Axioms, Volume 13, Issue 6 (June 2024) – 59 articles

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. Therefore, it presents an alternative for modeling this type of data. Several mathematical and statistical properties of this distribution have been derived, such as moments, hazard function, the Bonferroni curve, and entropy. Moreover, we investigate the characterizations of the proposed distribution based on its hazard function. Parameter estimation has been performed using both the maximum likelihood method and method of the moments. Because of this, we were able to determine the best critical region and the information matrix, facilitating the calculation of asymptotic confidence intervals. A simulation study is presented to analyze the behavior of the obtained estimators for different sample sizes. To demonstrate the suitability of the proposed distribution, applications and goodness-of-fit tests have been performed on two practical data sets. Full article

Sphere packing consists of placing several spheres in a container without mutual overlapping. While packing into regular-shape containers is well explored, less attention is focused on containers with nonlinear boundaries, such as ellipsoids or paraboloids. Packing n-dimensional spheres into a minimum-height container bounded by a parabolic surface is formulated. The minimum allowable distances between spheres as well as between spheres and the container boundary are considered. A normalized Φ-function is used for analytical description of the containment constraints. A nonlinear programming model for the packing problem is provided. A solution algorithm based on the feasible directions approach and a decomposition technique is proposed. The computational results for problem instances with various space dimensions, different numbers of spheres and their radii, the minimal allowable distances and the parameters of the parabolic container are presented to demonstrate the efficiency of the proposed approach. Full article

This article introduces the concept of generalized $(ff\mathcal{F},\mathtt{b},\breve{\varphi })$ contraction in the context of $\mathtt{b}$-metric spaces by utilizing the idea of $\mathcal{F}$ contraction introduced by Dariusz Wardowski. The main findings of the research focus on the existence of best proximity points for multi-valued $(ff\mathcal{F},\mathtt{b},\breve{\varphi })$ contractions in partially ordered $\mathtt{b}$-metric spaces. The article provides examples to illustrate the main results and demonstrates the existence of solutions to a second-order differential equation and a fractional differential equation using the established theorems. Additionally, several corollaries are presented to show that the results generalize many existing fixed-point and best proximity point theorems. Full article

This paper explores the inference for a constant-stress accelerated life test under a ranked set sampling scenario. When the lifetime of products follows the Fréchet distribution, and the failure times are collected under a maximum ranked set sampling with unequal samples, classical and Bayesian approaches are proposed, respectively. Maximum likelihood estimators along with the existence and uniqueness of model parameters are established, and the corresponding asymptotic confidence intervals are constructed based on asymptotic theory. Under squared error loss, Bayesian estimation and highest posterior density confidence intervals are provided, and an associated Monte-Carlo sampling algorithm is proposed for complex posterior computation. Finally, extensive simulation studies are conducted to demonstrate the performance of different methods, and a real-data example is also presented for applications. Full article

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study. Full article

Monitoring the parameter of discrete distributions is common in industrial production. Also, it is often crucial to monitor the parameter of geometric distribution, which is often regarded as the nonconforming item rate. To enhance the detection of nonconforming item, we designed an exponentially weighted moving average (EWMA) scheme based on the weighted likelihood ratio test (WLRT) method, and this scheme is denoted as the EWLRT scheme, specifically designed for monitoring the increase of the parameter in geometric distribution. Moreover, the optimal statistical design of the EWLRT scheme is presented when the shift is known. Results from numerical comparisons through Monte Carlo simulations indicates that the EWLRT scheme performs better than the competing schemes in some scenarios. Additionally, the designed scheme is characterized by its simplicity and ease of use, making it ideally suited for scenarios involving single observation. An example is illustrated to demonstrate the effectiveness of the EWLRT scheme. Full article

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation. Full article

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently,none provided a good shape for fast-contracting polyhedral control nets. This work improves the state-of-the-art of piecewise polynomial spline surfaces accommodating fast-contracting control nets. The new fast-contracting (FC) surface algorithm yields the industry-preferred uniform degree bi-3 (bi-cubic). The surfaces are by default differentiable, have an improved shape, measured empirically as to highlight the line distribution, and require fewer pieces compared to existing methods. Full article

This paper proposes a precise motion control strategy for a three-wheeled mobile robot with two driven rear wheels and one steered front wheel so that an obstacle avoidance motion task is able to be well implemented. Initially, the motion laws under nonholonomic constraints are expounded for the three-wheeled mobile robot in order to facilitate the derivation of its dynamic model. Subsequently, a prescribed target curve is converted into a speed target through the nonholonomic constraint of zero lateral speed. A modified dynamical tracking target that is aligned with the dynamic model is then developed based on the relative curvature of the prescribed curve. By applying this dynamical tracking target, path tracking precision is enhanced through appropriate selection of a yaw motion speed target, thus preventing speed errors from accumulating during relative curvature tracking. On this basis, integral sliding mode control and feedback linearization methods are adopted for designing robust controllers, enabling the accurate movement of the three-wheeled mobile robot along a given path. A theoretical analysis and simulation results corroborate the effectiveness of the proposed trajectory tracking control strategy in preventing off-target deviations, even with significant speed errors. Full article

Graph anomaly detection aims to identify unusual patterns or structures in graph-structured data. Most existing research focuses on anomalous nodes in ordinary graphs with pairwise relationships. However, complex real-world systems often involve relationships that go beyond pairwise relationships, and insufficient attention is paid to hypergraph anomaly detection, especially anomalous hyperedge detection. Some existing methods for researching hypergraphs involve transforming hypergraphs into ordinary graphs for learning, which can result in poor detection performance due to the loss of high-order information. We propose a new method for Anomalous Hyperedge Detection on Symmetric Line Expansion (AHD-SLE). The SLE of a hypergraph is an ordinary graph with pairwise relationships and can be backmapped to the hypergraph, so the SLE is able to preserve the higher-order information of the hypergraph. The AHD-SLE first maps the hypergraph to the SLE; then, the information is aggregated by Graph Convolutional Networks (GCNs) in the SLE. After that, the hyperedge embedding representation is obtained through a backmapping operation. Finally, an anomaly function is designed to detect anomalous hyperedges using the hyperedge embedding representation. Experiments on five different types of real hypergraph datasets show that AHD-SLE outperforms the baseline algorithm in terms of Area Under the receiver operating characteristic Curve(AUC) and Recall metrics. Full article

One of the most relevant topics in web theory is linearization. A particular class of linearizable webs is the Grassmannizable web. Akivis gave a characterization of such a web, showing that Grassmannizable webs are equivalent to isoclinic and transversally geodesic webs. The obstructions given by Akivis that characterize isoclinic and transversally geodesic webs are computed locally, and it is difficult to give them an interpretation in relation to torsion or curvature of the unique Chern connection associated with a web. In this paper, using Nagy’s web formalism, Frölisher—Nejenhuis theory for derivation associated with vector differential forms, and Grifone’s connection theory for tensorial algebra on the tangent bundle, we find invariants associated with almost-Grassmann structures expressed in terms of torsion, curvature, and Nagy’s tensors, and we provide an interpretation in terms of these invariants for the isoclinic, transversally geodesic, Grassmannizable, and parallelizable webs. Full article

This paper establishes a novel technique, which is called the G-double-Laplace transform. This technique is an extension of the generalized Laplace transform. We study its properties with examples and various theorems related to the G-double-Laplace transform that have been addressed and proven. Finally, we apply the G-double-Laplace transform decomposition method to solve the nonlinear sine-Gordon and coupled sine-Gordon equations. This method is a combination of the G-double-Laplace transform and decomposition method. In addition, some examples are examined to establish the accuracy and effectiveness of this technique. Full article

This paper investigates the state estimation and event-triggered control for positive nonlinear multi-agent systems. Firstly, a proportional–integral observer is established to estimate the states of the considered nonlinear positive multi-agent systems based on the matrix decomposition method. Then, a dynamic event-triggered mechanism is constructed, and a control protocol is proposed based on the proportional–integral observer and event-triggered mechanism. By combining linear programming with linear co-positive Lyapunov functions, the considered multi-agent systems are guaranteed to be positive and achieve consensus. Moreover, by introducing three new variables and a finite vector, the final convergence point can be changed based on the given vector. Finally, two illustrative examples demonstrate the validity of the proposed theoretical results. Full article

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ${▵}^{2}u=f\left(\right|x|,u,|\nabla u|,▵u)$ in an annular domain $\Omega =\{x\in {\mathbb{R}}^N:r_1<|x|<r_2\}$($N\ge 2$) with the boundary conditions ${u|}_{\partial \Omega }=0$ and ${▵u|}_{\partial \Omega }=0$, where $f:[r_1,r_2]\times \mathbb{R}\times {\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R}$ is continuous. Under certain inequality conditions on f involving the principal eigenvalue ${\lambda }_1$ of the Laplace operator $-▵$ with boundary condition ${u|}_{\partial \Omega }=0$, an existence result and a uniqueness result are obtained. The inequality conditions allow for $f(r,\xi ,\zeta ,\eta )$ to be a superlinear growth on $\xi ,\zeta ,\eta$ as $\left|\right(\xi ,\zeta ,\eta \left)\right|\to \infty$. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates. Full article

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and $\left\{b\right\}$-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where $d\ge 2$, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable. Full article

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group ${\mathrm{GL}}_3\left({\mathbb{F}}_q\right).$ By considering a cubic extension of the finite field ${\mathbb{F}}_q$, we define an analog of the upper half-space and an action of ${\mathrm{GL}}_3\left({\mathbb{F}}_q\right)$ on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation $\rho ={\mathrm{Ind}}_{\Gamma }^{G}1$ for $G={\mathrm{GL}}_3\left({\mathbb{F}}_q\right)$ and $\Gamma ={\mathrm{GL}}_3\left({\mathbb{F}}_p\right).$ Full article

In this paper, we define a cohomology theory of a modified $\lambda$-differential left-symmetric algebra. Moreover, we introduce the notion of modified $\lambda$-differential left-symmetric 2-algebras, which is the categorization of a modified $\lambda$-differential left-symmetric algebra. As applications of cohomology, we classify linear deformations and abelian extensions of modified $\lambda$-differential left-symmetric algebras using the second cohomology group and classify skeletal modified $\lambda$-differential left-symmetric 2-algebra using the third cohomology group. Finally, we show that strict modified $\lambda$-differential left-symmetric 2-algebras are equivalent to crossed modules of modified $\lambda$-differential left-symmetric algebras. Full article

A variety of process capability indices are applied to the quantitative measurement of the potential and performance of processes in manufacturing. As it is easy to understand the formulae of these indices, this method is easy to apply. Furthermore, a process capability index is frequently utilized by a manufacturer to gauge the quality of a process. This index can be utilized by not only an internal process engineer to assess the quality of the process but also as a communication tool for an external sales department. When the manufacturing process deviates from the target value T, the process capability index C_PMK can be quickly detected, which is conducive to the promotion of smart manufacturing. Therefore, this study applied the index C_PMK as an evaluation tool for process quality. As noted by some studies, process capability indices have unknown parameters and therefore must be estimated from sample data. Additionally, numerous studies have addressed that it is essential for companies to establish a rapid response mechanism, as they wish to make decisions quickly when using a small sample size. Considering the small sample size, this study proposed a 100 (1 − α)% confidence interval for the process capability index C_PMK based on suggestions from previous studies. Subsequently, this study built a fuzzy testing model on the 100 (1 − α)% confidence interval for the process capability index C_PMK. This fuzzy testing model can help enterprises make decisions rapidly with a small sample size, meeting their expectation of having a rapid response mechanism. Full article

Under the condition of non-classical distributed errors, the test for spatial dependence in spatial panel data models is still a problem waiting to be solved. In this paper, we apply the FDB (Fast Double Bootstrap) method to spatial panel data models to test spatial dependence. In order to research the validity of the Bootstrap LM-Error test in spatial random effect models under the condition that the error term obeys a normal distribution, heteroscedasticity, or time-series correlation, we construct Bootstrap LM-Error statistics and make use of Monte Carlo simulation from size distortion and power aspects to carry out our research. The Monte Carlo simulation results show that the asymptotic LM-Error test in the spatial random effects model has a large size of distortion when the error term disobeys classical distribution. However, the FDB LM-Error test can effectively correct the size distortion of the asymptotic test with the precondition that there is nearly no loss of power in the FDB test. Obviously, compared to the asymptotic LM-Error test, the FDB LM-Error test is a more valid method to test spatial dependence in a spatial random effects model. Full article

This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions. Full article

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given. Full article

The edge DP-chromatic number of G, denoted by ${\chi }_{DP}^{\prime }\left(G\right)$, is the minimum k such that G is edge DP-k-colorable. In 1999, Juvan, Mohar, and Thomas proved that the edge list chromatic number of $K_4$-minor free graph G with $\Delta \ge 3$ is $\Delta$. In this paper, we prove that if G is a $K_4$-minor free graph, then ${\chi }_{DP}^{\prime }\left(G\right)\in \{\Delta ,\Delta +1\}$, and equality ${\chi }_{DP}^{\prime }\left(G\right)=\Delta +1$ holds for some $K_4$-minor free graph G with $\Delta =3$. Moreover, if G is a planar graph with $\Delta \ge 9$ and with no intersecting triangles, then ${\chi }_{DP}^{\prime }\left(G\right)=\Delta$. Full article

The problem of inverting dynamic complex matrices remains a central and intricate challenge that has garnered significant attention in scientific and mathematical research. The zeroing neural network (ZNN) has been a notable approach, utilizing time derivatives for real-time solutions in noiseless settings. However, real-world disturbances pose a significant challenge to a ZNN’s convergence. We design an accelerated dual-integral structure zeroing neural network (ADISZNN), which can enhance convergence and restrict linear noise, particularly in complex domains. Based on the Lyapunov principle, theoretical analysis proves the convergence and robustness of ADISZNN. We have selectively integrated the SBPAF activation function, and through theoretical dissection and comparative experimental validation we have affirmed the efficacy and accuracy of our activation function selection strategy. After conducting numerous experiments, we discovered oscillations and improved the model accordingly, resulting in the ADISZNN-Stable model. This advanced model surpasses current models in both linear noisy and noise-free environments, delivering a more rapid and stable convergence, marking a significant leap forward in the field. Full article

The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map $\alpha$ and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general $(n+1)$-dimensional n-Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. Full article

A simple dynamic system with dry friction is studied theoretically and numerically. Models of systems including dry friction are not easily obtained, as defining the relationship between the friction force and the relative velocity presents a significant challenge. It is known that friction forces exhibit notable discontinuities when there is a change in the direction of motion. Additionally, when the relative motion ceases, the friction force can assume any value within a certain range. In the literature, numerous models of dry friction are presented, and most of them assume a biunivocal dependency of the friction force with respect to relative velocity. The dynamic system considered here is a tilted rod with spherical ends, initially at rest. Dry friction forces are evident at the contact point with the horizontal plane. The ball–plane contact highlights the rolling friction or/and sliding friction. The problem is theoretically solved after adopting one of the two cases of friction: rolling friction or sliding friction. The nonlinear differential equations of motion have been derived, along with expressions for the magnitude of the normal reaction and the friction force. The results of the model are displayed graphically for three different sets of values for the coefficient of friction. It is revealed that there is a critical value of the coefficient of friction that determines the transition from rolling to sliding regimes. To validate the theoretical model, dynamic simulation software was utilised. The excellent match between the theoretical predictions and the results from the numerical simulation confirms the accuracy of the proposed analytical solution. Full article

In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local $ϵ$-quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan’s lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. Full article

In this article, we study the Ricci soliton on slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Moreover, we derive a lower-bound-type inequality for the slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection in terms of gradient Ricci solitons. We also characterize anti-invariant, invariant, quasi-umbilical submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection for which the same inequality case holds. Finally, we deduce the above inequalities in terms of a scalar concircular field on submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Full article

We give the Karhunen–Loève expansion of the covariance function of a family of discrete weighted Brownian bridges, appearing as discrete analogues of continuous Gaussian processes related to Cramér –von Mises and Anderson–Darling statistics. This analogy enables us to introduce a discrete Cramér–von Mises statistic and show that this statistic satisfies a property of local asymptotic Bahadur optimality for a statistical test involving the classical hypergeometric distributions. Our statistic and the goodness-of-fit problem we deal with are based on basic properties of Hahn polynomials and are, therefore, subject to some extension to all families of classical orthogonal polynomials, as well as their q-analogues. Due probably to computational difficulties, the family of discrete Cramér–von Mises statistics has received less attention than its continuous counterpart—the aim of this article is to bridge part of this gap. Full article

Economic risk is a probability that measures the possible alterations, as well as the uncertainty, generated by multiple internal or external factors. Sometimes it could cause the impossibility of guaranteeing the level of compliance with organizational goals and objectives, which is why for their administration they are frequently divided into multiple categories according to their consequences and impact. Global indicators are dynamic and sometimes the correlation is uncertain because they depend largely on a combination of economic, social, and environmental factors. Thus, our proposal consists of a model for prediction and classification of multivariate risk factors such as birth rate and population growth, among others, using multiple neural networks and General Type-2 fuzzy systems. The contribution is the proposal to integrate multiple variables of several time series using both supervised and unsupervised neural networks, and a generalized Type-2 fuzzy integration. Results show the advantages of utilizing the method for the fuzzy integration of multiple time series attributes, with which the user can then prevent future threats from the global environment that impact the scheduled compliance process. Full article