Axioms, Volume 13, Issue 3 (March 2024) – 75 articles

In this study, the ill-conditioning of the iterative method for nonlinear models is discussed. Due to the effectiveness of ridge estimation for ill-conditioned problems and the lack of a combination of the H-K formula with the iterative method, the improvement of the LM algorithm is studied in this paper. Considering the LM algorithm for ill-conditioned nonlinear least squares, an improved LM algorithm based on the H-K formula is proposed for image distortion correction using self-calibration. Three finite difference methods are used to approximate the Jacobian matrix, and the H-K formula is used to calculate the damping factor in each iteration. The Brown model, quadratic polynomial model and Fourier model are applied to the self-calibration, and the improved LM algorithm is used to solve the model parameters. In the simulation experiment of space resection of a single image, we evaluate the performance of the LM algorithm based on the gain ratio (LM_h) and the improved LM algorithm based on the H-K formula (LM_HK), and the accuracy of different models and algorithms is compared. A ridge trace analysis is carried out on the damping factor to illustrate the effects of the improved algorithm in handling ill-conditioning. In the second experiment, the improved algorithm is applied to measure the diameter of a coin using a single camera. The experimental results show that the improved LM algorithm can reach the same or higher accuracy as the LM_h algorithm, and it can weaken the ill-conditioning to a certain extent and enhance the stability of the solution. Meanwhile, the applicability of the improved LM algorithm in self-calibration is verified. Full article

Sentiment analysis aims to study, analyse and identify the sentiment polarity contained in subjective documents. In the realm of natural language processing (NLP), the study of sentiment analysis and its subtask research is a hot topic, which has very important significance. The existing sentiment analysis methods based on sentiment lexicon and machine learning take into account contextual semantic information, but these methods still lack the ability to utilize context information, so they cannot effectively encode context information. Inspired by the concept of density matrix in quantum mechanics, we propose a sentiment analysis method, named Complex-valued Quantum-enhanced Long Short-term Memory Neural Network (CQLSTM). It leverages complex-valued embedding to incorporate more semantic information and utilizes the Complex-valued Quantum-enhanced Long Short-term Memory Neural Network for feature extraction. Specifically, a complex-valued neural network based on density matrix is used to capture interactions between words (i.e., the correlation between words). Additionally, the Complex-valued Quantum-enhanced Long Short-term Memory Neural Network, which is inspired by the quantum measurement theory and quantum long short-term memory neural network, is developed to learn interactions between sentences (i.e., contextual semantic information). This approach effectively encodes semantic dependencies, which reflects the dispersion of words in the embedded space of sentences and comprehensively captures interactive information and long-term dependencies among the emotional features between words. Comparative experiments were performed on four sentiment analysis datasets using five traditional models, showcasing the effectiveness of the CQLSTM model. Full article

The current values of many populations depend on the past values of the population. In many cases, this dependence is caused by the time certain processes take. This dependence on the past can be introduced into mathematical models by adding delays. For example, the growth rate of a population depends on the population $\tau$ time units ago, where $\tau$ is the maturation time. For an epidemic, there is a time $\tau$ between the contact of an infected individual and a susceptible one, and the time the susceptible individual actually becomes infected. This time $\tau$ is also a delay. So, the number of infected individuals depends on the population at the time $\tau$ units ago. A second way of introducing this dependence on past values is to use non-local operators in the description of the model. Fractional derivatives have commonly been used to provide non-local effects. In population growth models, it can also be done by introducing a new compartment, the immature population, and in epidemic models, by introducing an additional exposed population. In this paper, we study and compare these methods of adding dependence on past values. For models of processes that involve delays, all three methods include dependence on past values, but fractional-order models do not justify the form of the dependence. Simulations show that for the models studied, the fractional differential equation method produces similar results to those obtained by explicitly incorporating the delay, but only for specific values of the fractional derivative order, which is an extra parameter. But in all three methods, the results are improved compared to using ordinary differential equations. Full article

The objective of our study is to generalize the results on product states of the tensor product of two JC-algebras to infinite tensor product JC-algebras. Also, we characterize the tracial product state of the tensor product of two JC-algebras, and the tracial product state of infinite tensor products of JC-algebras. Full article

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form $Q_{\mathbb{C}}\left(p\right)=a{p}^n+mp+c$, where $n\ge 2$. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals. Full article

This paper shows that certain${ }_3{F}_4$ hypergeometric functions can be expanded in sums of pair products of${ }_2{F}_3$ functions, which reduce in special cases to${ }_2{F}_3$ functions expanded in sums of pair products of${ }_1{F}_2$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions,${ }_2{F}_1$ functions, and${ }_3{F}_2$ functions into the realm of${ }_p{F}_q$ functions where $p<q$ for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. Full article

In this paper, we explore a study focused on a two-variable extension of matrix Bessel polynomials. We initiate the discussion by introducing the matrix Bessel polynomials involving two variables and derive specific differential formulas and recurrence relations associated with them. Additionally, we present a segment detailing integral formulas for the extended matrix Bessel polynomials. Lastly, we introduce the Laplace–Carson transform for the two-variable matrix Bessel polynomial analogue. Full article

This article provides a detailed exploration of the SIR epidemic model, starting with its meticulous formulation. The study employs a novel approach called the upper and lower bounds technique to approximate the solution to the SIR model, providing insights into the dynamic interplay between susceptible S, infected I, and recovered R populations. A new parametric solution to this model has been presented. Applying the Adomian decomposition method (ADM) allows for the attaining of highly accurate approximate solutions in the context of the SIR epidemic model. To validate the accuracy and robustness of the proposed approach, a numerical exploration is conducted, considering a diverse range of experimental parameters. This numerical analysis provides valuable insights into the sensitivity and responsiveness of the SIR epidemic model under varying conditions, contributing to the broader understanding of infectious disease dynamics. The interplay between theoretical formulation and numerical exploration establishes a comprehensive framework for studying the SIR model, with implications for refining our ability to predict and manage the spread of infectious diseases. Full article

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model. Full article

Based on equivalence relation R on X, equivalence class $\left[x\right]$ of a point and equivalence class $\left[A\right]$ of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space $(X,R)$ depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology ${\tau }_R$ on X is equivalent to the generated topologies associated with metric d, proximity $\delta$ and uniformity $\mathcal{U}$ on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology ${\tau }_R$ on X. Full article

The Performance Evaluation Matrix (PEM) is an excellent decision-making tool for assessment and resource management. Satisfaction Index and Importance Index are two important evaluation indicators of construction and PEM. Managers can decide whether the service item needs to be improved based on the Satisfaction Index of the service item. When resources are limited, managers can determine the priority of improving the service item based on the Importance Index. In order to avoid the risk of misjudgment caused by sample errors and meet the needs of enterprises’ rapid decision-making, this study proposed a fuzzy test built on the confidence intervals of the above two key indicators to decide whether essential service items should be improved and determine the priority of improvement. Since the fuzzy test was relatively complex, this study further came up with fuzzy evaluation values and fuzzy evaluation critical values of service items following fuzzy testing rules. Besides, evaluation rules were established to facilitate industrial applications. This approach can be completed with any common word processing software, so it is relatively convenient in application and easy to manage. Finally, an application example was presented in this paper to explain the applicability of the proposed approach. Full article

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered. Full article

In this paper, we introduce and study new features for 2-variable $(p,q)$-Hermite polynomials, such as the $(p,q)$-diffusion equation, $(p,q)$-differential formula and integral representations. In addition, we establish some summation models and their $(p,q)$-derivatives. Certain parting remarks and nontrivial examples are also provided. Full article

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality for special contact slant submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric non-metric connection is obtained. Full article

For $S\subseteq V\left(G\right),{\kappa }_G\left(S\right)$ denotes the maximum number k of edge disjoint trees $T_1,T_2,\dots ,T_k$ in G, such that $V\left(T_i\right)\cap V\left(T_j\right)=S$ for any $i,j\in \{1,2,\dots ,k\}$ and $i\ne j$. For an integer $2\le r\le \left|V\right(G\left)\right|$, the generalized r-connectivity of G is defined as ${\kappa }_r\left(G\right)=min\{{\kappa }_G\left(S\right)|S\subseteq V\left(G\right)\mathrm{and}|S|=r\}$. In fact, ${\kappa }_2\left(G\right)$ is the traditional connectivity of G. Hence, the generalized r-connectivity is an extension of traditional connectivity. The exchanged folded hypercube $EFH(s,t)$, in which $s\ge 1$ and $t\ge 1$ are positive integers, is a variant of the hypercube. In this paper, we find that ${\kappa }_3\left(EFH(s,t)\right)=s+1$ with $3\le s\le t$. Full article

In this work, we will introduce the concept of ratio-covariety, as a family $\mathcal{R}$ of numerical semigroups that has a minimum, denoted by $\min \left(\mathcal{R}\right),$ is closed under intersection, and if $S\in \mathcal{R}$ and $S\ne \min \left(\mathcal{R}\right)$, then $S\backslash \left\{\mathrm{r}\right(S\left)\right\}\in \mathcal{R},$ where $\mathrm{r}\left(S\right)$ denotes the ratio of $S.$ The notion of ratio-covariety will allow us to: (1) describe an algorithmic procedure to compute $\mathcal{R}$; (2) prove the existence of the smallest element of $\mathcal{R}$ that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety $\mathcal{R}(F,m)=\{S\mid S \mathrm{is} \mathrm{a} \mathrm{numerical} \mathrm{semigroup} \mathrm{with} \mathrm{Frobenius} \mathrm{number} \mathrm{F} \mathrm{and} \mathrm{multiplicity}m\}.$ Full article

Index spaces serve as valuable metric models for studying properties relevant to various applications, such as social science or economics. These properties are represented by real Lipschitz functions that describe the degree of association with each element within the underlying metric space. After determining the index value within a given sample subset, the classic McShane and Whitney formulas allow a Lipschitz regression procedure to be performed to extend the index values over the entire metric space. To improve the adaptability of the metric model to specific scenarios, this paper introduces the concept of a composition metric, which involves composing a metric with an increasing, positive and subadditive function $\varphi$. The results presented here extend well-established results for Lipschitz indices on metric spaces to composition metrics. In addition, we establish the corresponding approximation properties that facilitate the use of this functional structure. To illustrate the power and simplicity of this mathematical framework, we provide a concrete application involving the modeling of livability indices in North American cities. Full article

In this paper, we consider blow-up solutions for the fourth-order nonlinear Schrödinger equation with mixed dispersions. We study the dynamical properties of blow-up solutions for this equation, including the ${\dot{H}}^{{\gamma }_c}$-concentration and limiting profiles, which extend and improve the existing results in the literature. Full article

New high-order weak schemes are proposed and simplified to solve stochastic differential equations with Markovian switching driven by pure jumps (PJ-SDEwMs). Using Malliavin calculus theory, it is rigorously proven that the new numerical schemes can achieve a high-order convergence rate. Some numerical experiments are provided to show the efficiency and accuracy. Full article

This paper deals with translational regular and rapid variations. By using a new method of proving the Galambos–Bojanić-Seneta type theorems, we prove two theorems of this type for translationally regularly varying and translationally rapidly varying functions and sequences, important objects in the asymptotic analysis of divergent processes. Also, we introduce and study the index functions for translationally regularly varying functions and sequences. For example, we prove that the index function of a translationally regularly varying function is also in the same class of functions. Full article

This paper presents a comparative analysis of the performance of Equivariant Quantum Neural Networks (EQNNs) and Quantum Neural Networks (QNNs), juxtaposed against their classical counterparts: Equivariant Neural Networks (ENNs) and Deep Neural Networks (DNNs). We evaluate the performance of each network with three two-dimensional toy examples for a binary classification task, focusing on model complexity (measured by the number of parameters) and the size of the training dataset. Our results show that the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ EQNN and the QNN provide superior performance for smaller parameter sets and modest training data samples. Full article

Models based on vision transformer architectures are considered state-of-the-art when it comes to image classification tasks. However, they require extensive computational resources both for training and deployment. The problem is exacerbated as the amount and complexity of the data increases. Quantum-based vision transformer models could potentially alleviate this issue by reducing the training and operating time while maintaining the same predictive power. Although current quantum computers are not yet able to perform high-dimensional tasks, they do offer one of the most efficient solutions for the future. In this work, we construct several variations of a quantum hybrid vision transformer for a classification problem in high-energy physics (distinguishing photons and electrons in the electromagnetic calorimeter). We test them against classical vision transformer architectures. Our findings indicate that the hybrid models can achieve comparable performance to their classical analogs with a similar number of parameters. Full article

This paper studies a particular type of planar Filippov system that consists of two discontinuity boundaries separating the phase plane into three disjoint regions with different dynamics. This type of system has wide applications in various subjects. As an illustration, a plant disease model and an avian-only model are presented, and their bifurcation scenarios are investigated. By means of the regularization approach, the blowing up method, and the singular perturbation theory, we provide a different way to analyze the dynamics of this type of Filippov system. In particular, the boundary equilibrium bifurcations of such systems are studied. As a consequence, the nonsmooth fold bifurcation becomes a saddle-node bifurcation, while the persistence bifurcation disappears after regularization. Full article

How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes. Full article

A highly relevant topic in the actuarial literature is so-called “claim reserving” or “loss reserving”, which involves estimating reserves to be provisioned for pending claims, as they can be deferred over various periods. This explains the proliferation of methods that aim to estimate these reserves and their variability. Regression methods are widely used in this setting. If we model error terms as random variables, the variability of provisions can consequently be modelled stochastically. The use of fuzzy regression methods also allows modelling uncertainty for reserve values using tools from the theory of fuzzy subsets. This study follows this second approach and proposes projecting claim reserves using a generalization of fuzzy numbers (FNs), so-called intuitionistic fuzzy numbers (IFNs), through the use of intuitionistic fuzzy regression. While FNs allow epistemic uncertainty to be considered in variable estimation, IFNs add bipolarity to the analysis by incorporating both positive and negative information regarding actuarial variables. Our analysis is grounded in the ANOVA two-way framework, which is adapted to the use of intuitionistic regression. Similarly, we compare our results with those obtained using deterministic and stochastic chain-ladder methods and those obtained using two-way statistical ANOVA. Full article

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples. Full article

In this thesis, we research quasilinear Schrödinger system as follows in which $3<N\in \mathbb{R}$, $2<p<N$, $2<q<N$, $V_1(x),V_2(x)$ are continuous functions, $k,\iota$ are parameters with $k,\iota >0$, and nonlinear terms $f,h\in C({\mathbb{R}}^N\times {\mathbb{R}}^2,\mathbb{R})$. We find a nontrivial solution $(u,v)$ for all $\iota >{\iota }_1(k)$ by means of the mountain-pass theorem and change of variable theorem. Our main novelty of the thesis is that we extend $\Delta$ to ${\Delta }_p$ and ${\Delta }_q$ to find the existence of a nontrivial solution. Full article

In the present work, in order to approximate integrable vector-valued functions, we study the Kantorovich version of vector-valued Shepard operators. We also display some applications supporting our results by using parametric plots of a surface and a space curve. Finally, we also investigate how nonnegative regular (matrix) summability methods affect the approximation. Full article

In this work, the notion of digital fiber homotopy is defined and its properties are given. We present some new results on digital fibrations. Moreover, we introduce digital h-fibrations. We prove some of the properties of these digital h-fibrations. We show that a digital fibration and a digital map p are fiber homotopic equivalent if and only if p is a digital h-fibration. Finally, we explore a relation between digital fibrations and digital h-fibrations. Full article