Axioms, Volume 13, Issue 1 (January 2024) – 72 articles

The vertex quadrangulation $QG$ of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the question was posed: does the spectrum of an arbitrary unweighted graph $QG$ include the full spectrum $\{3,{(-1)}^3\}$ of the tetrahedron graph (complete graph $K_4$)? Previously, many bipartite and nonbipartite graphs $QG$ with such a subspectrum have been found; for example, a nonbipartite variant of the graph $Q{K}_5$. Here, we present one of the variants of the nonbipartite vertex quadrangulation $QO$ of the octahedron graph O, which has eigenvalue $(-1)$ of multiplicity 2 in the spectrum, while the spectrum of the bipartite variant $QO$ contains eigenvalue $(-1)$ of multiplicity 3. Thus, in the case of nonbipartite graphs, the answer to the question posed depends on the particular graph $QG$. Here, we continue to explore the spectrum of graphs $QG$. Some possible connections of the mathematical theme to chemistry are also noted. Full article

In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function ${\zeta }^p$ in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta–Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method. Full article

A recent paper introduced the interpolating family (IF) of distributions, and they also derived various mathematical properties of the family. Some of the most important properties discussed were the integer order moments of the IF distributions. The moments were expressed as an integral (which were not evaluated) or as finite sums of the beta function. In this paper, more general expressions for moments of any integer order or any real order are derived. Apart from being more general, our expressions converge for a wider range of parameter values. The expressions for entropies are also derived, the maximum likelihood estimation is considered and the finite sample performance of maximum likelihood estimates is investigated. Full article

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems. Full article

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of $L^p$. Some known results in the literature are extended. Full article

The variation analysis is a key tool for ensuring the high quality assembly in the process of developing the technology for manufacturing of aircraft parts. One of the main factors in variations is the deviations in the positioning procedure. This paper is devoted to the development of an approach that allows taking into account the variations during positioning and merging it with the special algorithm of contact problem solving. The impact of varied boundary conditions is incorporated into an additional vector of forces that can be interpreted as reactions to the shift of supports. The obtained results are illustrated with a case of wing-to-fuselage assembly. Full article

The Sombor index is defined as $SO\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{d^2\left(u\right)+d^2\left(v\right)}$, where $d\left(u\right)$ and $d\left(v\right)$ represent the number of edges in the graph G connected to the vertices u and v, respectively. In this paper, we characterize the largest and second largest Sombor indexes with a given number of cut edges. Moreover, we determine the upper and lower sharp bounds of the Sombor index with a given number of clique numbers, and we characterize the extremal graphs. Full article

This study explores the problem of describing viscous fluid motion for Navier–Stokes equations in curved channels, which is important in applications like hemodynamics and pipeline transport. Channel curvature leads to vortex flows and closed vortex zones. Asymptotic models of the flux problem are useful for describing viscous fluid motion in long pipes, thus considering geometric parameters like pipe diameter and characteristic length. This study provides a representation for the vorticity vector and energy dissipation in the flow problem for a curved channel, thereby determining the magnitude of vorticity and energy dissipation depending on the channel’s central line curvature and torsion. The accuracy of the asymptotic formulas are estimated in terms of small parameter powers. Numerical calculations for helical tubes demonstrate the effectiveness of the asymptotic formulas. Full article

In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations. Full article

In this work, we propose an algorithm for the detection, measurement and classification of discontinuities in signals captured with noise. Our approach is based on the Harten’s subcell-resolution approximation adapted to the presence of noise. This technique has several advantages over other algorithms. The first is that there is a theory that allows us to ensure that discontinuities will be detected as long as we choose a sufficiently small discretization parameter size. The second is that we can consider different types of discretizations such as point values or cell-averages. In this work, we will consider the latter, as it is better adapted to functions with small oscillations, such as those caused by noise, and also allows us to find not only the discontinuities of the function, jumps in functions or edges in images, but also those of the derivative, corners. This also constitutes an advantage over classical procedures that only focus on jumps or edges. We present an application related to heart rate measurements used in sport as a physical indicator. With our algorithm, we are able to identify the different phases of exercise (rest, activation, effort and recovery) based on heart rate measurements. This information can be used to determine the rotation timing of players during a game, identifying when they are in a rest phase. Moreover, over time, we can obtain information to monitor the athlete’s physical progression based on the slope size between zones. Finally, we should mention that regions where heart rate measurements are abnormal indicate a possible cardiac anomaly. Full article

This paper focuses on the modeling of automatic pricing and replenishment strategies for perishable products with time-varying deterioration rates based on an improved SVR-LSTM-ARIMA hybrid model. This research aims to support supermarkets in planning future strategies, optimizing category structure, reducing loss rates, and improving profit margins and service quality. Specifically, the paper selects perishable vegetables as the research category and calculates the cost-plus ratio for each vegetable category. Correlation analysis is conducted with total sales, and a non-parametric relationship curve is obtained using support vector regression for nonlinear fitting. The long and short memory recurrent neural network is then used to predict sales volume, and a pricing strategy is calculated based on the fitting curve. Additionally, the paper establishes a correlation between loss rate and shelf life, corrects the daily average sales volume index, and solves the problem of quantity and category of replenishment using a backpack problem approach. By considering multiple constraints, a quantitative category replenishment volume and pricing strategy is obtained. The mathematical model proposed in this paper addresses the replenishment and pricing challenges faced by supermarkets, aiming to improve revenue and reduce loss while meeting market requirements. Full article

In the theory of gene networks, the mathematical apparatus that uses dynamical systems is fruitfully used. The same is true for the theory of neural networks. In both cases, the purpose of the simulation is to study the properties of phase space, as well as the types and the properties of attractors. The paper compares both models, notes their similarities and considers a number of illustrative examples. A local analysis is carried out in the vicinity of critical points and the necessary formulas are derived. Full article

Differential cohomology is a topic that has been attracting considerable interest. Many interesting applications in mathematics and physics have been known, including the description of WZW terms, string structures, the study of conformal immersions, and classifications of Ramond–Ramond fields, to list a few. Additionally, it is an interesting application of the theory of infinity categories. In this paper, we give an expository account of differential cohomology and the classification of higher line bundles (also known as $S^1$-banded gerbes) with a connection.We begin with how Čech cohomology is used to classify principal bundles and define their characteristic classes, introduce differential cohomology à la Cheeger and Simons, and introduce $S^1$-banded gerbes with a connection. Full article

Differential evolution (DE) is one of the most promising black-box numerical optimization methods. However, DE algorithms suffer from the problem of control parameter settings. Various adaptation methods have been proposed, with success history-based adaptation being the most popular. However, hand-crafted designs are known to suffer from human perception bias. In this study, our aim is to design automatically a parameter adaptation method for DE with the use of the hyper-heuristic approach. In particular, we consider the adaptation of scaling factor F, which is the most sensitive parameter of DE algorithms. In order to propose a flexible approach, a Taylor series expansion is used to represent the dependence between the success rate of the algorithm during its run and the scaling factor value. Moreover, two Taylor series are used for the mean of the random distribution for sampling F and its standard deviation. Unlike most studies, the Student’s t distribution is applied, and the number of degrees of freedom is also tuned. As a tuning method, another DE algorithm is used. The experiments performed on a recently proposed L-NTADE algorithm and two benchmark sets, CEC 2017 and CEC 2022, show that there is a relatively simple adaptation technique with the scaling factor changing between 0.4 and 0.6, which enables us to achieve high performance in most scenarios. It is shown that the automatically designed heuristic can be efficiently approximated by two simple equations, without a loss of efficiency. Full article

Firstly, Hilbert-type integral inequalities with best constant factors are established for two non-homogeneous kernels. Then, by utilizing the relationship between the Hilbert-type inequality and the integral operator of same kernel, the parameter conditions for two integral operators with non-homogeneous kernels in weighted Lebesgue space to be bounded and the formula for calculating the operator norm are obtained. Full article

In this paper, we introduce a new three-step iterative scheme for finding the common solutions of the variational inequality using the technique of updating the solution. We suggest, iterative algorithms involving three-steps for the predictor-corrector method of variational inequality in real Hilbert spaces H. Our results include the Takahashi and Toyoda, extra gradient, Mann and Noor iterations as special cases. We also investigate the convergence criteria of the three-step iterative scheme. As special cases, the earlier findings are included in our results, which can be seen as an advancement and improvement over the previous investigation. This is a new refinement in our existing literature and previously known algorithms. A numerical example is given to illustrate the efficiency and performance of the proposed self-adaptive scheme. Full article

Economic forecasting is crucial since it benefits many different parties, such as governments, businesses, investors, and the general public. This paper presents a novel methodology for forecasting business cycles using an autoregressive integrated moving average (ARIMA), a popular linear model in time series forecasting, and a neural network with weighted fuzzy membership functions (NEWFM) as a forecasting model generator. The study used a dataset that included seven components of the leading composite index, which is used to predict positive or negative trends in several economic sectors before the GDP is compiled. The preprocessed time series data comprising the leading composite index using ARIMA were used as input vectors for the NEWFM to predict comprehensive business fluctuations. The prediction capability significantly improved through the duplicated refining process of the dataset using ARIMA and NEWFM. The combined ARIMA and NEWFM techniques exceeded ARIMA in both classification and prediction, yielding an accuracy of 91.61%. Full article

As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are ${\mathcal{R}}^{+}$-simple and stable. Using this result, the parametric differential equation models in electrical power systems and engineering can be studied. Full article

With the vigorous development of emerging technology and the advent of the Internet generation, high-speed Internet and fast transmission 5G wireless networks contribute to interpersonal communication. Now, the Internet has become popular and widely available, and human life is inseparable from various experiences on the Internet. Many base stations and data centers have been established to convert and switch from electrical transmission to optical transmission; thus, it is entering the new era of optical fiber networks and optical communication technologies. For optical communication, the manufacturing of components for the purpose of high-speed networks is a key process, and the requirement for the stability of its production conditions is very strict. In particular, product yields are always low due to the restriction of high-precision specifications associated with the limitations of too many factors. Given these reasons, this study proposes a hybrid fuzzy control-based model for industry data applications to organize advanced techniques of box-and-whisker plot method, association rule, and decision trees to find out the determinants that affect the yield rate of products and then use the fuzzy control Proportional-Integral-Derivative (PID) method to manage the determinants. Since it is unrealistic to test the real machine online operation at the manufacturing stage, the simulation software supersedes this for improved results, and a mathematical neural network is used to verify the given data to confirm whether its result is similar to that of the simulation. The study suggests that excessive temperature differentials between substrate and cavity can lead to low yields. It suggests using fuzzy control technology for temperature management, which could increase yield, reduce labor costs, and accelerate the transition to high-speed networks by mass-producing high-precision optical filters. Full article

In this paper, we introduce the “Walk on Equations” (WE) Monte Carlo algorithm, a novel approach for solving linear algebraic systems. This algorithm shares similarities with the recently developed WE MC method by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. This method is particularly effective for large matrices, both real- and complex-valued, and shows significant improvements over traditional methods. Our comprehensive comparison with the Gauss–Seidel method highlights the WE algorithm’s superior performance, especially in reducing relative errors within fewer iterations. We also introduce a unique dominancy number, which plays a crucial role in the algorithm’s efficiency. A pivotal outcome of our research is the convergence theorem we established for the WE algorithm, demonstrating its optimized performance through a balanced iteration matrix. Furthermore, we incorporated a sequential Monte Carlo method, enhancing the algorithm’s efficacy. The most-notable application of our algorithm is in solving a large system derived from a finite-element approximation in constructive mechanics, specifically for a beam structure problem. Our findings reveal that the proposed WE Monte Carlo algorithm, especially when combined with sequential MC, converges significantly faster than well-known deterministic iterative methods such as the Jacobi method. This enhanced convergence is more pronounced in larger matrices. Additionally, our comparative analysis with the preconditioned conjugate gradient (PCG) method shows that the WE MC method can outperform traditional methods for certain matrices. The introduction of a new random variable as an unbiased estimator of the solution vector and the analysis of the relative stochastic error structure further illustrate the potential of our novel algorithm in computational mathematics. Full article

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element $f\in X$. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence $\left\{u_n\right\}\subset X$, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws. Full article

In this paper, we investigate a nonlinear coupled integro-differential system involving generalized Hilfer fractional derivative operators ($(k,\psi )$-Hilfer type) of different orders and equipped with non-local multi-point ordinary and fractional integral boundary conditions. The uniqueness results for the given problem are obtained by applying Banach’s contraction mapping principle and the Boyd–Wong fixed point theorem for nonlinear contractions. Based on the Laray–Schauder alternative and the well-known fixed-point theorem due to Krasnosel’skiĭ, the existence of solutions for the problem at hand is established under different criteria. Illustrative examples for the main results are constructed. Full article

We study two quite different types of Terracini loci for the order d-Veronese embedding of an n-dimensional projective space: the minimal one and the primitive one (defined in this paper). The main result is that if $n=4$, $d\ge 19$ and $x\le 2d$, no subset with x points is a minimal Terracini set. We give examples that show that the result is sharp. We raise several open questions. Full article

Applications of gradient method for nonlinear optimization in development of Gradient Neural Network (GNN) and Zhang Neural Network (ZNN) are investigated. Particularly, the solution of the matrix equation $AXB=D$ which changes over time is studied using the novel GNN model, termed as GGNN$(A,B,D)$. The GGNN model is developed applying GNN dynamics on the gradient of the error matrix used in the development of the GNN model. The convergence analysis shows that the neural state matrix of the GGNN$(A,B,D)$ design converges asymptotically to the solution of the matrix equation $AXB=D$, for any initial state matrix. It is also shown that the convergence result is the least square solution which is defined depending on the selected initial matrix. A hybridization of GGNN with analogous modification GZNN of the ZNN dynamics is considered. The Simulink implementation of presented GGNN models is carried out on the set of real matrices. Full article

A Hybrid Quantum Genetic Algorithm with Fuzzy Adaptive Rotation Angle (HQGAFARA) is introduced in this work to determine the optimal placements for Unmanned Aerial Vehicles (UAVs) aimed at maximizing coverage in disaster-stricken areas. The HQGAFARA is a hybrid quantum fuzzy meta-heuristic that uses the Deutsch–Jozsa quantum circuit to generate quantum populations synergistically working as haploid recombination and mutation operators that take advantage of quantum entanglement, providing exploitative and explorative features to produce new individuals. In place of the conventional lookup table or mathematical equation, we introduced a fuzzy heuristic to adapt the rotation angle employed in quantum gates. The hybrid nature of this algorithm becomes evident through its utilization of both classical and quantum computing components. Experimental evaluations were conducted using two distinct test sets. The first set, termed the “best case”, represents conditions that are the most favorable for determining the UAV positions, while the second set, the “worst-case”, simulates highly challenging conditions for locating the UAV positions, thereby posing a significant test for the proposed algorithm. We carried out statistical comparative analyses, assessing the HQGAFARA against other hybrid quantum algorithms that employ different rotation angles and against the classical genetic algorithm. The experimental results demonstrated that the HQGAFARA performed comparably, if not better, to the classical genetic algorithm regarding precision. Furthermore, quantum algorithms showcased their computational prowess in experiments related to the convergence time. Full article

This tutorial delves into the application of proportional odds-type ordinal logistic regression to assess the impact of incorporating both fixed and random effects when predicting the rankings of Brazilian universities in a well-established international academic assessment utilizing authentic data. In addition to offering valuable insights into the estimation of ordinal logistic models, this study underscores the significance of integrating random effects into the analysis and addresses the potential pitfalls associated with the inappropriate treatment of phenomena exhibiting categorical ordinal characteristics. Furthermore, we have made the R language code and dataset available as supplementary resources for the replication. Full article

This study introduces a new two-parameter Liu estimator (PMTPLE) for addressing the multicollinearity problem in the Poisson regression model (PRM). The estimation of the PRM is traditionally accomplished through the Poisson maximum likelihood estimator (PMLE). However, when the explanatory variables are correlated, thus leading to multicollinearity, the variance or standard error of the PMLE is inflated. To address this issue, several alternative estimators have been introduced, including the Poisson ridge regression estimator (PRRE), Liu estimator (PLE), and adjusted Liu estimator (PALE), each of them relying on a single shrinkage parameter. The PMTPLE uses two shrinkage parameters, which enhances its adaptability and robustness in the presence of multicollinearity between explanatory variables. To assess the performance of the PMTPLE compared to the four existing estimators (the PMLE, PRRE, PLE, and PALE), a simulation study is conducted that encompasses various scenarios and two empirical applications. The evaluation of the performance is based on the mean square error (MSE) criterion. The theoretical comparison, simulation results, and findings of the two applications consistently demonstrate the superiority of the PMTPLE over the other estimators, establishing it as a robust solution for count data analysis under multicollinearity conditions. Full article

In this work, the time-fractional Navier–Stokes equation is discussed using a calculational method, which is called the Sumudu-generalized Laplace transform decomposition method (DGLTDM). The fractional derivatives are defined in the Caputo sense. The (DGLTDM) is a hybrid of the Sumudu-generalized Laplace transform and the decomposition method. Three examples of the time-fractional Navier–Stokes equation are studied to check the validity and demonstrate the effectiveness of the current method. The results show that the suggested method succeeds remarkably well in terms of proficiency and can be utilized to study more problems in the field of nonlinear fractional differential equations (FDEs). Full article

This paper considers polynomial characteristics useful for a better understanding of the behaviour of these functions. Taylor series for the polynomials are described by the items with even and odd derivatives and powered changes in the argument, which leads to more specific studying of their properties. Connections between the derivative and antiderivative of the polynomial functions are defined. The structure of polynomial functions reveals their specific characteristic that the mean value of their roots equals the mean value of the locations of the critical points such as the extrema and inflection points. Derivatives of the quadratic exponent in relation to an interesting connection of two transcendental numbers are also described. The discussed properties of the polynomials can be helpful for practical implementations and educational purposes. Full article