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Automatic test paper design is critical in education to reduce workloads for educators and facilitate an efficient teaching process. However, current designs fail to satisfy the realistic teaching requirements of educators, including the consideration of both test quality and efficiency. This is the main reason why teachers still manually construct tests in most teaching environments. In this paper, the quality of tests is quantitatively defined while considering multiple objectives, including a flexible coverage of knowledge points, cognitive levels, and question difficulty. Then, a model based on the technique of linear programming is delicately designed to explore the optimal results for this newly defined problem. However, this technique is not efficient enough, which cannot obtain results in polynomial time. With the consideration of both test quality and generation efficiency, this paper proposes a genetic algorithm (GA) based method, named dynamic programming guided genetic algorithm with adaptive selection (DPGA-AS). In this method, a dynamic programming method is proposed in the population initialization part to improve the efficiency of the genetic algorithm. An adaptive selection method for the GA is designed to avoid prematurely falling into the local optimal for better test quality. The question bank used in our experiments is assembled based on college-level calculus questions from well-known textbooks. The experimental results show that the proposed techniques can construct test papers with both high effectiveness and efficiency. The computation time of the test assembly problem is reduced from 3 hours to 2 seconds for a 5000-size question bank as compared to a linear programming model with similar test quality. The test quality of the proposed method is better than the other baselines.

A scheme for the integration of $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type fractional differential equations (FDEs) is presented in this paper. The approach is based on the expansion of solutions to FDEs via fractional power series. It is proven that $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type FDEs can be transformed into equivalent $ \left(\, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)}\right)^n $-type FDEs via operator calculus techniques. The efficacy of the scheme is demonstrated by integrating the fractional Riccati differential equation.

Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.

This paper contributes to the classification of flag-transitive 2-$ (v, k, \lambda) $ designs. Let $ \mathcal{D} $ be a non-trivial and non-symmetric $ 2 $-$ (v, k, \lambda) $ design with $ \lambda $ prime and $ G $ be a flag-transitive point-primitive automorphism group of $ \mathcal{D} $. A recent work by the first author and Chen has proven that the socle of $G$ is either a nonabelian simple group or an elementary abelian $ p $-group for some prime $ p $. In this paper, we focus on the case where the socle of $G$ is an exceptional group of Lie type and give all possible parameters of such 2-designs.

Let $ R $ be a commutative ring with the identity $ 1_{R} $, and let $ R^* $ be the multiplicative group of units in $ R $. An element $ a\in R^* $ is called an exceptional unit if there exists a $ b\in R^* $ such that $ a+b = 1_{R} $. We set $ R^{**} $ to be the set of all exceptional units in $ R $. In this paper, we consider the residue-class ring $ \mathbb{Z}_n $. For any positive integers $ n, s $, and $ c\in\mathbb{Z}_n $, let $ {\mathcal N}_{s}(n, c): = \sharp\big\{(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s : x_1+...+x_s\equiv c \pmod n\big\} $. In 2016, Sander (J.Number Theory 159 (2016)) got a formula for $ {\mathcal N}_{2}(n, c) $. Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander's theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for $ {\mathcal N}_{s}(n, c) $. This extends and improves earlier results.

This paper investigates rumor propagation in a multilingual environment, taking into account language usage variations. Firstly, a 2I2S2R model is proposed within a heterogeneous network framework that incorporates both immunologic and cross-transmitted mechanisms. Secondly, the paper calculates the basic reproduction number $ R_0 $ by the next-generation matrix method. Thirdly, the local asymptotic stability and the global asymptotic stability are further explored, which indicate that whether the rumor continuously spreads or becomes extinct is determined by the threshold. Finally, the numerical simulation and sensitivity analysis are given to illustrate the effectiveness of theoretical results and the influence of model parameters on rumor spreading.

Cholera, as an endemic disease around the world, has imposed great harmful effects on human health. In addition, from a microscopic viewpoint, the interference of random factors exists in the process of virus replication. However, there are few theoretical studies of viral infection models with biologically reasonable stochastic effects. This paper studied a stochastic cholera model used to describe transmission dynamics in China. In this paper, we adopted a special method to simulate the effect of environmental perturbations to the system instead of using linear functions of white noise, i.e., the transmission rate of environment to human was satisfied Ornstein–Uhlenbeck processes, which is a more practical and interesting. First, it was theoretically proved that the solution to the stochastic model is unique and global, with an ergodic stationary distribution. Moreover, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. Finally, several numerical simulations are provided to confirm our analytical results.

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

The strong geodetic number of a graph and its edge counterpart are recent variations of the pioneering geodetic number problem. Covering every vertex and edge of $ G $, respectively, using a minimum number of vertices and the geodesics connecting them, while ensuring that one geodesic is fixed between each pair of these vertices, is the objective of the strong geodetic number problem and its edge version. This paper investigates the strong geodetic number of the lexicographic product involving graph classes that include complete graph $ K_{m} $, path $ P_{m} $, cycle $ C_{m} $ and star $ K_{1, \, m} $ paired with $ P_{n} $ and with $ C_{n} $. Furthermore, the parameter is studied in the lexicographic product of, arbitrary trees with diameter-2 graphs whose geodetic number is equal to 2, $ K_{n}-e $ with $ K_{2} $ and their converses. Upper and lower bounds for the parameter are established for the lexicographic product of general graphs and in addition, the edge variant of the aforementioned problem is studied in certain lexicographic products. The strong geodetic parameters considered in this paper have pivotal applications in social network problems, thereby making them indispensable in the realm of graph theoretical research. This work contributes to the expansion of the current state of research pertaining to strong geodetic parameters in product graphs.

The study of Large Language Models (LLMs), as an interdisciplinary discipline involving multiple fields such as computer science, artificial intelligence, and linguistics, has diverse collaborations within its field. In this study, papers related to LLMs in the SSCI and SCI sub-collections of the Web of Science core database from January 2020 to April 2024 are selected, and a mixed linear regression model is used to assess the impact of scientific collaborations on the application of LLMs. On this basis, the paper further considers factors such as financial support and dominant countries to deeply explore the heterogeneous impact of scientific collaborations on the application of LLMs. The findings show that (1) excessive involvement of academic institutions limits the research and application of LLMs, and the number of authors does not have a significant effect on the application of LLMs; (2) with or without financial support, the role played by scientific collaborations in the application of LLMs does not significantly change; and (3) differences in the dominant countries of scientific collaborations have a slightly heterogeneous effect on the role of LLMs applications, which are mainly reflected in the number of collaborators.

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $ {L}_{2} $ and $ {L}_{\infty } $ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-$ W_8 $ flat Lorentzian Kähler space-time manifolds.

The filled function method is a deterministic algorithm for finding a global minimizer of global optimization problems, and its effectiveness is closely related to the form of the constructed filled function. Currently, the filled functions mainly have three drawbacks in form, namely, parameter adjustment and control (if any), inclusion of exponential or logarithmic functions, and properties that are discontinuous and non-differentiable. In order to overcome these limitations, this paper proposed a parameter-free filled function that does not include exponential or logarithmic functions and is continuous and differentiable. Based on the new filled function, a filled function method for solving unconstrained global optimization problems was designed. The algorithm selected points in the feasible domain that were far from the global minimum point as initial points, and improved the setting of the step size in the stage of minimizing the filled function to enhance the algorithm's global optimization capability. In addition, tests were conducted on 14 benchmark functions and compared with existing filled function algorithms. The numerical experimental results showed that the new algorithm proposed in this paper was feasible and effective.

This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.

Let $ X $ be an infinite Tychonoff space, and $ Y $ be a topological subspace of $ X $. In this paper, we study some covering properties of the subspace $ C_{p}\left(Y|X\right) $ of $ C_{p}\left(Y\right) $ consisting of those functions $ f\in C\left(Y\right) $ which admit a continuous extension to $ X $ equipped with the relative topology of $ C_{p}\left(Y\right) $. Among other results, we show that $ \left(i\right) $ $ C_{p}(Y|X) $ has a fundamental bounded resolution if and only if $ Y $ is countable; when $ X $ is realcompact and $ Y $ is closed in $ X $, we have $ \left(ii\right) $ if $ C_{p}(Y|X) $ admits a resolution of convex compact sets that swallows the local null sequences in $ C_{p}(Y|X) $, then $ Y $ is countable and discrete; $ \left(iii\right) $ if $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets, then $ Y $ is also countable and discrete, and, as a corollary, we deduce that $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets if and only if $ C_{p}(Y|X) $ is a Polish space. We also prove that $ \left(iv\right) $ for a metrizable space $ X $, $ C_{p}\left(X\right) $ is a quasi-$ \left(LB\right) $-space if and only if $ X $ is $ \sigma $-compact, and hence for a subspace $ Y $ of $ X $, the space $ C_{p}\left(Y|X\right) $ is a quasi-$ \left(LB\right) $-space. We include some examples and observations that answer natural questions raised in this paper.

In this paper, we studied the stability of the Cauchy problem for a class of first-order linear quasi-periodically forced PDEs on the $ m $-dimensional torus: $ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+(\xi+f(x, \omega t, \xi))\cdot \partial_x u = 0, \\ u(x, 0) = u_0(x), \ \end{array} \right. \end{eqnarray*} $ where $ \xi\in \mathbb{R}^m, x\in \mathbb{T}^m, \omega\in\mathbb{R}^d, $ in the case of multidimensional Liouvillean forced frequency. We proved that for each compact set $ \mathcal{O}\in\mathbb{R}^m $ there exists a Cantor subset $ \mathcal{O}_\gamma $ of $ \mathcal{O} $ with positive Lebesgue measure such that if $ \xi\in\mathcal{O}_\gamma, $ then for a perturbation $ f $ being small in some analytic Sobolev norm, there exists a bounded and invertible quasi-periodic family of linear operator $ \Psi(\omega t) $, such that the above PDEs are reduced by the transformation $ v: = \Psi(\omega t)^{-1}[u] $ into the following PDE: $ \begin{eqnarray*} \partial_t v+ (\xi+ m_\infty(\omega t))\cdot\partial_x v = 0, \end{eqnarray*} $ provided that the forced frequency $ \omega\in \mathbb{R}^d $ possesses finite uniform Diophantine exponent, which allows Liouvillean frequency. The reducibility can immediately cause the stability of the above Cauchy problem, that is, the analytic Sobolev norms of the Cauchy problem are controlled uniformly in time. The proof is based on a finite dimensional Kolmogorov-Arnold-Moser (KAM) theory for quasi-periodically forced linear vector fields with multidimensional Liouvillean forced frequency. As we know, the results on Liouvillean frequency existing in the literature deal with two-dimensional frequency and exploit the theory of continued fractions to control the small divisor problem. The results in this paper partially extend the analysis to higher-dimensional frequency and impose a weak nonresonance condition, i.e., the forced frequency $ \omega $ possesses finite uniform Diophantine exponent. Our result can be regarded as a generalization of analytic cases in the work [R. Feola, F. Giuliani, R. Montalto and M, Procesi, Reducibility of first order linear operators on tori via Moser's theorem, J. Funct. Anal., 2019] from Diophantine frequency to Liouvillean frequency.

This paper is concerned with the attraction-repulsion chemotaxis system (1.1) define on a bounded domain $ \Omega \subset \mathbb{R}^N(N\geq 1) $ with no-flux boundary conditions. The source function $ f $ in this system is a smooth function $ f $ that satisfies $ f(u)\leq a-bu^\eta $ for $ u\geq0 $. It is proven that $ \eta\geq1 $ is sufficient to ensure the boundedness of the solution when $ r < \frac{4(N+1)}{N(N+2)} $ is in the balance case $ \chi\alpha = \xi\gamma $, which improve the relevant results presented in papers such as Li and Xiang (2016), Xu and Zheng (2018), Xie and Zheng (2021), and Tang, Zheng and Li (2023).

The purpose of this paper is to establish a numerical method for solving time-fractional diffusion equations. To obtain the numerical solution, a binary reproducing kernel space is defined, and the orthonormal basis is constructed based on Legendre polynomials in this space. In order to find the $ {\varepsilon} $-approximation solution of time-fractional diffusion equations, which is defined in this paper, the algorithm is designed using the constructed orthonormal basis. Some numerical examples are analyzed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that the presented method is considerably accurate and effective, as expected.

In this paper, we investigated $ n $-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $ 3 $-dimensional space at the end of the paper.

In this paper, we investigate the asymptotic stability of the trajectories governed by some delay differential inequalities on time scales. Based on time scale theory and the fixed-point theorem, some sufficient conditions are obtained for guaranteeing asymptotic stability. It is interesting that the inequalities studied in this paper include the generalized Halanay inequalities. Due to the fact that dynamic systems on a time scale unify discrete and continuous systems, the results of this paper have wider application value. Furthermore, some numerical examples verify the main results.

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.

Skeleton-based action recognition is an important but challenging task in the study of video understanding and human-computer interaction. However, existing methods suffer from two deficiencies. On the one hand, most methods usually involve manually designed convolution kernel which cannot capture spatial-temporal joint dependencies of complex regions. On the other hand, some methods just use the self-attention mechanism, ignoring its theoretical explanation. In this paper, we proposed a unified spatio-temporal graph convolutional network with a self-attention mechanism (SA-GCN) for low-quality motion video data with fixed viewing angle. SA-GCN can extract features efficiently by learning weights between joint points of different scales. Specifically, the proposed self-attention mechanism is end-to-end with mapping strategy for different nodes, which not only characterizes the multi-scale dependencies of joints, but also integrates the structural features of the graph and an ability of self-learning fusion features. Moreover, the attention mechanism proposed in this paper can be theoretically explained by GCN to some extent, which is usually not considered in most existing models. Extensive experiments on two widely used datasets, NTU-60 RGB+D and NTU-120 RGB+D, demonstrated that SA-GCN significantly outperforms a series of existing mainstream approaches in terms of accuracy.

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research.

This paper delves into three research directions, leveraging the Lovász $ \vartheta $-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that $ n \geq 14 $, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on $ n $ vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász $ \vartheta $-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the $ \vartheta $-function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the $ \vartheta $-function variant by Schrijver does not possess the property of the Lovász $ \vartheta $-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory.

In recent years, there has been a notable shift in focus towards the analysis of non-stationary time series, driven largely by the complexities associated with delineating significant asymptotic behaviors inherent to such processes. The genesis of the theory of locally stationary processes arises from the quest for asymptotic inference grounded in nonparametric statistics. This paper endeavors to formulate a comprehensive framework for conducting inference within the realm of locally stationary functional time series by harnessing the conditional $ U $-statistics methodology as propounded by W. Stute in 1991. The proposed methodology extends the Nadaraya-Watson regression function estimations. Within this context, a novel estimator was introduced for the single index conditional $ U $-statistics operator, adept at accommodating the non-stationary attributes inherent to the data-generating process. The primary objective of this paper was to establish the weak convergence of conditional $ U $-processes within the domain of locally stationary functional mixing data. Specifically, the investigation delved into scenarios of weak convergence involving functional explanatory variables, considering both bounded and unbounded sets of functions while adhering to specific moment requirements. The derived findings emanate from broad structural specifications applicable to the class of functions and models under scrutiny. The theoretical insights expounded in this study constitute pivotal tools for advancing the domain of functional data analysis.

In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $ O(h_{s}^{3+\sigma_{1} }+h_{t}^{3+\sigma_{2} }) $ with $ 0 < \sigma_{1}, \sigma_{2} < 1 $ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $ \psi $ to support the theoretical results.

This paper addresses the challenge of adaptive control for nonstrict-feedback nonlinear systems that involve input delay, actuator faults, and external disturbance. To deal with the complexities arising from input delay and unknown functions, we have incorporated Pade approximation and radial basis function neural networks, respectively. An adaptive controller has been developed by utilizing the Lyapunov stability theorem and the backstepping approach. The suggested method guarantees that the tracking error converges to a compact neighborhood that contains the origin and that every signal in the closed-loop system is semi-globally uniformly ultimately bounded. To demonstrate the efficacy of the proposed method, an electromechanical system application example, and a numerical example are provided. Additionally, comparative analysis was conducted between the Pade approximation proposed in this paper and the auxiliary systems in the existing method. Furthermore, error assessment criteria have been employed to substantiate the effectiveness of the proposed method by comparing it with existing results.

The asymptotic synchronization problem of chaotic Lur'e systems in the master-slave framework was explored in this paper. A time-varying delay feedback controller with quantization considerations and a delay-product-type Lyapunov-Krasovskii functional technique were employed to tackle this problem. Consider an error system based on master and slave systems, for which sufficient asymptotic stability requirements are developed to assure that the addressed system achieves proper synchronization. Following that, the desired control gain was determined by finding a feasible solution to these stability requirements. The results of this paper were validated using a numerical example with simulations, which revealed that they were superior to previously published ones.

Accuracy is an important factor to consider when evaluating the performance of a manipulator. The accuracy of a manipulator is determined by its ability to accurately move and position objects in a precise manner. This research paper aims to evaluate the performance of different methods for the kinematic analysis of manipulators. The study employs four distinct techniques, namely mathematical modeling using the closed form solutions method, roboanalyzer, Peter Corke toolbox, and particle swarm optimization, to perform kinematic analysis for manipulators. The KUKA industrial manipulator is used as an illustrative case study in this research due to its widespread use in various industrial applications in addition to its high precision and stability. Its wide usage in the industry makes the results of this research highly relevant and allows for a thorough evaluation of the performance of the different methods being studied. Furthermore, understanding the kinematic analysis of the manipulator can also help in improving the performance and increasing the efficiency of the robot in different tasks. This paper conducts a comparison of the accuracy of the four methods, and the results indicate that particle swarm optimization is the most accurate method. The RoboAnalyzer approach achieved the fastest execution time.

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper.

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order $ \beta_{\xi} $ instead of $ \beta_{1} $; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.

In this paper, we dealt with the tracking control problem of a class of fractional-order uncertain systems with time delays. In order to handle the effects brought by the uncertainties, external disturbances, time-delay terms, and to overcome the obstacles caused by inputs saturation, the tracking controller, which consisted of linear control law, nonlinear law, and robust control law proposed in this paper, was designed by combining the composite nonlinear feedback control method and the properties of fractional order operators. Furthermore, the validation of this tracking controller was proved.

In demanding application scenarios such as clinical psychotherapy and criminal interrogation, the accurate recognition of micro-expressions is of utmost importance but poses significant challenges. One of the main difficulties lies in effectively capturing weak and fleeting facial features and improving recognition performance. To address this fundamental issue, this paper proposed a novel architecture based on a multi-scale 3D residual convolutional neural network. The algorithm leveraged a deep 3D-ResNet50 as the skeleton model and utilized the micro-expression optical flow feature map as the input for the network model. Drawing upon the complex spatial and temporal features inherent in micro-expressions, the network incorporated multi-scale convolutional modules of varying sizes to integrate both global and local information. Furthermore, an attention mechanism feature fusion module was introduced to enhance the model's contextual awareness. Finally, to optimize the model's prediction of the optimal solution, a discriminative network structure with multiple output channels was constructed. The algorithm's performance was evaluated using the public datasets SMIC, SAMM, and CASME Ⅱ. The experimental results demonstrated that the proposed algorithm achieves recognition accuracies of 74.6, 84.77 and 91.35% on these datasets, respectively. This substantial improvement in efficiency compared to existing mainstream methods for extracting micro-expression subtle features effectively enhanced micro-expression recognition performance and increased the accuracy of high-precision micro-expression recognition. Consequently, this paper served as an important reference for researchers working on high-precision micro-expression recognition.

This paper presents a novel hybrid algorithm that combines the Butterfly Optimization Algorithm (BOA) and Quantum-behavior Particle Swarm Optimization (QPSO) algorithms, leveraging $ gbest $ to establish an algorithm communication channel for cooperation. Initially, the population is split into two equal subgroups optimized by BOA and QPSO respectively, with the latter incorporating the Lévy flight for enhanced performance. Subsequently, a hybrid mechanism comprising a weight hybrid mechanism, a elite strategy, and a diversification mechanism is introduced to blend the two algorithms. Experimental evaluation on 12 benchmark test functions and the Muskin model demonstrates that the synergy between BOA and QPSO significantly enhances algorithm performance. The hybrid mechanism further boosts algorithm performance, positioning the new algorithm as a high-performance method. In the Muskingum model experiment, the algorithm proposed in this article can give the best sum of the square of deviation (SSQ) and is superior in the comparison of other indicators. Overall, through benchmark test function experiments and Muskin model evaluations, it is evident that the algorithm proposed in this paper exhibits strong optimization capabilities and is effective in addressing practical problems.

Convolutional neural networks (CNNs) play a prominent role in solving problems in various domains such as pattern recognition, image tasks, and natural language processing. In recent years, neural architecture search (NAS), which is the automatic design of neural network architectures as an optimization algorithm, has become a popular method to design CNN architectures against some requirements associated with the network function. However, many NAS algorithms are characterised by a complex search space which can negatively affect the efficiency of the search process. In other words, the representation of the neural network architecture and thus the encoding of the resulting search space plays a fundamental role in the designed CNN performance. In this paper, to make the search process more effective, we propose a novel compact representation of the search space by identifying network blocks as elementary units. The study in this paper focuses on a popular CNN called DenseNet. To perform the NAS, we use an ad-hoc implementation of the particle swarm optimization indicated as PSO-CNN. In addition, to reduce size of the final model, we propose a segmentation method to cut the blocks. We also transfer the final model to different datasets, thus demonstrating that our proposed algorithm has good transferable performance. The proposed PSO-CNN is compared with 11 state-of-the-art algorithms on CIFAR10 and CIFAR100. Numerical results show the competitiveness of our proposed algorithm in the aspect of accuracy and the number of parameters.

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper.

This paper delves into fixed point findings within a complete partially ordered $ b $-metric space, focusing on mappings that adhere to weakly contractive conditions in the presence of essential topological characteristics. These findings represent modifications of established results and further extend analogous outcomes in the existing literature. The conclusions are substantiated by illustrative examples that strengthen the conclusion of the paper.

In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area.

In this paper, we expanded the concept of tempered fractional derivatives within both the Riemann-Liouville and Caputo frameworks, introducing a novel class of fractional operators. These operators are characterized by their dependence on a specific arbitrary smooth function. We then investigated the existence and uniqueness of solutions for a particular class of fractional differential equations, subject to specified initial conditions. To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.

This study employed an event-triggered control (ETC) strategy to investigate the problems of fixed-time stabilization (FTS) and preassigned-time stabilization (PTS) for state-dependent switching neural networks (SDSNNs) that involved mixed time delays. To enhance the network's generalization capability and accelerate convergence stabilization, a more intricate weight-switching mechanism was introduced, then to mitigate transmission energy consumption, this paper proposed a tailored event-triggering rule that triggered the ETC solely at predetermined time points. This rule ensured the stability of the system while effectively reducing energy consumption. Using the Lyapunov stability theory and various inequality techniques, this paper presented new results for FTS and PTS of SDSNNs. The validity of these findings was supported by conducting data simulations in two illustrative examples.

This paper addresses the new concatenation model incorporating quintic-order dispersion, incorporating four well-known nonlinear models. The concatenated models are the nonlinear Schrödinger equation, the Hirota equation, the Lakshmanan-Porsezian-Daniel equation, and the nonlinear Schrödinger equation with quintic-order dispersion. The model itself is innovative and serves as an encouragement for investigating and analyzing the extracted optical solitons. These models play a crucial role in nonlinear optics, especially in studying optical fibers. Three integration algorithms are implemented to investigate the optical solitons with the governing model. These techniques are the Weierstrass-type projective Riccati equation expansion method, the addendum to Kudryashov's method, and the new mapping method. The solutions obtained include various solitons, such as bright, dark, and straddled solitons. Additionally, the paper reports hyperbolic solutions and Weierstrass-type doubly periodic solutions. These solutions are novel and have never been reported before. Visual depictions of some selected solitons illustrate these solutions' dynamic behavior and wave structure.

In many studies dealing with mathematical models, the subject is examining the fitting between actual data and the solution of the mathematical model by applying statistical processing. However, if there is a solution that fluctuates greatly due to a small perturbation, it is expected that there will be a large difference between the actual phenomenon and the solution of the mathematical model, even in a short time span. In this study, we address this concern by considering Ulam stability, which is a concept that guarantees that a solution to an unperturbed equation exists near the solution to an equation with bounded perturbations. Although it is known that Ulam stability is guaranteed for the standard von Bertalanffy growth model, it remains unsolved for a model containing the Allee effect. This paper investigates the Ulam stability of a von Bertalanffy growth model with the Allee effect. In a sense, we obtain results that correspond to conditions of the Allee effect being very small or very large. In particular, a more preferable Ulam constant than the existing result for the standard von Bertalanffy growth model, is obtained as the Allee effect approaches zero. In other words, this paper even improves the proof of the result in the absence of the Allee effect. By guaranteeing the Ulam stability of the von Bertalanffy growth model with Allee effect, the stability of the model itself is guaranteed, and, even if a small perturbation is added, it becomes clear that even a small perturbation does not have a large effect on the solutions. Several examples and numerical simulations are presented to illustrate the obtained results.

The global robust stability of uncertain delayed fuzzy cellular neural networks (UDFCNNs) was analyzed in this paper. The major results of this paper provided some new criteria for the existence and uniqueness of the equilibrium point of UDFCNN. Furthermore, suitable Lyapunov-Krasovskii functionals was designed for obtaining the adequate conditions for the global asymptotic robust stability and global exponential robust stability of UDFCNN. Finally, several numerical examples was provided to verify the validity of the results.

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.

The cotton-picking robot needs to locate the target object in space in the process of picking in the field and other outdoor strong light complex environments. The difficulty of this process was binocular matching. Therefore, this paper proposes an accurate and fast binocular matching method. This method used the deep learning model to obtain the position and shape of the target object, and then used the matching equation proposed in this paper to match the target object. Matching precision of this method for cotton matching was much higher than that of similar algorithms. It was 54.11, 45.37, 6.15, and 12.21% higher than block matching (BM), semi global block matching (SGBM), pyramid stereo matching network (PSMNet), and geometry and context for deep stereo regression (GC-net) respectively, and its speed was also the fastest. Using this new matching method, the cotton was matched and located in space. Experimental results show the effectiveness and feasibility of the algorithm.

This paper explores the application of rough set theory in analyzing ambiguous data within complete information systems. The study extends topological structures using equivalence relations, establishing an extension of topological lattice within lattices. Various relations on topological spaces generate different forms of exact and rough lattices. Building on Zhou's work, the research investigates rough sets within the extension topological lattice and explores the isomorphism between topology and its extension. Additionally, the paper investigates the integration of lattices and rough sets, essential mathematical tools widely used in problem-solving. Focusing on computer science's prominent lattices and Pawlak's rough sets, the study introduces extension lattices, emphasizing lower and upper extension approximations' adaptability for practical applications. These approximations enhance pattern recognition and model uncertain data with finer granularity. While acknowledging the benefits, the paper stresses the importance of empirical validations for domain-specific efficacy. It also highlights the isomorphism between topology and its extension, revealing implications for data representation, decision-making, and computational efficiency. This isomorphism facilitates accurate data representations and streamlines computations, contributing to improved efficiency. The study enhances the understanding of integrating lattices and rough sets, offering potential applications in data analysis, decision support systems, and computational modeling.