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• The referees' comments have been incorporated into the new version of the manuscript, correcting many typos. • The structure of the paper has been improved and the number of figures has been reduced. • Some maintained graphs have been combined to reduce the number of figures. • Some tables have been combined for the same purpose. • The notation used for the quasi-interpolation operators has been improved. • In the quadratic case, a very interesting comment by reviewer 3 has been incorporated, which relates the quasi-interpolation scheme introduced in the article to a quasi-interpolant obtained by discretisation of the linear functional of the classical quadratic differential quasi-interpolant. • The constructive method introduced in the article has the potential to give rise to new quasi-interpolants that incorporate specific properties. In this paper we propose the construction of univariate low-degree quasi-interpolating splines in the Bernstein basis, considering C 1 and C 2 smoothness, specific polynomial reproduction properties and different sets of evaluation points. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values. Moreover, we get quasi-interpolating splines with special properties, imposing particular requirements in case of free parameters. Finally, we provide numerical tests showing the performances of the proposed methods. [ABSTRACT FROM AUTHOR]

• Given an edge-colored graph G , if any two edges of G receive distinct colors, then we call G a r a i n b o w graph. The anti-Ramsey number A R (G ; F) is the maximum integer k in a k -edge-coloring of G that there is no any rainbow graph in F. • An interesting topic about edge-colored graphs is to study the order of monochromatic or rainbow component. Here we focus on anti-Ramsey problem on rainbow component. This equals to study the existence of rainbow trees. We call a tree of order k + 1 a k -tree, since it just contains k edges. • The anti-Ramsey problem on k-trees obviously received less attention. The value of anti-Ramsey number of k -trees in K n has been determined by Jiang and West. Later, Jin and Li considered its anti-Ramsey number in complete bipartite graph. Recently, Zhang and Dong gave a formula to determine the value of A R (G ; k -trees), where G is a complete multipartite graph of order n , for 4 n − 2 5 ≤ k ≤ n − 1. When k is small, it is difficult to determine the exact value. In this paper, we aim to determine the exact value of A R (G ; k -trees) when G is a balanced complete tripartite graph, especially for k much less than the order of host graph. The anti-Ramsey number A R (K , F) is the largest color number in an edge-coloring of K that contains no rainbow copies of graphs in the family F. In this paper, we determine the exact value of the anti-Ramsey number for trees in a balanced complete tripartite graph. [ABSTRACT FROM AUTHOR]

• Unlike [7,8] and [31] , which designs the controller under the assumption that the system states are available and the Lipschitz condition is needed for the nonlinear term. This paper designs the NNs-based observer to estimate unknown states and approximate nonlinear terms simultaneously, which guarantees the robustness and stability of the MASs can be improved. • Compared to Yang et al. [20] , He et al. [21] , Ma et al. [22] , Liu et al. [23] , Fu et al. [24] , Zhao and Yang [25] dealing with deception attacks by using the global topology information, the fully distributed intrusion-tolerance control scheme is designed in this paper to achieve the consensus control for multi-agent systems that suffer from deception attacks, which only uses adjacent agent information in the design process of controller. In addition, the impact of the error signal injected by the attacker on the adaptive parameters are considered. This paper proposes a novel resilient intrusion-tolerance control scheme to solve the consensus control problem of multi-agent systems (MASs) under deception attacks. Consider the state unavailability of MASs with unknown nonlinear terms, the unmeasurable state is estimated by designing the neural-networks(NNs)-based observer. Then, a fully distributed intrusion-tolerant adaptive protocol is developed to implement the consensus control task. The designed control strategy only involves the agent states that interact with the neighboring agent and does not relate to the whole communication topology information. The designed controller can ensure that the consensus error of the nonlinear MASs is eventually ultimately bounded under deception attacks. Finally, a numerical simulation is provided to verify the feasibility of the designed scheme. [ABSTRACT FROM AUTHOR]

• Fitness-based mobility is defined in a pair of species. • Considering fitness to mobility can promote coexistence. • Pairwise interaction is governed by an active individual. • Strong sensitivity of fitness can produce empty sites. In ecosystems, species are willing to stay longer when they are in good environments and, in this regard, explaining biodiversity and understanding the relative effect of species' behavior is a fundamental issue in ecological sciences. Staying or moving stands for the movement of the population, and mobility has been accepted as one of the important mechanisms in systems of cyclic competition. In this paper, we investigate the biodiversity in the system of cyclic competition when an individual's mobility is affected by environmental fitness. We found that fitness-based mobility, which is defined by the ratio of the difference between predator and prey among species, can lead to the slow behavior of species when local fitness is higher than the average fitness and enhance the biodiversity of species. The fitness-based mobility behavior is successful in reducing the extinction probability on the lattice network. However, when the average degree of the network increases, the extinction probability increases significantly. In addition, when we consider the same model on heterogeneous networks, the extinction probability is always 1 regardless of the sensitivity to the fitness. The finding indicates that the heterogeneous network destroys species coexistence. Through extensive numerical investigations, we provided solid evidence supporting the great importance of fitness-based mobility in maintaining biodiversity. We further highlighted the significant relationship among fitness, the fraction of empty sites, and the tunable parameter by investigating the correlation that the relationship is not influenced by mobility. [ABSTRACT FROM AUTHOR]

• The paper discusses the relationship of state jump at the switching moment, a state jump lemma with disturbance input and time-varying delay is proposed, using differential and algebraic systems. Current results such as [15] do not consider the disturbance input problem. • A new bounding lemma is established to solve the reachable state boundary problem of continuous SSSs under all initial states. Compared with existing lemmas such as studying the discrete system under zero initial state in [32] and researching the continuous system under zero initial state in [33]. Our proposed lemma overcomes these limitations. • The paper provides criteria for the estimation of the reachable set for continuous SSSs with or without time-varying delay based on the Lyapunov theory. Only one work has studied the reachable set of discrete-time SSSs [34]. Our proposed criteria significantly contribute to the body of knowledge on the estimation of the reachable set for continuous SSSs. This paper addresses the estimation problem of the reachable set for switched singular systems with state jump under initial condition and bounded peak disturbance. The aim is to determine a bounded set that contains all reachable states. Initially, a state jump lemma is presented for switched singular systems. Then, a new bounding lemma is shown by analyzing the variation of piecewise continuous functional in subintervals and referencing the definition of the average dwell-time. Afterwards, using lemmas and linear matrix inequalities, sufficient conditions are derived for switched singular systems with and without time-varying delay, which ensure that the state trajectory of the system remains within a closed bounded set. Finally, the obtained results are validated through numerical examples. [ABSTRACT FROM AUTHOR]

• Different from the nonlinear MAS in [8] whose all agents are second-order, in this paper, the considered MAS consists of first-order and second-order agents simultaneously. Moreover, all agents are considered to have multiple modes and then exhibit the switching feature. Therefore, the MAS model of this paper is more general. • To cope with the issue of DoS attacks on communication channels of systems, an underlying topology switching mechanism is proposed to ensure tolerance for network disconnections. Meanwhile, the DoS attack frequency and attack time ratio are given to characterize the features of DoS attacks. Furthermore, an efficient consensus algorithm is presented by employing the back-stepping technique and the Lyapunov function method. • In contrast to [38], the hysteretic quantization input is considered for switched nonlinear MASs under DoS attacks in this paper, which attenuates the impact of hysteretic quantizers on control performances. Besides, global information is not used in the protocol design, and the dynamics of agents in this study need not to strictly abide by the nonlinear Lipschitz condition. Hence, the developed control scheme in this paper is distributed. In this paper, the consensus tracking control problem is investigated for a class of heterogeneous switched nonlinear multiagent systems (MASs) with input quantization under denial-of-service (DoS) attacks, where first-order agents and second-order agents exist simultaneously, and each agent shows the switching characteristic. Due to the prevalence of DoS attacks in communication channels, a DoS attack model is constructed in this study, and the upper bounds on the duration and frequency of DoS attacks are given. Meanwhile, taking the chattering phenomena of control inputs into account, a hysteresis quantizer is introduced to overcome the issue of the undesirable input chattering. Hence, a new anti-attack distributed consensus protocol is developed such that MASs with input quantization can effectively resist the impact of DoS attacks. Based on the Lyapunov stability theory, the boundedness of all signals in the closed-loop system is verified and the consensus target can be achieved. Finally, the effectiveness of the proposed control scheme is demonstrated by two simulation examples. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this short note is to show on a numerical example that one of the two optimization algorithms proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] may not lead to the optimal solution in some cases. Besides it will be pointed out that the other algorithm (correct, but unfortunately ineffective for larger problems) proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] can be extended to a wider class of non-convex optimization problems. [Copyright &y& Elsevier]

This paper considers the time-varying formation (TVF) tracking issue of a class of multiagent systems (MASs) with input saturation. Meanwhile, the prescribed transient and steady-state performance of MASs is guaranteed. Compared to the existing prescribed performance control (PPC) approaches, the maximum convergence time of the designed algorithm is preassigned. To save network resources, an event-based protocol is introduced to achieve lower information communication frequency, where Zeno behavior is excluded. Furthermore, an improved control strategy is employed where an auxiliary system is designed to improve the performance under actuator saturation. According to Lyapunov stability theory, the expected practical TVF tracking control problem is solved for the MASs. Finally, the effectiveness of the designed schemes is demonstrated by the simulation examples. • A low-complexity, static, state-feedback control algorithm is proposed. • The prescribed performance control law achieves the appointed-time convergence. • The tracking precision of the control law is guaranteed under actuator saturation. [ABSTRACT FROM AUTHOR]

As a social welfare, pension insurance has provided a great assistance to retiree in maintaining their livelihoods. In order to reward the contributions made by cooperators, an additional pension income is provided to them. In this paper, we study a public goods game based on the pension system on population structure, in which only cooperators can purchase pension insurance and receive pension benefits upon meeting specified conditions. In the simulation, we demonstrate the evolution of cooperation frequency over time and investigate the influence of insurance premium, pension baseline, the number of defected neighbors of cooperators, and the enhancement factor on the cooperation frequency. Additionally, we examine the impact of the pension system on cooperative behavior in triangular and hexagonal networks. Our results suggest that in the public goods game on networked structure, the pension system not only fails to promote cooperation but actually inhibits it, even when cooperators do not have to pay any costs and can receive pensions when conditions are met. [ABSTRACT FROM AUTHOR]

In graph theory, a packing within graph G is defined as a collection H = { H 1 , H 2 } , where H 1 and H 2 are both equivalent to G , yet they exist as edge-disjoint subgraphs within K n. Considering a positive integer k , we introduce D k , a set comprising graphs within G , each with a maximum degree not exceeding k. Extending the concept of a packing, a near packing that accommodates D k in graph G emerges, realized by overlapping two iterations of G. In this configuration, the intersecting subgraph formed by edges shared in both G copies is a member of D k. The notation m (n , D k) represents the highest possible value of m , ensuring each graph with order n and size at most m can incorporate a near packing that accommodates D k. A notable case is m (n , D 0) = n − 2 , highlighting that a near packing which admits D 0 equates to a standard packing. This paper focuses on establishing the value of m (n , D 1) = ⌊ 3 n − 5 2 ⌋. • We show that each graph of order n and size at most ⌊ 3 n − 5 2 ⌋ has a near packing. • Our proof is constructive. • The bound of our result is sharp in general. [ABSTRACT FROM AUTHOR]

Given a non-negative integer g , the g - e x t r a c o n n e c t i v i t y in a connected graph G is defined as the minimum number of vertices that, when removed, results in G becoming disconnected and each remaining component containing more than g vertices. As an extension of the connectivity, the g - e x t r a c o n n e c t i v i t y provides a better measurement for the fault tolerance of an interconnection network. The strong product G 1 ⊠ G 2 of graphs G 1 and G 2 is the graph with vertex set V (G 1) × V (G 2) , where two distinct vertices (a 1 , b 1) and (a 2 , b 2) are adjacent if and only if a 1 = a 2 and b 1 b 2 ∈ E (G 2) , or b 1 = b 2 and a 1 a 2 ∈ E (G 1) , or a 1 a 2 ∈ E (G 1) and b 1 b 2 ∈ E (G 2). In this paper, we focus on networks modeled by the s t r o n g p r o d u c t G 1 ⊠ G 2. We determine the g (≤ 3) - e x t r a c o n n e c t i v i t y of G 1 ⊠ G 2 , where G 1 and G 2 are regular and maximally connected graphs with girth at least g + 4. Additionally, we give the g (≤ 3) - e x t r a conditional diagnosability of G 1 ⊠ G 2 under PMC model. • Determined the g (≤3)- extra connectivity of the strong product of regular and maximally connected graphs. • Gave the g (≤ 3) - extra conditional diagnosability of a class of strong product graphs under PMC model. • Provided two reliability-measurements for the interconnection networks modeled by a class of strong product graphs. [ABSTRACT FROM AUTHOR]

This paper focuses on non-instantaneous impulsive systems with double state-dependent delays. The finite-time stability criteria of such systems are obtained via impulsive control theory and Lyapunov function method, and the effects of destabilizing impulse and stabilizing impulse on the finite-time stability are considered, respectively. Meanwhile, the obtained results are applied to finite-time stability of genetic regulatory networks with state-dependent delays and non-instantaneous impulses as an example. • The relationship between state-dependent delay and non-instantaneous impulse is revealed. • The restrictions of state-dependent delays are reduced in order to make the conditions more applicable. • The effects of destabilizing and stabilizing impulse on the finite-time stability are considered. • The obtained results are applied to genetic regulatory networks. [ABSTRACT FROM AUTHOR]

This study proposes a search algorithm for the compact uncertainty region of system elements with affine constraints and interval information. While existing studies have considered only the interval information of uncertainty, the proposed algorithm reflects both the interval and affine constraints of uncertain elements to reduce conservatism due to uncertainties. First, the affine constraints of uncertainty are plotted in an N -dimensional space, where N is the number of elements that should be considered. Due to the properties of the affine constraints, the affine constraints are expressed as an (N -1) dimensional structure. The interval information is then expressed as an N -dimensional structure. The algorithm iteratively checks the edges of the N -dimensional structure between two geometrical structures to determine the intersection region. The algorithm then collects the vertices of the convex polytope that overlap the two structures. To show the effectiveness of proposed algorithm, this study proposes a consensus criterion for a multi-agent time-delayed system with uncertain switching topologies as an application of a system with uncertainties in system elements. • We clarify the motivation and contribution of this paper. • We update figures in Example 1. • We add the Example 2 to better show the efficiency of proposed algorithm. • To help us understand, we added Remarks 1, 8. [ABSTRACT FROM AUTHOR]

This paper specializes in exponential output synchronization of reaction-diffusion neural networks (RDNNs) for two different cases of output couplings. The proposed model is novel for incorporating multiple output couplings related to their own output states as well as output spatial diffusion couplings in the absence of ordinary differential equations (ODE) systems. Generalized intermittent control based on spatial sampled-data is adopted for the first time to handle error systems consisting of the states of RDNNs with different output couplings scenarios and the average of the sum of their output states to achieve exponential stability. Some sufficient conditions for determining the proposed network models are established by control protocols based on the Lyapunov functional method. Finally, the obtained theoretical results are demonstrated to be valid by numerical simulations. • A novel RDNNs with multiple output spatial diffusion coupling is proposed. • The method of intermittent spatial sampled-data control is first employed in the output-coupled model. • The output synchronization criterion is established by means of Lyapunov functions and intermittent control. [ABSTRACT FROM AUTHOR]

Let C m be a cycle with length m. The k -uniform hypercycle with length m is obtained by adding k − 2 new vertices in every edge of C m , denoted by C m , k. In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four. • By the Matrix-Tree Theorem and BEST Theorem, etc., we present some trace formulas of uniform hypercycles with length four. • By applying the trace formulas, we study the characteristic polynomials of uniform hypercycles with length four. • We study the traces and obtain an explicit characteristic polynomial expression of uniform hypercycles with length four. [ABSTRACT FROM AUTHOR]

The problem of stochastic input-to-state stability in probability (ISSiP) is studied in this paper for a class of constrained impulsive switched systems with unstable subsystems, where the switching signal and impulsive signal are confined to the subsets of all modes. Especially, the impulsive intensity and impulsive density are both stochastic, which triggered by a renewal process. Taking into account asynchrony of switching and impulsive signal, a new comprehensive admissible edge-dependent average impulsive interval (CAED-AII) concept is proposed to characterize the impulsive signal. Under constrained switching and stochastic impulses, we investigate ISSiP of impulsive switched systems in two cases: the impulsive intensity is supposed to be stochastic and impulsive density is fixed, and the impulsive intensity and density are both stochastic. In these cases, the situation of the subsystem is divided into two scenarios for discussion as well. The first scenario is that all subsystems are unstable. The second scenario is that some subsystems are unstable and some subsystems are stable. With the help of the admissible edge-dependent average dwell time (AED-ADT), CAED-AII, and Lyapunov-like function, sufficient conditions are established for ISSiP of the whole system. Finally, two numerical examples are given to illustrate the developed results. • A more general framework of nonlinear systems is constructed in the presence of switching signals and impulsive signals, which contains constrained switchings and impulses, and stabilizing or destabilizing impulses with stochastic intensity and density. Due to the asynchronous switchings and impulses, two kinds of time sequence need to be tackled separately. • A novel CAED-AII concept is proposed for impulsive signals. Since the impulsive signals occur independently of the switching signals, it is less conservative. For impulsive intensity and density of stochastic impulses, the AII of impulse mode with different intensity and density is related to the current and historical impulse mode. • The ISSiP problem is addressed for two cases. When all subsystems are unstable but the impulses are stabilizing, the system is ISSiP while the impulses only have stochastic intensity or the impulses have stochastic intensity and density. When some subsystems are stable and the impulses are destabilizing, the system is ISSiP with the impulses are as before. • A comprehensive modeling paradigm that can describe stochastic impulses from the method of CAED-AII is constructed through the impulsive effects of stochastic impulsive intensity and density. [ABSTRACT FROM AUTHOR]

Let G ˙ be an unbalanced signed graph. In this paper, we establish that e (G ˙) ⩽ 1 2 (n 2 − n) − (n − 3) and ρ (G ˙) ⩽ n − 2 if G ˙ does not contain unbalanced K 4 as a signed subgraph. Moreover, comprehensive characterizations of the extremal signed graphs have been obtained. • The situation that the index of signed graph equals to the spectral radius of underlying graph has been characterized. • We determine the Turán number of K 4 − in unbalanced signed graph, and we characterize all the extremal signed graphs. • We solve the spectral Turán problem among all K 4 − -free unbalanced signed graphs. [ABSTRACT FROM AUTHOR]

In this paper, we explore proper vertex colorings of graphs with the additional constraint that the colors used on the closed neighborhood of every vertex form an integer interval; such colorings are called proper closed interval vertex colorings (PCIV colorings for short). As not every graph admits such a coloring, we are concerned with the problem of deciding which graphs admit PCIV colorings, describing several sufficient conditions (among them unique and 3-colorability, partial decomposability into two bipartite graphs, and density constraints) as well as some constructions of PCIV uncolorable graphs. We also present an algorithmic support for PCIV colorability based on a modification of the standard greedy algorithm for standard proper coloring. • Propose a new proper coloring (PCIV coloring) where colors on closed neighborhoods of vertices form integer intervals. • Particular sufficient conditions for PCIV colorability are proved. • Several constructions of PCIV uncolorable graphs are presented. • Greedy algorithm for finding a PCIV coloring is developed. [ABSTRACT FROM AUTHOR]

This paper focuses on solving stochastic linear quadratic nonzero-sum differential games with completely unknown system matrices and long-time average costs. Firstly, for the case that the system matrices are known, we design a model-based value iteration (VI) algorithm and a model-based robust VI algorithm to solve the coupled algebraic Riccati equations (AREs), which are directly related to the construction of the feedback Nash equilibrium solution. Then, we present a model-free VI algorithm, based on which feedback control strategies for players are designed using only the data of states and inputs, without requiring the prior knowledge of the system matrices. Moreover, we show that the designed feedback control strategies make the closed-loop game system stable and reach the Nash equilibrium. Finally, a numerical simulation is given to demonstrate the effectiveness of the proposed method. • A model-based robust VI algorithm is proposed to solve stochastic linear quadratic nonzero-sum games. • A novel model-free VI algorithm is proposed for unknown stochastic linear quadratic nonzero-sum differential games. • Feedback control strategies for players are designed by using only the data of states and inputs. • The designed feedback control strategies can make the closed-loop game system stable and reach the Nash equilibrium. [ABSTRACT FROM AUTHOR]

The stabilized collocation method (SCM) is a promising meshless collocation method that can overcome the instability defects in the classical direct collocation method. To improve the performance of the SCM, a superconvergent stabilized collocation method (SSCM) is developed in this paper for linear and nonlinear elliptic problems through the use of the moving least squares (MLS) approximation and its smoothed derivatives. Accuracy of the SSCM and the SCM is analyzed with an emphasis on the influence of boundary conditions, and precise error measures are presented for different types of boundary conditions. Numerical results validate the superconvergence of the SSCM and confirm the theoretical analysis. • A superconvergent stabilized collocation method is developed for linear and nonlinear elliptic problems. • Theoretical accuracy of the method is analyzed detailedly. • Theoretical and numerical results reveal that the method possesses superconvergence property. [ABSTRACT FROM AUTHOR]

In this paper, the Lagrange stability and passivity of a class of discrete-time stochastic Markovian switching neural networks with mixed delays are studied. The mixed delays include time-varying transmission and distribution delays. Global exponential Lagrange stability criterion and passivity criterion in the mean square sense are derived. A numerical example illustrates the usefulness of these criteria. The originality of the proposed method lies in: (i) the criteria are derived directly from the definitions rather than constructing Lyapunov–Krasovskii functional; (ii) the inequalities in the criteria contain only a few decision variables, which is easy to solve and leads to low computational complexity. • Lagrange stability and passivity of discrete-time SM-SNNs with mixed delays are studied. • The time-varying transmission and distribution delays are involved. • Global exponential Lagrange stability criterion and passivity criterion in MMS are derived. • The inequalities in the criteria are easy to solve and lead to low computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we examine the solvability of a functional equation in a Lipschitz space. As an application, we use our result to determine the existence and uniqueness of solutions to an equation describing a specific type of choice behavior model for the learning process of the paradise fish. Finally, we present some concrete examples where, using numerical techniques, we obtain approximations to the solution of the functional equation. As the straightforward Picard's iteration can be very expensive, we show that an analytical suboptimal least-squares approximation can be chosen in practice, resulting in very good accuracy. • Functional equation from behavioral science is considered. • A unique solution is found in the Lipschitz space. • An analytical accurate approximate solution is devised. [ABSTRACT FROM AUTHOR]

As a fundamental task, medical image reconstruction has attracted growing attention in clinical diagnosis. Aiming at promising performance, it is critical to deeply understand and effectively design advanced model for image reconstruction. Indeed, one possible solution is to integrate the deep learning methods with the variational approaches to absorb benefits from both parts. In this paper, to protect more details and a better balance between the computational burden and the numerical performance, we carefully choose the multi-level wavelet convolutional neural network (MWCNN) for this issue. As the tight frame regularizer has the capability of maintaining edge information in image, we combine the MWCNN with the tight frame regularizer to reconstruct images. The proposed model can be solved by the celebrated proximal alternating minimization (PAM) algorithm. Furthermore, our method is a noise-adaptive framework as it can also handle real-world images. To prove the robustness of our strategy, we address two important medical image reconstruction tasks: Magnetic Resonance Imaging (MRI) and Positron Emission Tomography (PET). Extensive numerical experiments show clearly that our approach achieves better performance over several state-of-the-art methods. • We explore a medical image reconstruction framework with deep learning-based method. • To solve the hybrid model, we introduce the PAM algorithm. • We analyze the convergence of the proposed method with numerical results. • The robustness of our method can be illustrated by the PET and MRI reconstruction. [ABSTRACT FROM AUTHOR]

As an important part of the evolution of cooperation, interaction diversity has attracted extensive attention. Traditional studies mainly focus exclusively on either time or space, and the comprehensive impact of these two dimensions on the evolution of cooperation deserves further exploration. In this paper, the interaction willingness is abstracted as the network weight, and an adaptive mechanism for interaction willingness based on the delay effect is proposed by combining the network weight and accumulated payoff. Specifically, individuals adjust their willingness to participate in the game with a delay, considering the change in accumulated payoffs within a certain period, rather than the change in immediate payoffs. The degree of delay is reflected in the length of individuals' memory of historical payoffs. Simulation results deviate from the intuition that more information facilitates cooperation, and in certain modes of interaction willingness adjustment, the correlation between cooperation level and memory–length is non–monotonic. Long–memory is conducive to promoting cooperation under the condition of a relatively large range adjustment of interaction willingness. Conversely, with a narrow adjustment range of interaction willingness, cooperation can be effectively established with short–memory. These results hold substantial implications for both theory and practice, as slight adjustment often corresponds to low loss and high operability. In addition, the underlying mechanism is furthermore explored, revealing that the 'isolation zones' composed of individuals with low interaction willingness play an important role in promoting cooperation. • The self-adaptive mechanism based on memory influences cooperation evolution. • The relationship between cooperation level and memory-length is non-monotonic. • Memory-length affect cooperation differently based on willingness adjustment mode. [ABSTRACT FROM AUTHOR]

In this paper, we explore and analyze a stochastic epidemiological model with vaccination and two mean-reverting Ornstein-Uhlenbeck processes to describe the dynamics of HBV transmission. Our study begins by deriving a sufficient condition for the extinction of hepatitis B. Additionally, we determine the presence of a stationary distribution using the Markov process stability theory. Next, we utilize algebraic equation theory and the associated Fokker-Planck equation to derive an explicit expression for the unique probability density function. To validate our theoretical findings, we conduct several numerical simulations using hepatitis B data provided by the World Health Organization (WHO). The results of these simulations support and complement our analytical approach. • A stochastic hepatitis B epidemic model with vaccination and OU process is proposed. • The dynamics of the stochastic model and its corresponding deterministic version are consistent. • The existence of ergodic stationary distribution for the proposed model is proved. • An explicit expression for the unique probability density function is derived. • Numerical simulations are taken to support and complement the analytical approach. [ABSTRACT FROM AUTHOR]

Hedetniemi's conjecture, a significant problem in graph theory, was disproved by Shitov in 2019. This breakthrough stimulates us to consider the signed graphs version of Hedetniemi's conjecture, where signed graphs are graphs with an assignment of { ± 1 } to the edges. In this paper, we provide counterexamples to the signed analog of Hedetniemi's conjecture. Nevertheless, we show that the chromatic number of the product is determined by its factors in some instances, i.e., if Γ → Θ and (Θ , π) is balanced, then χ ((Γ , σ) × (Θ , π)) = χ ((Γ , σ)). As a byproduct, we investigate the homomorphisms from a factor of (Γ , σ) × (Θ , π) to the other, say from (Γ , σ) to (Θ , π). We find that if such homomorphisms exist, then the chromatic number of (Γ , +) provides a lower bound for χ ((Γ , σ) × (Θ , π)). Moreover, if Γ is a core, then the number of balanced subgraphs isomorphic to (Γ , +) in (Γ , σ) × (Θ , π) is equal to the number of such homomorphisms. [ABSTRACT FROM AUTHOR]

• A self-propelled particle model considering leadership and experience is proposed. • Group queuing and stable movement are achieved under the guidance of leaders. • The optimal leadership and vision intervals are obtained. The coordinated and ordered collective behavior arising from dispersed and local self-organizing interactions among individuals is widely observed in many biological communities. For populations engaging in group foraging or migration, complex societies with multiple hierarchical levels are common. A prevalent scenario involves two dominant levels, one corresponding to leaders and the other composed of followers. In this paper, we explore how a group can be influenced by a set of leaders to organize and move in a coordinated manner towards a preferred direction. Specifically, we balance social interactions among group members through two weighted factors, corresponding to the leadership and experience of leaders. Through computer simulations, we observe a linear relationship between the proportion of leaders guiding group movement and the group size, with larger groups requiring a higher proportion of leaders while maintaining initial density. Additionally, we identify an optimal leadership interval where group movement performance enhances with increasing leadership up to a certain point, beyond which it starts to decline. Similar patterns are observed concerning individual visual field, indicating the existence of an optimal interval for visual angle. Our findings not only contribute to understanding collective motion in natural biological systems but also provide new insights into effective leadership mechanisms in artificial swarm systems. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time cooperative output regulation (FTCOR) problem with unknown exosystem of second-order nonlinear multiagent systems. To solve this problem, it is necessary to ensure that each agent can obtain the reference trajectory originating from an unknown exosystem and reconstruct the solution of the regulator equation containing the exogenous signal and unknown parameter vectors within a fixed time, and the unknown exosystem makes it difficult to solve these two problems with existing tools. By using auxiliary filtering variables, we propose a fixed-time adaptive distributed observer (FTADO) and design a fixed-time distributed internal model (FTDIM) based on this FTADO. These two new tools ensure the solvability of the two problems just mentioned. Then, based on these two tools, we transform the problem into a fixed-time stabilization problem of a so-called augmented system. Through the backstepping technique, we propose a distributed controller that solves this fixed-time stabilization problem while relaxing the assumptions of similar references. • More general and difficult controlled objects and research problem are considered. • A fixed-time adaptive distributed observer for an unknown leader system is designed. • A compensator is designed to reconstruct the ideal value of the system in fixed time. • Additional assumption for similar work is avoided. [ABSTRACT FROM AUTHOR]

The motivation of this paper is inspired by the research of Zanette et al. (2011) [4] , which reveals that the perception of predation risk can reduce the number of juvenile prey. To investigate how fear affects the behavior of prey population, we formulate a predator-prey model with a stage structure for prey in which adult prey is undergoing the interference of fear. First, the dynamic properties of the model are analyzed, including the existence and stability of equilibria, bifurcations dynamics, such as transcritical, saddle-node, and Hopf bifurcations. Our results reveal that fear can promote the formation of the stability of the model by transforming it from an unstable state to a stable state. Numerical simulations and theoretical analysis indicated that when the level of fear reaches 25.4, the number of juvenile prey affected by fear decreases by 40% compared to those not affected by fear. Terrifyingly, the predators are on the brink of extinction in high levels of fear. Moreover, we examine the impact of time delay on fear in the previous model as prey needs some time to assess the predation risk. Some dynamic characteristics of this delayed model are also analyzed. Results indicate that the fear effect is conducive to generating a stable state, while delay is detrimental to generating a stable state, leading to limit cycles or even chaos. • A stage-structure predator-prey model with fear and delay is proposed. • The feasibility equilibria and bifurcation dynamics of the model are studied. • Fear can transform a model from an unstable state to a stable state. • Delay is detrimental to generating a stable state, leading to limit cycles or chaos. [ABSTRACT FROM AUTHOR]

This paper investigates practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise (MHFNLs) via aperiodically intermittent discrete-time observation control (AIDOC). It is worth pointing out that AIDOC is considered into MHFNLs for the first time. Due to the difficulty of satisfying linear conditions in real life, polynomial growth conditions are used instead of linear growth conditions. Meanwhile, a global Lyapunov function is constructed and sufficient conditions which can remain MHFNLs practically stable are gained. Finally, an application and its numerical simulations are presented to verify the effectiveness and availability of the theoretical results. • Polynomial growth conditions are used instead of linear growth conditions. • We consider Lévy noise and construct a global Lyapunov function. • Aperiodically intermittent discrete-time observation control is designed. • The theoretical result is applied to Van der Pol-Duffing oscillator. [ABSTRACT FROM AUTHOR]

This paper presents a new interface sampling and reduction method for varying parameters within the framework of the component mode synthesis (CMS) method. In the conventional interpolation-based parametric reduced-order model (IB-PROM) based on CMS, reduced subsystems are synthesized, and interface degrees-of-freedom are reduced via the secondary eigenvalue analysis in the online stage, which is essential to obtain highly reduced system matrices. However, it inhibits rapid parameter updating and requires extra computational resources in the online stage. In the present study, the pre-reduced interface sampling method is proposed by synthesizing substructures and reducing the interface before beginning the online stage. The interface solution is obtained through offline and online adaptation, likewise the procedure of computing the interior solution of previous studies. Additionally, congruence transformations are applied to each reduced interface, which is mandatory for interpolating symmetric positive definite matrices with respect to a new input parameter. As a result, the online performance of IB-PROM is greatly enhanced, showing rapid and near real-time adaptations to the variations in system parameters. • A new interface sampling and reduction method for varying parameters. • Efficient online reduced-order model with pre-reduced interface degrees of freedom. • System-level interface sampling and reduction in the offline stage. • Online interpolation of pre-reduced interfaces via congruence transformation. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V and edge set E. A topological index has the form T I (G) = ∑ u v ∈ E f (d u , d v) , where f = f (x , y) is a pertinently chosen function which must be symmetric and real-valued for all x , y pertaining to vertex degrees of the graph G. Particularly interesting are the Sombor index SO and the elliptic Sombor index ESO , induced by the functions f (x , y) = x 2 + y 2 and f (x , y) = (x + y) x 2 + y 2 , respectively. In this paper we analyze the ordering relations in benzenoid systems with respect to these two important topological indices. Also, we extend the results to general Sombor index SO α , β and general elliptic Sombor index ESO α. [ABSTRACT FROM AUTHOR]

The issue of composite adaptive exponential tracking control for a class of large-scale nonlinear systems under model uncertainties, external disturbances, along with multiplicative and additive time-varying sensor faults is considered in this paper. The command filtered backstepping approach is employed to address the "explosion of terms" issue inherent in standard backstepping method. A novel compensating system is incorporated to mitigate the effects of filtering errors and enhance the convergence of tracking errors. The composite estimation laws are designed by integrating the compensated tracking errors, prediction errors stemming from output estimators, along with the proportional and integral estimation errors of faulty terms. This on-line estimation framework enables achieving fast, robust and accurate estimation, even when employing low learning and modification gains. By incorporating modification terms with appropriate time-varying gains, it is demonstrated that the resulting system is globally exponentially stable. Finally, the effectiveness of the presented FTC approach is illustrated through two simulation examples. • A composite adaptive FTC strategy is developed under time-varying sensor faults. • The proposed method incorporates composite estimation laws into the CFB framework. • A novel compensating system is utilized to mitigate the effects of filtering errors. • The on-line estimation architecture enables fast, robust and accurate estimation. • The closed-loop system is rigorously proven to be globally exponentially stable. [ABSTRACT FROM AUTHOR]

A signed graph G ˙ is defined to be harmonic if its net-degree vector d ± (G ˙) is a zero vector or an eigenvector of adjacency matrix A (G ˙) corresponding to an eigenvalue λ. In this paper, we demonstrate the existence of λ -harmonic signed trees with arbitrary diameter and the nonexistence of finite 0-harmonic signed trees. And we determine all harmonic signed trees with diameter at most 5. Moreover, we give some general properties of finite harmonic signed trees. In particular, a 1-harmonic signed tree must be 1 net-regular. • The existence of harmonic signed tree with arbitrary diameter is proved. • The nonexistence of finite 0-harmonic signed trees is proved. • All harmonic signed trees with diameter less than 6 are determined. • Some general properties of finite harmonic signed trees are given. • 1-harmonic signed tree must be 1 net-regular. [ABSTRACT FROM AUTHOR]

The Collatz conjecture is a famous conjecture aiming to estimate the behaviors of positive integers after a series of iterations of the Collatz function f (n) , where f (n) = { 3 n + 1 2 if n is odd , n 2 if n is even. Collatz conjecture has attracted a great deal of attention since it was put forward in 1930s due to the surprising nature. However, after strenuous efforts, many mathematicians think the current knowledge seems insufficient to solve Collatz conjecture, although it appears in a very simple form. As a replacement, a "periodic" (weaker) version of Collatz conjecture (called Collatz periodic conjecture) was proposed and discussed. In this paper, we would like to explore what properties Collatz (periodic) conjecture could imply. Maybe it is another possible direction to confirm or disprove Collatz (periodic) conjecture. • Provide some equivalent forms of Collatz (periodic) conjecture. • Consider the Collatz (periodic) conjecture in the aspect of digraphs and matrices. • Reveal some further properties what Collatz (periodic) conjecture could imply. [ABSTRACT FROM AUTHOR]

In the paper, an unconditional energy-stable scheme is developed with the help of the scalar auxiliary variable (SAV) approach and the leapfrog time-discretization. The fully discrete scheme gives a linear and decoupled system, whereas previous SAV-type methods usually gave a coupled linear system. Therefore, the proposed scheme is decoupled and thus easier to be implemented. Moreover, it is proved that the proposed scheme has second-order time accuracy in the temporal direction. Several numerical examples are provided to support the proposed theoretical results. • An unconditional energy-stable scheme is developed to numerically solve the Swift-Hohenberg equations. • The proposed scheme is linearly implicit and decoupled and thus easier to be implemented. • Convergence of the semi-discrete scheme is obtained. [ABSTRACT FROM AUTHOR]

In this paper, the cost function of a single-server queueing system with Poisson input stream and Erlang- k service time will be analyzed. Treated as a function of the traffic intensity ρ , with respect to some known constants, we will show that its stationary points are solutions of a fourth-degree polynomial equation with real coefficients. Moreover, an explicit form of these solutions is given and it is shown the function reaches a minimum value at some of these points. For illustration, a numerical analysis of the cost function is carried out by changing the values of the costs, which are changed according to the principle of arithmetic progression. Also, a statistical analysis of the relationship between optimal solutions ρ ⁎ and Φ (ρ ⁎) is done. • A cost function is formed for single-server queueing system with Poisson input stream and Erlang-k service time. • Stationary points of the proposed cost function are solutions of fourth-degree polynomial equation with real coefficients. • The sensitivity of the parameters is tested according to the principle of arithmetic progression. • A statistical analysis of the optimal solutions is made and an economic interpretation of the results is given. [ABSTRACT FROM AUTHOR]

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed. • Extending the Lyapunov first method to systems of Volterra integral equations and Volterra integro-differential equations. • Using the logarithmic norm to obtain sufficient stability conditions for integral and integro-differential Volterra equations. • Common approach to constructing the sufficient stability conditions for integral and integro-differential Volterra equations. • Explicit sufficient conditions via kernels and coefficients of the equations. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the stochastic Volterra integral equations (SVIEs) with double weakly singular kernels. Our primary objective is to examine the well-posedness of the proposed equation. Specifically, we explore the presence of existence, uniqueness, boundedness, and the continuous dependence of the exact solution on the initial data. Additionally, we develop a stochastic θ -scheme as a numerical solution for the equation and demonstrate that the convergence rate of the scheme is influenced by the kernel parameters. To validate the accuracy and reliability of our theoretical findings, we present two numerical examples. Notably, one of these examples concentrates on estimating the price of a European call option using the Heston stochastic volatility model with a singular kernel. Our results, when compared to the corresponding findings by Li et al. [3] , not only relax the integrable limitations of singular kernels but also establish a precise convergence order. In addition, we propose an improved scheme, based on the efficient sum-of-exponentials (SOE) approximation, to address the low computational efficiency of the stochastic θ -scheme. The results confirm that our approach aligns significantly with the expected physical interpretations. • We introduce the novel double singular stochastic Volterra integral equation (SVIE). • We investigate the well-posedness of SVIEs with double weakly singular kernels. • Additionally, the boundedness of the p th moment of the proposed equation is achieved. • The proposed scheme, which is built upon the SOE approximation, effectively reduces the computational cost and storage requirements to O (N log ⁡ N) and O (log ⁡ N) when T ≫ 1 , respectively. When T ≈ 1 , the computational cost and storage requirements are further reduced to O (N log 2 ⁡ N) and O (log 2 ⁡ N) , respectively. • In addition, through our numerical experiments, we have substantiated that the computational efficiency of the proposed scheme, utilizing the SOE approximation, outperforms conventional methods such as the Euler method and the Galerkin approximation. [ABSTRACT FROM AUTHOR]

In this article, the null controllability of a coupled ODE-heat system is considered by backstepping method. First, according to a backstepping transformation, the system is transferred into an intermediate system which has an equivalence of null controllability with the original system. Second, we divide the finite time interval into a sequence of time intervals, and on each time interval, transform the intermediate system into a target system through another backstepping transformation. Consequently, we obtain the crucial norm estimates of states and control laws. Finally, based on the invertibility of the target system, we prove null controllability of the intermediate system and ultimately, show the null controllability of the original coupled ODE-heat system. In addition, numerical experiment results are presented to examine the validity for the presented controller. • We construct a piecewise constant feedback controller for the null controllability of coupled ODE-heat system coupled at boundary and internal term. • We obtain the crucial norm estimates for the kernels, states and control. Especially, we obtain the kernels estimate by using suitable induction hypothesis expression in successive approximation. • We obtain the uniform estimate of the decay rate β for the target system, which is independent of the time intervals. • The procedure of this paper avoiding some technical skills for constructing continuous feedback control in references, which can be extended to similar problems for some other PDE-ODE system. [ABSTRACT FROM AUTHOR]

This paper designs the asynchronous finite-time extended dissipative filter on neutral semi-Markov jump systems subject to sensor saturation and dual-channel denial-of-service attacks. Based on event-triggering scheme and hidden Markov model, the asynchronous filter is designed. Sufficient conditions which filter error system is extended dissipative stochastic finite-time bound are derived. In the meantime, we present novel results on extended dissipative stochastic finite-time contractive bound. Finally, the partial element equivalent circuits (PEECs) are employed to evidence the validness of results. • Because system modes might be unavailable, hidden Markov model is used to explain non-synchronous phenomenon between system and filter, which subsystems could be detected with the conditional probability matrix. • As for dual-channel DoS attacks, two mutually independent Bernoulli variables are utilized to depict whether two channels are attacked or not. • Both finite-time bound and finite-time contractive bound are considered to reflect the transient performance of filter error system. By adjusting parameter matrices of extended dissipative, different performances could be obtained, such as passive, (Q , S , R)-dissipative, etc. [ABSTRACT FROM AUTHOR]

Due to their dynamic properties such as irregular sampling rate and high-frequency sampling, Continuous Time Series (CTS) are found in many applications. Since CTS with irregular sampling rate are difficult to model with standard Recurrent Neural Networks (RNNs), RNNs have been generalised to have continuous-time hidden dynamics defined by a Neural Ordinary Differential Equation (Neural ODE), leading to the ODE-RNN model. Another approach that provides a better modelling is that of the Latent ODE model, which constructs a continuous-time model where a latent state is defined at all times. The Latent ODE model uses a standard RNN as the encoder and a Neural ODE as the decoder. However, since the RNN encoder leads to difficulties with missing data and ill-defined latent variables, a Latent ODE-RNN model has recently been proposed that uses a ODE-RNN model as the encoder instead. Both the Latent ODE and Latent ODE-RNN models are difficult to train due to the vanishing and exploding gradients problem. To overcome this problem, the main contribution of this paper is to propose and illustrate a new model based on a new Latent ODE using an ODE-LSTM (Long Short-Term Memory) network as an encoder - the Latent ODE-LSTM model. To limit the growth of the gradients, the Norm Gradient Clipping strategy was embedded on the Latent ODE-LSTM model. The performance evaluation of the new Latent ODE-LSTM (with and without Norm Gradient Clipping) for modelling CTS with regular and irregular sampling rates is then demonstrated. Numerical experiments show that the new Latent ODE-LSTM performs better than Latent ODE-RNNs and can avoid the vanishing and exploding gradients during training. Code implementations developed in this work are available at github.com/CeciliaCoelho/LatentODELSTM. [ABSTRACT FROM AUTHOR]

Hexagonal chains are graph representations of benzenoid polymers which are common but important molecular structures in chemistry. A spanning tree of a graph is a subgraph which is a tree and contains all the vertices of the graph. A perfect matching of a graph is a set of disjoint edges covering all vertices of the graph. In this paper, we give exact expression for the number of spanning trees containing perfect matchings in hexagonal chains with a unique kink. • The number of spanning trees containing perfect matchings in hexagonal chains with a unique kink is derived. • The number of spanning trees in the graph obtained from hexagonal chain by contracting a perfecting matching is obtained. • Graphs obtained from the hexagonal chain with a unique kink by contracting a perfecting matching are characterized. [ABSTRACT FROM AUTHOR]

This paper is concerned with the protocol-based fault detection (FD) issue for state-saturated delayed nonlinear systems with both redundant channels and sensor nonlinearities. Specifically, in pursuit of relieving the communication pressure, the stochastic communication protocol is adopted, which can regulate the transmission of output signal during the communication process. Moreover, for the purpose of decreasing the negative impact of one channel failure on network communication and enhancing data reliability, the redundant channels transmission is introduced. Then, the protocol-based FD filter is constructed such that the resultant FD dynamics system attains the global asymptotic stability in mean-square sense and satisfies the H ∞ performance index. Moreover, an implement-to-easy algorithm is developed to facilitate the solvability of the desirable FD filtering method. In the end, two illustrative examples are utilized to demonstrate the validity of the developed FD filtering algorithm. • The stochastic communication protocol is adopted to regulate the network communication. • The redundant channels transmission mechanism is introduced into the signal communication process. • New sufficient criteria are derived to assure the existence of desirable FD filtering scheme for state-saturated systems. [ABSTRACT FROM AUTHOR]

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω = (2 − α) / α shall give an optimal temporal convergence O (N − (2 − α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme. • Efficient spatial semi-discretization using matrix transfer technique for time-space fractional diffusion problem is established. • Stability and error bounds of full discrete scheme based on fast time-stepping L1 method on graded time meshes are discussed. • Fast and accurate solver for system of equations is designed by discrete sine transform at per time level. • Commendably concise and implementation-friendly double fast algorithm is proposed. [ABSTRACT FROM AUTHOR]

In this paper, the robust security protection control for switched systems is studied. A robust anti-disturbance mechanism using the Radial Basis Function Neural Network (RBFNN) is proposed for switched systems, which can approximate and compensate for the impact of unknown disturbance on the system state. Then, a network security protection mechanism based on encoder and decoder is presented, which has the ability to resist the dual impact on the feedback information caused by the network privacy snooping and data injection attacks. Accordingly, stability analysis and state-feedback controller design are given for the switched systems under unknown disturbance, network privacy snooping and injection attacks. Finally, simulation results illustrate the effectiveness of the proposed method. • A data-based robust anti-disturbance mechanism is adopted to approximate any unknown disturbance. • A novel network security protection mechanism is proposed to resist data injection attacks and prevent privacy snooping. • For the first time in the control field, the privacy protection level of encoder and decoder is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, a type of delayed Clifford-valued memristor-based neural networks (CLVMNNs) with impulsive disturbances is established, and the global exponential stability is investigated by using generalized norm. Firstly, the n-dimensional Clifford-valued systems are decomposed into 2 m n -dimensional real-valued systems to address the non-commutativity problem of the multiplication of Clifford numbers. Secondly, the generalized ∞-norm and 1-norm are introduced to induce the global exponential stability for CLVMNNs, and two special Lyapunov functionals are established to prove the stability. Thirdly, the strict assumption of the boundedness of activation function in previous research is loosened, and some less conservative conditions of stability are obtained based on the constructed Lyapunov functionals. Finally, the theoretical results are verified by two numerical simulations, and an image encryption scheme is proposed to show the application in real world situation based on the delayed CLVMNNs. • Impulsive disturbances and memristor are taken into account for Clifford-valued NNs. • The generalized { ξ , 1 } -norm and { ξ , ∞ } -norm are introduced to investigate the global exponential stability of CLVMNNs. • The global exponential stability of CLVMNNs is discussed and some new criteria are derived. [ABSTRACT FROM AUTHOR]

The Fantope-constrained sparse principal subspace estimation problem is initially proposed by Vu et al. (Vu et al., 2013). This paper investigates a semismooth Newton based proximal point (Ppassn) algorithm for solving the equivalent form of this problem, where a semismooth Newton (Ssn) method is utilized to optimize the inner problems involved in the Ppassn algorithm. Under standard conditions, the Ppassn algorithm is proven to achieve global convergence and an asymptotic superlinear convergence rate. Computationally, we derive nontrivial expressions for the Fantope projection and its generalized Jacobian, which are key ingredients for the Ppassn algorithm. Some numerical results on synthetic and real data sets are presented to illustrate the effectiveness of the proposed Ppassn algorithm for large-scale problems and superiority over the alternating direction method of multipliers (ADMM). [ABSTRACT FROM AUTHOR]

• This work first time deploys the radial basis function neural network (RBFNN)-based control with norm estimation in a new way to facilitate the control development. • Inequality constraint of using norm estimation is relaxed but still retaining the same properties of the original system. • A nonlinear disturbance state observer (NDSO)-based RBFNN is employed to not only estimate unmeasured states but also lumped disturbances and uncertainties. • An observer-based NN finite time control is established to achieve the system output tracking regulation. Due to the difficulty encountered in dealing with unstructured system dynamics with unmeasured system state variables, this paper presents a novel observer-based neural network finite-time output control strategy for general high-order nonlinear systems (HNSs). The suggested technique is performed based on the backstepping-like control (BSC) scheme with a hybrid nonlinear disturbance-state observer and norm estimation-based radial basis function neural network (RBFNN). This helps not only reduce the number of estimated parameters but also relax the restriction of using inequality when exploiting the norm estimation concept in a conventional way; thus, retaining the same properties of the original system. Therefore, an observer-based finite-time output feedback control is established to deal with the unstructured dynamical behaviors and satisfying the output tracking regulation with the semi-global practically finite-time stability (SGPFS) guaranteed for the closed-loop system. The effectiveness and workability of the proposed algorithm is verified by a numerical simulation on a specific practical application. [ABSTRACT FROM AUTHOR]