Axioms, Volume 13, Issue 2 (February 2024) – 62 articles

The Bessel function of the first kind $J_N\left(kx\right)$ is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind $I_N\left(kx\right)$. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for $J_N\left(kx\right)$ useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving ${ }_1{F}_2$ hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k. Full article

In this study, we present the generalized form of the higher-order nonlinear fractional Bratu-type equation. In this generalization, we deal with a generalized fractional derivative, which is quite useful from an application point of view. Furthermore, some special cases of the generalized fractional Bratu equation are recognized and examined. To solve these nonlinear differential equations of fractional order, we employ the homotopy perturbation transform method. This work presents a useful computational method for solving these equations and advances our understanding of them. We also plot some numerical outcomes to show the efficiency of the obtained results. Full article

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron. Full article

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives. Full article

We propose a new tool for estimating the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighboring values in a time series. This makes it possible to describe the signal as efficiently as prior proposed parameters, such as permutation entropy (PE) or modified permutation entropy (mPE). Firstly, this method reduces the size of the sample that is necessary to estimate the parameter value, and secondly it enables the use of the Kullback–Leibler divergence to estimate the “distance” between the time series data and random signals. Full article

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. Full article

In this paper, we analyze the identification of the amount of pollutant discharged problem by each source in a heat system when the dynamics of the state are governed by a parameterized unknown operator. In this way, we introduce the notion of average sentinel. The decomposition method is used to solve the equation of this problem, the gradient method is used to calculate the averaged control, and the combination of the two methods is used to estimate the pollution terms. Numerical example is given to confirm this result. Full article

In this paper, the generalized widest path problem (or generalized maximum capacity problem) is studied. This problem is denoted by the GWPP. The classical widest path problem is to find a path from a source (s) to a sink (t) with the highest capacity among all possible s-t paths. The GWPP takes into account the presence of loss/gain factors on arcs as well. The GWPP aims to find an s-t path considering the loss/gain factors while satisfying the capacity constraints. For solving the GWPP, three strongly polynomial time algorithms are presented. Two algorithms only work in the case of losses. The first one is less efficient than the second one on a CPU, but it proves to be more efficient on large networks if it parallelized on GPUs. The third algorithm is able to deal with the more general case of losses/gains on arcs. An example is considered to illustrate how each algorithm works. Experiments on large networks are conducted to compare the efficiency of the algorithms proposed. Full article

In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times. Full article

Count data consists of both observed and unobserved events. The analysis of count data often encounters overdispersion, where traditional Poisson models may not be adequate. In this paper, we introduce a tractable one-parameter mixed Poisson distribution, which combines the Poisson distribution with the improved second-degree Lindley distribution. This distribution, called the Poisson-improved second-degree Lindley distribution, is capable of effectively modeling standard count data with overdispersion. However, if the frequency of the unobserved events is unknown, the proposed distribution cannot be directly used to describe the events. To address this limitation, we propose a modification by truncating the distribution to zero. This results in a tractable zero-truncated distribution that encompasses all types of dispersions. Due to the unknown frequency of unobserved events, the population size as a whole becomes unknown and requires estimation. To estimate the population size, we develop a Horvitz–Thompson-like estimator utilizing truncated distribution. Both the untruncated and truncated distributions exhibit desirable statistical properties. The estimators for both distributions, as well as the population size, are asymptotically unbiased and consistent. The current study demonstrates that both the truncated and untruncated distributions adequately explain the considered medical datasets, which are the number of dicentric chromosomes after being exposed to different doses of radiation and the number of positive Salmonella. Moreover, the proposed population size estimator yields reliable estimates. Full article

In this paper, we investigate an application of the statistical concept of causality, based on Granger’s definition of causality, on raw increasing processes as well as on optional and predictable measures. A raw increasing process is optional (predictable) if the bounded (left-continuous) process X, associated with the measure ${\mu }_A(X)$, is self-caused. Also, the measure ${\mu }_A(X)$ is optional (predictable) if an associated process X is self-caused with some additional assumptions. Some of the obtained results, in terms of self-causality, can be directly applied to defining conditions for an optional stopping time to become predictable. Full article

This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. In this paper, two models of fuzzy equivalence operators are obtained by the composition of overlap functions, grouping functions and fuzzy negations. Their main properties are also studied. Full article

The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank 2, we derive the Hessian configuration $({12}_3,{16}_4)$ of points and lines. Finally, we introduce and investigate the de Vries configuration of points and lines in a cubic structure. Full article

Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its possible applications to other types of evolution equations are discussed. Full article

We extend previous research to derive three additional M-1-dimensional integral representations over the interval $[0,1]$. The prior version covered the interval $[0,\infty ]$. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) $|{\mathbf{x}}_1-{\mathbf{x}}_2|=\sqrt{x_1^2-2x_1{x}_{2}cos\theta +x_2^2}$ to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval $[0,1]$ are numerically stable, as was the prior version, having integrals running over the interval $[0,\infty ]$, and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over $[0,1]$ than over $[0,\infty ]$. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form $(a{x}^2+bx+c)/x$, for which a single tabled integral exists for the integrals from running over the interval $[0,\infty ]$ of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval $[0,1]$. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over $[0,1]$. Full article

We investigate the existence of solutions to the scalar field equation $-\Delta u=g\left(u\right)-\lambda u\mathrm{in}{\mathbb{R}}^N,$ with mass constraint ${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{2}dx=a>0,u\in H^1\left({\mathbb{R}}^N\right).$ Here, $N\ge 3$; g is a continuous function satisfying the conditions of the Berestycki–Lions type; $\lambda$ is a Lagrange multiplier. Our results supplement and generalize some of the results in L. Jeanjean, S.-S. Lu, Calc. Var. Partial Differential Equations. 61 (2022), Paper No. 214, 18, and J. Hirata, K. Tanaka, Adv. Nonlinear Stud. 19 (2019), 263–290. Full article

This research introduces a rule-based decision-making model to investigate corporate governance, which has garnered increasing attention within financial markets. However, the existing corporate governance model developed by the Security and Future Institute of Taiwan employs numerous indicators to assess listed stocks. The ultimate ranking hinges on the number of indicators a company meets, assuming independent relationships between these indicators, thereby failing to reveal contextual connections among them. This study proposes a hybrid rough set approach based on multiple rules induced from a decision table, aiming to overcome these constraints. Additionally, four sample companies from Taiwan undergo evaluation using this rule-based model, demonstrating consistent rankings with the official outcome. Moreover, the proposed approach offers a practical application for guiding improvement planning, providing a basis for determining improvement priorities. This research introduces a rule-based decision model comprising ten rules, revealing contextual relationships between indicators through if–then decision rules. This study, exemplified through a specific case, also provides insights into utilizing this model to strengthen corporate governance by identifying strategic improvement priorities. Full article

A mutator is a variant in a population of organisms whose mutation rate is higher than the average mutation rate in the population. For genetic and population dynamics reasons, mutators are produced and survive with much greater frequency than anti-mutators (variants with a lower-than-average mutation rate). This strong asymmetry is a consequence of both fundamental genetics and natural selection; it can lead to a ratchet-like increase in the mutation rate. The rate at which mutators appear is, therefore, a parameter that should be of great interest to evolutionary biologists generally; for example, it can influence: (1) the survival duration of a species, especially asexual species (which are known to be short-lived), (2) the evolution of recombination, a process that can ameliorate the deleterious effects of mutator abundance, (3) the rate at which cancer appears, (4) the ability of pathogens to escape immune surveillance in their hosts, (5) the long-term fate of mitochondria, etc. In spite of its great relevance to basic and applied science, the rate of mutation to a mutator phenotype continues to be essentially unknown. The reasons for this gap in our knowledge are largely methodological; in general, a mutator phenotype cannot be observed directly, but must instead be inferred from the numbers of some neutral “marker” mutation that can be observed directly: different mutation-rate variants will produce this marker mutation at different rates. Here, we derive the expected distribution of the numbers of the marker mutants observed, accounting for the fact that some of the mutants will have been produced by a mutator phenotype that itself arose by mutation during the growth of the culture. These developments, together with previous enhancements of the Luria–Delbrück assay (by one of us, dubbed the “Jones protocol”), make possible a novel experimental protocol for estimating the rate of mutation to a mutator phenotype. Simulated experiments using biologically reasonable parameters that employ this protocol show that such experiments in the lab can give us fairly accurate estimates of the rate of mutation to a mutator phenotype. Although our ability to estimate mutation-to-mutator rates from simulated experiments is promising, we view this study as a proof-of-concept study and an important first step towards practical empirical estimation. Full article

In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback loop. In such a control framework, the nominal plant is firstly right coprime factorized and a feed forward and a feedback controllers are designed based on Bezout identity to ensure the overall stability. Subsequently, considering the sundry disturbed open loop dynamics, a new condition formulated as a disturbed Bezout identity is put forward to achieve the closed loop stability of the system, even in the presence of disturbances existing in sundry open loops, where in the feedback loop a disturbed identity operator is defined. This approach guarantees the system robustness if a specific inequality condition is satisfied. And, it should be noted that the proposed approach is applicable to both linear and nonlinear systems with sundry disturbed open loop dynamics. Simulations demonstrate the effectiveness of our methodology. Full article

The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, $S\left(t\right)$ and $R\left(t\right)$, respectively. This feature allows a substantially complete elucidation of qualitative properties. The model exhibits three modes of behaviour classified in terms of the sign of $-S^{\prime }\left(0\right)$, the initial value of the epidemic curve. Model behaviour is similar to that of the SIS model if $S^{\prime }\left(0\right)>0$ and to the SIR model if $S^{\prime }\left(0\right)<0$. The separating case is completely soluble and $S\left(t\right)$ is constant-valued. Long-term outcomes are determined for all cases, together with determination of the rate of convergence. Determining the shape of the epidemic curve motivates an investigation of curvature properties of all three state functions and quite complete results are obtained that are new, even for the SIR model. Finally, the second threshold theorem for the SIR model is extended in refined and generalised forms. Full article

In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain pre-existing inequalities found in the literature. Finally, we provide applications to the newly established inequalities to verify their usefulness. Full article

Electro-optical detection systems face numerous challenges due to the complexity and difficulty of targeting controls for “low, slow and tiny” moving targets. In this paper, we present an optimal model of an advanced n-step adaptive Kalman filter and gyroscope short-term integration weighting fusion (nKF-Gyro) method with targeting control. A method is put forward to improve the model by adding a spherical coordinate system to design an adaptive Kalman filter to estimate target movements. The targeting error formation is analyzed in detail to reveal the relationship between tracking controller feedback and line-of-sight position correction. Based on the establishment of a targeting control coordinate system for tracking moving targets, a dual closed-loop composite optimization control model is proposed. The outer loop is used for estimating the motion parameters and predicting the future encounter point, while the inner loop is used for compensating the targeting error of various elements in the firing trajectory. Finally, the modeling method is substituted into the disturbance simulation verification, which can monitor and compensate for the targeting error of moving targets in real time. The results show that in the optimal model incorporating the nKF-Gyro method with targeting control, the error suppression was increased by up to 36.8% compared to that of traditional KF method and was 25% better than that of the traditional nKF method. Full article

Organized conflict, while confined by the laws of physics—and, under profound strategic incompetence, by the Lanchester equations—is not a physical process but rather an extended exchange between cognitive entities that have been shaped by path-dependent historical trajectories and cultural traditions. Cognition itself is confined by the necessity of duality, with an underlying information source constrained by the asymptotic limit theorems of information and control theories. We introduce the concept of a ‘basic underlying probability distribution’ characteristic of the particular cognitive process studied. The dynamic behavior of such systems is profoundly different for ‘thin-tailed’ and ‘fat-tailed’ distributions. The perspective permits the construction of new probability models that may provide useful statistical tools for the analysis of observational and experimental data associated with organized conflict, and, in some measure, for its management. Full article

In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if $A^{\overline{)\mathrm{S}}}B=0$ and $B{A}^{\overline{)\mathrm{S}}}=0$, where $A^{\overline{)\mathrm{S}}}$ is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized. Full article

As digital technologies continue to reshape economic landscapes, the comprehensive evaluation of digital economy (DE) development in provincial regions becomes a critical endeavor. This article proposes a novel approach, integrating the linear programming method, fuzzy logic, and the alternative ranking order method accounting for two-step normalization (AROMAN), to assess the multifaceted facets of DE growth. The primary contribution of the AROMAN is the coupling of vector and linear normalization techniques in order to produce accurate data structures that are subsequently utilized in calculations. The proposed methodology accommodates the inherent uncertainties and complexities associated with the evaluation process, offering a robust framework for decision-makers. The linear programming aspect optimizes the weightings assigned to different evaluation criteria, ensuring a dynamic and context-specific assessment. By incorporating fuzzy logic, the model captures the vagueness and imprecision inherent in qualitative assessments, providing a more realistic representation of the DE’s multifaceted nature. The AROMAN further refines the ranking process, considering the interdependencies among the criteria and enhancing the accuracy of the evaluation. In order to ascertain the efficacy of the suggested methodology, a case study is undertaken pertaining to provincial areas, showcasing its implementation in the evaluation and a comparison of DE progress in various geographical settings. The outcomes illustrate the capacity of the model to produce perceptive and implementable insights for policymakers, thereby enabling them to make well-informed decisions and implement focused interventions that promote the expansion of the DE. Moreover, managerial implications, theoretical limitations, and a comparative analysis are also given of the proposed method. Full article

We investigate $L_2$-approximation problems in the worst case setting in the weighted Hilbert spaces $H\left(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}\right)$ with weights $R_{d,\mathit{\alpha },\mathit{\gamma }}$ under parameters $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$ and $1<{\alpha }_1\le {\alpha }_2\le \cdots$. Several interesting weighted Hilbert spaces $H\left(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}\right)$ appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of $L_2$-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ${\epsilon }^{-1}$. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and $(t_1,t_2)$-weak tractability ($(t_1,t_2)$-WT) for all $t_1>1$ and $t_2>0$ in terms of the introduced weights under the absolute error criterion or the normalized error criterion. Full article

In this note, we show that, for any real number $\tau \in [\frac{1}{2},1)$, any finite set of positive integers K and any integer $s_1\ge 2$, the sequence of integers $s_1,s_2,s_3,\dots$ satisfying $s_{i+1}-s_i\in K$ if $s_i$ is a prime number, and $2\le s_{i+1}\le \tau s_i$ if $s_i$ is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on $\tau ,s_1$ and the maximal element of K only. In particular, if K is a singleton set and for each composite $s_i$ the integer $s_{i+1}$ in the interval $[2,\tau s_i]$ is chosen by some prescribed rule, e.g., $s_{i+1}$ is the largest prime divisor of $s_i$, then the sequence $s_1,s_2,s_3,\dots$ is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either $K=\left\{1\right\}$ and $\tau \in [\frac{1}{2},\frac{3}{4})$ or $K=\left\{2\right\}$ and $\tau \in [\frac{1}{2},\frac{5}{9})$. Full article

The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold $\Omega$, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold $\Omega$ near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity. Full article

This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings. Full article