. Using Monte Carlo simulations, all theoretical conclusions are tested and contrasted. For illustration purposes, an actual sample from comparative populations is analysed. Full article