Axioms

This article primarily investigates nonlinear disturbance observer-based bearing-only formation tracking control for unmanned aerial vehicle (UAV) systems that encounter uncertainties and disturbances. The employed distributed control strategy relies solely on the relative bearing information of neighboring UAVs. To tackle the challenges posed by unknown disturbances and system uncertainties, a novel nonlinear disturbance observer is proposed to effectively mitigate their impact. Moreover, the issue of unknown controller orientation arising from controller singularities is addressed by introducing a Butterworth low-pass filter. This filter ensures a consistent controller gain and enhances disturbance suppression, ultimately transforming the controller gain function into a constant value of 1. Subsequently, a bearing-only formation tracking controller is developed using the backstepping control approach. The stability of the closed-loop control systems is rigorously proven using Lyapunov theory. Finally, numerical simulations are conducted to validate the effectiveness of the proposed scheme in achieving formation control objectives. Full article

To solve the problems of the original sparrow search algorithm’s poor ability to jump out of local extremes and its insufficient ability to achieve global optimization, this paper simulates the different learning forms of students in each ranking segment in the class and proposes a customized learning method (CLSSA) based on multi-role thinking. Firstly, cube chaos mapping is introduced in the initialization stage to increase the inherent randomness and rationality of the distribution. Then, an improved spiral predation mechanism is proposed for acquiring better exploitation. Moreover, a customized learning strategy is designed after the follower phase to balance exploration and exploitation. A boundary processing mechanism based on the full utilization of important location information is used to improve the rationality of boundary processing. The CLSSA is tested on 21 benchmark optimization problems, and its robustness is verified on 12 high-dimensional functions. In addition, comprehensive search capability is further proven on the CEC2017 test functions, and an intuitive ranking is given by Friedman's statistical results. Finally, three benchmark engineering optimization problems are utilized to verify the effectiveness of the CLSSA in solving practical problems. The comparative analysis shows that the CLSSA can significantly improve the quality of the solution and can be considered an excellent SSA variant. Full article

Matrix energy is a valid mathematical tool for representing collective information. However, it is not used in the existing literature for fuzzy sets, intuitionistic fuzzy sets, and multi-attribute group decision making (MAGDM) problems, which highlights research gaps. Motivated by both matrix energy and the research gaps, this study aims to extend matrix energy to the energy of an intuitionistic fuzzy matrix (IFM) and to utilize the IFM energy method in the MAGDM problem, which fully contains all the IFM information on attribute weights, decision maker weights, and attribute values. To achieve these objectives, this paper first proposes IFM energy in terms of the true matrix energy and false matrix energy; then, it develops a MAGDM model using the IFM energy method and the score and accuracy equations of IFM energy. Then, the developed MAGDM model is applied to the selection problem of hospital locations in Shaoxing City, China to demonstrate the practicality and validity of the developed model. Compared with existing intuitionistic fuzzy MAGDM methods, the developed MAGDM model using the IFM energy method reveals its superiority and novelty in complete IFM information expressions and the MAGDM method because the existing MAGDM methods containing intuitionistic fuzzy set information have difficulty tackling MAGDM problems containing all the IFM information on attribute weights, decision maker weights, and attribute values. Full article

As a novel neural network learning framework, Twin Extreme Learning Machine (TELM) has received extensive attention and research in the field of machine learning. However, TELM is affected by noise or outliers in practical applications so that its generalization performance is reduced compared to robust learning algorithms. In this paper, we propose two novel distance metric optimization-driven robust twin extreme learning machine learning frameworks for pattern classification, namely, CWTELM and FCWTELM. By introducing the robust Welsch loss function and capped $L_{2,p}$-distance metric, our methods reduce the effect of outliers and improve the generalization performance of the model compared to TELM. In addition, two efficient iterative algorithms are designed to solve the challenges brought by the non-convex optimization problems CWTELM and FCWTELM, and we theoretically guarantee their convergence, local optimality, and computational complexity. Then, the proposed algorithms are compared with five other classical algorithms under different noise and different datasets, and the statistical detection analysis is implemented. Finally, we conclude that our algorithm has excellent robustness and classification performance. Full article

The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind. Let ${\phi }_{BS}\left(\xi \right)={\left(\frac{\xi }{e^{\xi }-1}\right)}^2={\sum }_{n=0}^{\infty }\frac{{\xi }^n{B}_n^2}{n!}$, where the coefficients of $B_n^2$ are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions $𝟊$ such that $\frac{\xi 𝟊\prime \left(\xi \right)}{𝟊\left(\xi \right)}\prec {\phi }_{BS}\left(\xi \right).$ We found out the coefficient bounds, several radii problems, structural formulas, and inclusion relations. We also found sharp Hankel determinant problems of this class. Full article

In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of $H^1$ 2$\pi$-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact upper bounds of the best approximations of this class by trigonometric polynomials. In addition, we obtain the sets of all of extremal functions for approximations of the class $H^1$ by linear methods of summation of Fourier series. Furthermore, we receive the set of all extremal functions for the class $H^1$ in the Korneichuk–Stechkin lemma and its analogue, the Stepanets lemma, for the Hölder class $H^{1,1}$ functions of two variables being 2$\pi$-periodic in each variable. Full article

The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated. Full article

Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails, and weighted by three harmonic-like elementary symmetric functions are examined. By means of integral representations of zeta tails together with the telescopic approach, twelve general summation theorems are established that express these series as coefficients of the bivariate beta function $\mathrm{Beta}(u,v)$. By further expanding $\mathrm{Beta}(u,v)$ into Laurent series in u and v, several explicit summation formulae are shown as consequences. Full article

The sum range $\mathrm{SR}\left[\mathbf{x};X\right]$, for a sequence $\mathbf{x}={\left(x_n\right)}_{n\in \mathbb{N}}$ of elements of a topological vector space X, is defined as the set of all elements $s\in X$ for which there exists a bijection (=permutation) $\pi :\mathbb{N}\to \mathbb{N}$, such that the sequence of partial sums ${\left({\sum }_{k=1}^n{x}_{\pi \left(k\right)}\right)}_{n\in \mathbb{N}}$ converges to s. The sum range problem consists of describing the structure of the sum ranges for certain classes of sequences. We present a survey of the results related to the sum range problem in finite- and infinite-dimensional cases. First, we provide the basic terminology. Next, we devote attention to the one-dimensional case, i.e., to the Riemann–Dini theorem. Then, we deal with spaces where the sum ranges are closed affine for all sequences, and we include some counterexamples. Next, we present a complete exposition of all the known results for general spaces, where the sum ranges are closed affine for sequences satisfying some additional conditions. Finally, we formulate two open questions. Full article

The goal of the present study is to find a method for improving the predictive capabilities of feedforward neural networks in cases where values distant from the input–output sample interval are predicted. This paper proposes an iterative prediction algorithm based on two assumptions. One is that predictions near the statistical sample have much lower error than those distant from the sample. The second is that a neural network can generate additional training samples and use them to train itself in order to get closer to a distant prediction point. This paper presents the results of multiple experiments with different univariate and multivariate functions and compares the predictions made by neural networks before and after their training with the proposed iterative algorithm. The results show that, having passed through the stages of the algorithm, artificial neural networks significantly improve their interpolation performance in long-term forecasting. The present study demonstrates that neural networks are capable of creating additional samples for their own training, thus increasing their approximating efficiency. Full article

In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the integral equation and fractional differential equation to support our main result. Full article

Omega rings ($\mathsf{\Omega }$-rings) (and other related structures) are lattice-valued structures (with $\mathsf{\Omega }$ being the codomain lattice) defined on crisp algebras of the same type, with lattice-valued equality replacing the classical one. In this paper, $\mathsf{\Omega }$-ideals are introduced, and natural connections with $\mathsf{\Omega }$-congruences and homomorphisms are established. As an application, a framework of approximate solutions of systems of linear equations over $\mathsf{\Omega }$-fields is developed. Full article

The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different phase portraits in the Poincaré disc of two of these classes. Full article

In the context of an increasingly interconnected global economy, deciphering the complex ripple effects of external financial disruptions on national economies is a task of utmost significance. This article dives deep into the intricate repercussions of such disturbances on the macroeconomic dynamics of China using the example of the potential insolvency of a Silicon Valley bank. Grounded in empirical scrutiny, we leverage data spanning from Q1 2000 to Q1 2022 and the analytical utility of the impulse response function to illuminate our findings. We find that external financial tumult triggers a global recession, adversely impacting China’s export-driven economy while simultaneously unsettling aggregate output, employment levels, and wage stability. Simultaneously, these disruptions induce variability in consumption tendencies, investment trajectories, and import volumes and inject instability into interest rate paradigms. We also acknowledge the potential for currency depreciation and bank insolvency incidents to induce inflationary stresses, primarily by escalating the costs of imports. However, these inflationary tendencies may be offset by the concomitant economic slowdown and diminished demand inherent to global recessions. Importantly, the tightening of global credit conditions, coupled with existing financial ambiguities, may obstruct investment initiatives, curtail imports, and exert influence on both risk-free and lending interest rates. Our investigation also probes into the response of the Chinese government’s monetary policy to these external financial shocks. Despite the vital role of monetary policy in alleviating the impacts of these shocks, the potential secondary effects on China’s domestic economy warrant attention. Our study underscores the imperative of proper policy design rooted in a profound understanding of the intricate economic interdependencies for effective management and mitigation of the potentially detrimental consequences of such financial upheavals on China’s macroeconomic resilience within the tapestry of a tightly knit global financial ecosystem. Full article

It is shown that for any integers $k\ge 2$, $q\ge 2k$ and $N\ge k+q+2$, there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations $\mathfrak{t}$ possessing the eigenvalue spectrum $\mathrm{spec}\left(\mathfrak{t}\right)=\left(1,2,\cdots ,k,q,q+1\cdots ,N\right)$, a nilradical of the nilpotence index $N-k$ and a characteristic sequence $(N-k,1^k)$. Full article

Let $\mathfrak{A}$ be a prime *-algebra. A product defined as $U•V=U{V}^{\ast }+V{U}^{\ast }$ for any $U,V\in \mathfrak{A}$, is called a bi-skew Jordan product. A map $\xi :\mathfrak{A}\to \mathfrak{A}$, defined as $\xi \left(p_n\left(U_1,U_2,\cdots ,U_n\right)\right)={\sum }_{k=1}^n{p}_n\left(U_1,U_2,...,U_{k-1},\xi \left(U_k\right),U_{k+1},\cdots ,U_n\right)$ for all $U_1,U_2,...,U_n\in \mathfrak{A}$, is called a non-linear bi-skew Jordan n-derivation. In this article, it is shown that $\xi$ is an additive ∗-derivation. Full article

The present study focuses on estimating the stress–strength parameter when the parent distribution is the alpha power exponential model and the available data are progressively Type-II censored. As a starting point, the usual maximum likelihood approach is applied to obtain point and interval estimates of the model parameters, as well as the stress–strength parameter. Another competing strategy employed in this paper is the maximum product of spacing method, which may be thought of as a rival to the maximum likelihood method. The product of spacing approach is used to obtain point and interval estimates for the various parameters. The asymptotic properties of both methods are used to obtain interval estimates of the model parameter and the stress–strength parameter, and the variance of the stress–strength parameter is approximated using the well-known delta method. Two parametric bootstrap confidence intervals are provided based on the suggested classical estimation procedures. A simulation study is also used to assess the performance of various point and interval estimations. For illustrative purposes, the proposed methods are applied to two real data sets, one for the kidney patients’ recurrence times to infection and the other for breaking the strength of jute fibers. Full article

This article utilizes improved adaptive progressively Type-II censored data to estimate the entropy of the inverse Weibull distribution. Rényi, q, and Shannon entropy measurements are used to define entropy to achieve this objective. Both point and interval estimations of the entropy quantities are investigated through the maximum likelihood and maximum product of spacing methods. Two parametric bootstrap confidence intervals based on the two estimation techniques are also considered for the various entropy measures. A Monte Carlo simulation study is conducted to investigate how estimates behave at various sample sizes and different censoring schemes based on some statistical measurements. The simulations demonstrate that, as anticipated, when the sample size grows, the estimation accuracy also grows. Furthermore, they show that the estimated entropy measures get closer to the actual entropy values when the censoring level decreases. For purposes of explanation, two applications to actual datasets are taken into consideration. The results verified that the adaptive or improved adaptive progressive censoring schemes give more information about data than the conventional progressive censoring scheme in terms of minimum entropy measures. Full article

Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in recent years. But studies have shown that there is a gradient pathology in PINNs. That is, there is an imbalance gradient problem in each regularization term during back-propagation, which makes it difficult for neural network models to accurately approximate partial differential equations. Based on the depth-weighted residual neural network and neural attention mechanism, we propose a new mixed-weighted residual block in which the weighted coefficients are chosen autonomously by the optimization algorithm, and one of the transformer networks is replaced by a skip connection. Finally, we test our algorithms with some partial differential equations, such as the non-homogeneous Klein–Gordon equation, the (1+1) advection–diffusion equation, and the Helmholtz equation. Experimental results show that the proposed algorithm significantly improves the numerical accuracy. Full article