Axioms, Volume 13, Issue 4 (April 2024) – 71 articles

This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto’s theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations. Full article

Sample selection is one of the most important factors in estimating the unknown parameters of distributions, as it saves time, saves effort, and gives the best results. One of the challenges is deciding on a suitable distribution estimate technique and adequate sample selection to provide the best results in comparison with earlier research. The method of moments (MOM) was decided on to estimate the unknown parameters of the Gumbel distribution, but with four changes in the sample selection, which were simple random sample (SRS), ranked set sampling (RSS), maximum ranked set sampling (MRSS), and ordered maximum ranked set sampling (OMRSS) techniques, due to small sample sizes. The MOM is a traditional method for estimation, but it is difficult to use when dealing with RSS modification. RSS modification techniques were used to improve the efficiency of the estimators based on a small sample size compared with the usual SRS estimator. A Monte Carlo simulation study was carried out to compare the estimates based on different sampling. Finally, two datasets were used to demonstrate the adaptability of the Gumbel distribution based on the different sampling techniques. Full article

This paper aims to unify q-derivative/q-integrals and h-derivative/h-integrals into a single definition, called $q-h$-derivative/$q-h$-integral. These notions are further extended on the finite interval $[a,b]$ in the form of left and right $q-h$-derivatives and $q-h$-integrals. Some inequalities for $q-h$-integrals are studied and directly connected with well known results in diverse fields of science and engineering. The theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified using the $q-h$-derivative/$q-h$-integral concept. Full article

The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using Schauder’s fixed point theorem; the uniqueness of solutions is obtained by adding an additional hypothesis and applying Banach’s contraction principle. An example is provided to demonstrate the results. Full article

To address public participation-oriented, large group decision-making problems with uncertain attribute weights, we propose a multi-attribute decision-making method considering public satisfaction. Firstly, a large group is organized to provide their opinions in the form of linguistic variables. Public opinions can be categorized into two types based on their content: one reflects the effectiveness of an alternative implementation and the other reflects the public expectations. Secondly, the two types of public opinions are sorted separately by linguistic variables. The evaluation of alternatives and the evaluation of expectations in different attributes are determined, both of which are expressed in the form of linguistic distributions. These two evaluations are then compared to determine the public satisfaction of the attributes in different alternatives. Thirdly, based on the deviation of public satisfaction in different attributes, a weight optimization model is constructed to determine the attribute weights. Fourthly, leveraging the interval credibility of attribute satisfaction for various alternatives, an evidential reasoning non-linear optimization model is established to obtain the comprehensive utility evaluation value for each alternative, which is used for ranking. Finally, a numerical example is employed to validate the feasibility and effectiveness of the proposed approach. According to the results of the numerical example, it can be concluded that the proposed approach can be effectively applied to large group decision-making problems that consider public satisfaction. Based on the comparison of methods, the proposed approach has certain advantages in reflecting public opinions and setting reference points, which can ensure the reliability of the decision results. Full article

In this paper, we introduce the concept of monotonicity according to a direction for a set of random variables. This concept extends well-known multivariate dependence notions, such as corner set monotonicity, and can be used to detect dependence in multivariate distributions not detected by other known concepts of dependence. Additionally, we establish relationships with other known multivariate dependence concepts, outline some of their salient properties, and provide several examples. Full article

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem $Ly=\lambda y$, for an algebraic parameter $\lambda$ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve $\Gamma$. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve $\Gamma$, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, $\Gamma$ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of $L-\lambda$ becomes explicit over a new coefficient field containing $\Gamma$. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. Full article

The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics. Full article

Consider a supercritical Galton–Watson process with immigration $({\mathrm{X}}_n;n\ge 0)$. The Lotka–Nagaev estimator $\frac{{\mathrm{X}}_{n+1}}{{\mathrm{X}}_n}$ is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cramér moderate deviation results for the Lotka–Nagaev estimator. To satisfy our needs, we employed the well-known Cramér approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals. Full article

We continue the study of Terracini loci formed by x points of a variety embedded in a projective space. Our main results are a refined study of Terracini loci arising from linear projections, the description of the maximal x with a non-empty Terracini locus for Hirzebruch surfaces, and the maximal “weight”, “corank”, or “defect” in several cases. For low x, we even show which defects can occur. Full article

The objective of this research is to describe and investigate a novel class of separation axioms and discuss some of their fundamental characteristics using a nano weakly generalized closed set. In nano topological space, $\mathcal{N}\mathcal{w}\mathcal{g}$-closed graph and strongly $\mathcal{N}\mathcal{w}\mathcal{g}$-closed graph functions are introduced and explored. We also analyse some of the characterizations of closed graph functions with the separation axioms via a nano weakly generalized closed set. Full article

The Wiener index is one of the most classic and widely used indicators in topology. It reflects the average distance of any node pair in the graph. It not only makes the boundaries of given graphs clearer but also continuously generates topological indices that are more suitable for new fields, such as the Gutman index. The Wiener index and Gutman index are two important topological indices, which are commonly used to describe the characteristics of molecular structure. They are closely related to the physical and chemical properties of molecular compounds. And they are widely used to predict the physical and chemical properties and biological activity of molecular compounds. In this paper, we study the vertex Gutman index and Gutman index and describe the structural characteristics of all cases of two simple cycles intersecting. We comprehensively analyze the Gutman index and vertex Gutman index in these cases in detail by means of classification discussion and analogical reasoning and characterize their maximum and minimum accordingly. Full article

Let $\mu$ be a self-similar measure with compact support K. The Hausdorff dimension of K is $\alpha$. The Cauchy transform of $\mu$ is denoted by $F\left(z\right)$. For $0<\beta <1,$ we define the function $F^{\left[\beta \right]},$ which compares with the fractional derivative of F of order $\beta .$ Let $\Phi \left(z\right)=F(1/z),\left|z\right|<1$. In this paper, we prove that ${\Phi }^{\left[\beta \right]}$ belongs to $A^p$ for $0<p<1/(\beta +1)$, and ${\left({\Phi }^{\prime }\right)}^{\left[\beta \right]}$ belongs to $A^p$ for $1\le p<1/\beta \le 1/(2-\alpha )$, where $A^p$ is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem. Full article

Many papers have been devoted to applying fuzzy sets to algebraic structures. In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory, lattice theory, and coding theory points of view, and some applications of fuzzy sets in residuated lattices are presented. Since ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic, first, we investigate the lattice of fuzzy ideals in residuated lattices and study some connections between fuzzy sets associated to ideals and Hadamard codes. Finally, we present applications of fuzzy sets in coding theory. Full article

We are honoured to present this Special Issue of Axioms with the title “Topology and Functional Analysis” to showcase recent work on this and related topics and to provide an opportunity for María Jesús Chasco’s friends and colleagues to pay tribute to her mathematical career on the occasion of her 65th birthday [...] Full article

A 10-field theory for second-grade viscoelastic fluids is developed in the framework of Rational Extended Thermodynamics. The field variables are the density, the velocity, the temperature and the stress tensor. The particular case of an adiabatic fluid is considered. The field equations are determined by use of physical universal principles such as the Galileian and the Entropy Principles. As already proved, Rational Extended Thermodynamics is able to eliminate some inconsistencies with experiments that arise in Classical Thermodynamics. Moreover, the paper shows that, if the quadratic terms are taken into account, the classical constitutive relations for a second-grade fluid can be obtained as a limit case of the field equations of the present theory. Full article

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points. Full article

This paper considers a multi-product, multi-criteria supply–demand network equilibrium model with capacity constraints and uncertain demands. Strict network equilibrium principles are proposed both in the case of a single criterion and multi-criteria, respectively. Based on a single criterion, it proves that strict network equilibrium flows are equivalent to vector variational inequalities, and the existence of strict network equilibrium flows is derived by virtue of the Fan–Browder fixed point theorem. Based on multi-criteria, the scalarization of strict network equilibrium flows is given by using Gerstewitz’s function without any convexity assumptions. Meanwhile, the necessary and sufficient conditions of strict network equilibrium flows are derived in terms of vector variational inequalities. Finally, an example is given to illustrate the application of the derived theoretical results. Full article

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants. Full article

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by $x\left(s\right)=1+\lambda x\left(s\right){\int }_0^1\frac{s}{t+s}x\left(t\right)dt,$ can be very often encountered in many applications, where x is the function to be determined, $\lambda$ is a parameter, and $t,s\in [0,1]$. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form $\chi \left(l\right)=\varrho \left(l\right)+\chi \left(l\right){\int }_p^{q}k(l,z)\left(V\chi \right)\left(z\right)dz$ are investigated in the space $C_{\omega }\left[p,q\right],$ where $\chi$ is the unknown function to be determined, V is a given operator, and $\varrho ,k$ are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided. Full article

Non-negative integer-valued time series are usually encountered in practice, and a variety of integer-valued autoregressive processes based on various thinning operators are commonly used to model these count data with temporal dependence. In this paper, we consider a first-order integer-valued autoregressive process constructed by the negative binomial thinning operator with random coefficients, to address the problem of constant thinning parameters which might not always accurately represent real-world settings because of numerous external and internal causes. We estimate the model parameters of interest by the two-step conditional least squares method, obtain the asymptotic behaviors of the estimators, and furthermore devise a technique to test the constancy of the thinning parameters, which is essential for determining whether or not the proposed model should consider the parameters’ randomness. The effectiveness and dependability of the suggested approach are illustrated by a series of thorough simulation studies. Finally, two real-world data analysis examples reveal that the suggested approach is very useful and flexible for applications. Full article

In this article, a distributed charging strategy problem for plug-in electric vehicles (PEVs) with feeder constraints based on generalized Nash equilibria (GNE) in a novel smart charging station (SCS) is investigated. The purpose is to coordinate the charging strategies of all PEVs in SCS to minimize the energy cost of SCS. Therefore, we build a non-cooperative game framework and propose a new price-driven charging control game by considering the overload constraint of the assigned feeder, where each PEV minimizes the fees it pays to satisfy its optimal charging strategy. On this basis, the existence of GNE is given. Furthermore, we employ a distributed algorithm based on forward–backward operator splitting methods to find the GNE. The effectiveness of the employed algorithm is verified by the final simulation results. Full article

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included. Full article

Vegetables have a short period of freshness, and therefore, the purchase of vegetables has to be carefully matched with sales, especially in the “small production and big market” setting prevalent in China. Therefore, it is worthwhile to develop a systematic and comprehensive mathematical model of replenishment plans and pricing strategies for each category of vegetables and individual products. In this paper, we analyze the following three questions: Question One: What is the distribution law and relationship between the sales volume of vegetable categories and single products? Question Two: What is the relationship between total sales volume and cost-plus pricing of vegetable categories? And is it possible to provide the daily total replenishment and pricing strategy of each vegetable category for the following week to maximize supermarket profit? Question Three: How can we incorporate the market demand for single vegetable products into a profit-maximizing program for supermarkets? Is it possible to further formulate the replenishment plan requirements for single products? To answer the first question, we created pivot tables to analyze occupancy. We found that mosaic leaves, peppers, and edible mushrooms accounted for a larger proportion of occupacy, while cauliflowers, aquatic rhizomes, and eggplants accounted for a smaller proportion. For the single items, lettuce, cabbage, green pepper, screw pepper, enoki mushroom, and shiitake mushroom accounted for a large proportion of their respective categories. We used the Pearson correlation coefficient and the Mfuzz package based on fuzzy c-means (FCM) algorithm to analyze the correlation between vegetable categories and single products. We found that there was a strong correlation between vegetable categories. Moreover, the sale of vegetable items belonging to the same category exhibited the same patterns of change over time. In order to address the second question, we established the LightGBM sales forecasting model. Combined with previous sales data, we forecasted and planned an efficient daily replenishment volume for each vegetable category in the coming week. In addition, we developed a pricing strategy for vegetable categories to maximize supermarket profits. For the third question, we built a dynamic programming model combining an optimal replenishment volume with a product pricing strategy for single items, which let the supermarket maximize its expected profits. Full article

In a series of papers, we have discussed the concept of a modal topological structure modified, extended and illustrated by examples from intuitionistic fuzzy sets. Here, the concept of a temporal modal topological structure is introduced and illustrated with four different intuitionistic fuzzy temporal modal topological structures. These structures are based on intuitionistic fuzzy topological, temporal and modal operators. They are extensions of the temporal topological structures as well as of the modal topological structures. Full article

This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson’s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark’s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator. Full article

The distributed training of federated machine learning, referred to as federated learning (FL), is discussed in models by multiple participants using local data without compromising data privacy and violating laws. In this paper, we consider the training of federated machine models with uncertain participation attitudes and uncertain benefits of each federated participant, and to encourage all participants to train the desired FL models, we design a fuzzy Shapley value incentive mechanism with supervision. In this incentive mechanism, if the supervision of the supervised mechanism detects that the payoffs of a federated participant reach a value that satisfies the Pareto optimality condition, the federated participant receives a distribution of federated payoffs. The results of numerical experiments demonstrate that the mechanism successfully achieves a fair and Pareto optimal distribution of payoffs. The contradiction between fairness and Pareto-efficient optimization is solved by introducing a supervised mechanism. Full article

This paper presents the Slash-Exponential-Fréchet distribution, which is an expanded version of the Fréchet distribution. Through its stochastic representation, probability distribution function, moments and other relevant features are obtained. Evidence supports that the updated model displays a lighter right tail than the Fréchet model and is more flexible as for skewness and kurtosis. Results on maximum likelihood estimators are given. Our proposition’s applicability is demonstrated through a simulation study and the evaluation of two real-world datasets. Full article

The purpose of this paper is to generalize the concept of classical fuzzy set to vector-valued fuzzy set which can attend values not only in the real interval $[0, 1]$, but in an ordered interval of a Banach algebra as well. This notion allows us to introduce the concept of vector-valued fuzzy metric space which generalizes, extends and unifies the notion of classical fuzzy metric space and complex-valued fuzzy metric space and permits us to consider the fuzzy sets and metrics in a larger domain. Some topological properties of such spaces are discussed and some fixed point results in this new setting are proved. Multifarious examples are presented which clarify and justify our claims and results. Full article

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform $\mathcal{Z}\left(s\right)$, $s=\sigma +it$, with fixed $1/2<\sigma <1$, of the square ${\left|\zeta (1/2+it)\right|}^2$ of the Riemann zeta-function. We consider probability measures defined by means of $\mathcal{Z}(\sigma +i\phi (t\left)\right)$, where $\phi \left(t\right)$, $t⩾t_0>0$, is an increasing to $+\infty$ differentiable function with monotonically decreasing derivative ${\phi }^{\prime }\left(t\right)$ satisfying a certain normalizing estimate related to the mean square of the function $\mathcal{Z}(\sigma +i\phi (t\left)\right)$. This allows us to extend the distribution laws for $\mathcal{Z}\left(s\right)$. Full article