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A [ 1 , 2 ] -set of a graph G is a set S of vertices of G if every vertex not in S has one or two neighbors in S. The [ 1 , 2 ] -domination number γ [ 1 , 2 ] (G) of G equals the minimum cardinality of a [ 1 , 2 ] -set of G. A Roman [ 1 , 2 ] -dominating function (R [ 1 , 2 ] DF) on a graph G is a function f from the vertex set V of G to the set { 0 , 1 , 2 } such that any vertex assigned 0 under f has one or two neighbors assigned 2. The weight of an R [ 1 , 2 ] DF f is the sum ∑ x ∈ V f (x). The Roman [ 1 , 2 ] -domination number γ R [ 1 , 2 ] (G) of G equals the minimum weight of an R [ 1 , 2 ] DF on G. In this paper, we prove that the decision problem on the Roman [ 1 , 2 ] -domination is NP-complete for bipartite and chordal graphs. Moreover, we give some bounds on the Roman [ 1 , 2 ] -domination number. In particular, we show that for any nontrivial tree T , γ R [ 1 , 2 ] (T) ≥ γ [ 1 , 2 ] (T) + 1 and characterize all trees obtaining equality in this bound. • The decision problem on Roman [1,2]-domination is NP-complete for bipartite and chordal graphs. • Some bounds on the Roman [1,2]-domination number are given. • The Roman [1,2]-domination number of any nontrivial tree is no less than its [1,2]-domination number plus one. [ABSTRACT FROM AUTHOR]

In this paper, using two special types of rise-dimensional operators based on existing fractal functions, we construct new fractal surfaces with any value of the Hausdorff and Box dimension between two and three. Further, we demonstrate that the lower and upper Box dimension of such fractal surfaces may be unequal to each other. This result could be useful to the research on creating various fractal surfaces with the required fractal dimensions in the future. • The relationship between the Hausdorff and the Box dimensions of the original graph in R 2 and its transformations in R 3 has been studied. • New fractal surfaces with any value of the Hausdorff and the Box dimension between two and three have been constructed. • New fractal surfaces whose lower and upper Box dimensions are unequal to each other have been constructed. [ABSTRACT FROM AUTHOR]

• Analyzing the temporal dynamics of swarmalators' movements in the 2-D plane. • High sensitivity of swarmalators' movements to the initial conditions. • The positive largest Lyapunov exponent of swarmalators' dynamic spatial time series. • Chaotic nature of swarmalators' movements in their dynamic states. The simultaneous interactions of synchronization and swarming phenomena are the primary basis for the functioning of many natural systems, such as vinegar eels, sperms, and magnetic particles. Investigating the dual behavior of these particles, which are called swarmalators, has recently attracted the attention of many researchers. However, more needs to be done to analyze the dynamics of swarmalators separately. This paper explores the dynamics of the swarmalators' movements individually over time in two-dimensional plane. This study's findings represent that based on the two rules of high sensitivity to the initial conditions and the positive largest Lyapunov exponent, the swarmalators' movements in the dynamic states have a chaotic nature. This result can lead to a deeper understanding of the complex dynamics of natural swarmalators. It may provide an opportunity for other researchers to analyze the individual behavior of swarmalators along with their collective behavior. [ABSTRACT FROM AUTHOR]

In this article, the hybrid cyber-attacks for singular Markov jump systems (MJSs) are mainly solved by designing the asynchronous mixed H ∞ and passive controller. A kind of dynamic event-triggered (ETM) based strategy is adopted to maximize the savings of network bandwidth resources. And the hybrid cyber-attacks including two common attacks, the spoofing attack and the denial of service (DoS) attack are considered, which are assumed to be mutually independent and meet Bernoulli distribution. The singular MJSs under cyber-attacks are converted into the singular Markov jump time-delay systems. Then, the criteria of stochastically stable for the closed-loop singular MJSs meeting the mixed H ∞ and passive performance are obtained by linear matrix inequalities (LMIs). And by optimizing the obtained LMIs, the gains of asynchronous resilient controller are further derived, and the weighting matrices of dynamic event-triggered based strategy are also gained. Eventually, the feasibility of the offered way is verified by an industrial example of DC motor control system. • In this paper, the resilient controller for the singular Markov jump systems is considered. • A dynamic event-triggered based strategy is introduced to further save network bandwidth. • Occurrence of hybrid cyber-attacks are described by two mutually unrelated random variables satisfying Bernoulli distribution. • Criteria of stochastically admissible for singular Markov jump CPSs under hybrid cyber-attacks are obtained by a set of LMIs. • The codesign of resilient controller and the dynamic event-triggered based strategy is presented. [ABSTRACT FROM AUTHOR]

In this paper, two reaction-diffusion models describing the interaction among susceptible people (ordinary citizens), infective people (drug users/dealers), and law enforcement personnel are analyzed. The models here considered are a generalization of the crimo-taxis model originally proposed by Epstein in 1997. The modifications allow us to describe various scenarios. We analyze the equilibrium points, together with their stability, of the homogeneous system. Moreover, according to the Turing approach to reaction–diffusion models, we investigate the instability driven processes and the emergence of patterns in the complete models. • Reaction-diffusion equations. • Crimo-taxis model. • Turing instability and pattern formation. [ABSTRACT FROM AUTHOR]

In the United States the laws regulating the carrying of firearms in public vary state-to-state. In a highly publicized event, the Governor of New Mexico recently issued an emergency order temporarily banning the carrying of firearms in some areas – and thus rescinding the right-to-carry law in New Mexico – after a spate of gun violence, citing a statistical threshold of general societal violence under which right-to-carry laws should be superseded. In this paper we frame this policy intervention as a dynamical systems model that measures the incidence of gun violence as a function of gun prevalence. We show that the Governor's emergency order – fittingly issued as an emergency health order – is effectively like trying to stop an epidemic after it has become viral in that with this kind of instantaneous stopping condition, under simple assumptions, such a regulation is too little, too late. On the other hand, a graduated response that scales with increasing violence can drive equilibrium gun prevalence to zero. This is a new mathematical model relating gun violence with gun prevalence which we hope continues to spur the modeling community to bring its tools and techniques to bear on this important and challenging social problem. More importantly, our model, despite its simplicity, exhibits complex dynamics and has substantial research and educational value in promoting the application of mathematics in the social sciences, extending beyond gun control issues. • We use a dynamical systems model to assess the effectiveness of threshold-dependent executive orders on curbing gun violence. • Our study finds that sudden step-function like government interventions cannot halt gun violence. • Graduated response is key to mitigating gun violence before it becomes too little, too late to control. [ABSTRACT FROM AUTHOR]

This paper focuses on the advancement of weak Galerkin (WG) finite element methods for addressing two-dimensional and three-dimensional Biharmonic interface problems with polygonal/polyhedral meshes. The WG method has been demonstrated to be accurate and efficient, providing optimal order error estimates in discrete H 2 and standard L 2 norms. A series of extensive numerical tests are conducted to validate the WG solutions, showcasing the flexibility, stability, and robustness of the proposed method for handling both smooth and complicated interfaces. [ABSTRACT FROM AUTHOR]

In a (proper) edge-coloring of a bridgeless cubic graph G an edge e is rich (resp. poor) if the number of colors of all edges incident to end-vertices of e is 5 (resp. 3). An edge-coloring of G is normal if every edge of G is either rich or poor. In this paper we consider snarks G ˜ obtained by a simple superposition of edges and vertices of a cycle C in a snark G. For an even cycle C we show that a normal coloring of G can be extended to a normal coloring of G ˜ without changing colors of edges outside C in G. An interesting remark is that this is in general impossible for odd cycles, since the normal coloring of a Petersen graph P 10 cannot be extended to a superposition of P 10 on a 5-cycle without changing colors outside the 5-cycle. On the other hand, as our colorings of the superpositioned snarks introduce 18 or more poor edges, we are inclined to believe that every bridgeless cubic graph distinct from P 10 has a normal coloring with at least one poor edge and possibly with at least 6 if we also exclude the Petersen graph with one vertex being truncated. • The Petersen Coloring Conjecture is equivalent to stating that every snark has a normal 5-edge- coloring. • The most general known method for constructing snarks is superposition. • We consider a class of superpositioned snarks and construct a normal 5-edge-coloring for them. [ABSTRACT FROM AUTHOR]

In this paper we develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a Markovian (regime switching) environment. In particular, by expressing the Gerber-Shiu function in terms of potential measures of an upward (downward) skip-free discrete-time and discrete-space Markov Additive Process (MAP), we derive closed form expressions for the Gerber-Shiu function in terms of the so-called (discrete) W v and Z v scale matrices, which were introduced in [27]. We show that the discrete scale matrices allow for a unified approach for identifying the Gerber-Shiu function as well as the value function of the associated constant dividend barrier problems. [ABSTRACT FROM AUTHOR]

The loyalty of individual strategies is an important factor affecting cooperative behavior in human society. Previous researches on the loyalty have mostly focused on the loyalty of individual to a single strategy, that is, the loyalty of individual to cooperative strategy. But from the perspective of defective strategy, individual holding defective strategy can also be seen as so-called loyalty to defective strategy. This paper presents a spatial prisoner's dilemma game model with the local loyalty of two-strategy, in which the loyalty to cooperative strategy and loyalty to defective strategy are considered simultaneously. To distinguish the loyalties of these two types of strategies, the loyalty value corresponding to defective strategy is represented as a negative number. A fitness of an individual is defined as weighted average of individual's payoff and power of average loyalty of local nearest neighbors. In strategy learning, if the loyalty of an individual and the loyalty of the target neighbor are different signs, a discount coefficient is added to the strategy transition probability function to reduce the probability. Numerical simulations show that considering individual loyalty to the two strategies can promote cooperative behavior in the system when the strength index of local loyalty in fitness function is positive, and there are optimal values for the weighting coefficient and the strength index of local loyalty, which can enable the system to reach full cooperators state. The discount coefficient to the fitness of neighbor can impact on the frequency of cooperators surviving and the proportion of surviving cooperators during phase transition. • A prisoner's dilemma game considering the loyalty of two-strategy is presented. • The weight coefficient and the strength index of local loyalty are introduced to the fitness. • Considering individual loyalty to the two strategies can promote cooperative behavior. • There exists optimal value of weight coefficient and the strength index for cooperative behavior. [ABSTRACT FROM AUTHOR]

In this paper, we show the connections of inequalities involving fractional integrals with Ohlin lemma and Levin-Stechkin theorem. First, we give a new proof of the fractional version of the inequality of Hermite Hadamard type and then we extend it in two directions. Thus we compare the fractional expression occurring in this inequality with the usual integral and we obtain stronger inequalities (one of them is related to Bullen inequality). • The use of stochastic orderings tools in the theory of inequalities for fractional integrals. • A new proof of the fractional version of the Hermite-Hadamard inequality. • Inequalities between the fractional integral and the regular one. • Refinements of Hermite-Hadamard inequality (in particular a fractional version of Bullen inequality). [ABSTRACT FROM AUTHOR]

This paper examines the mechanism design for a two-facility-location game involving two types of agents. Type I agents only prioritize the facility closer to their location, while type II agents are concerned with the overall distance to both facilities. The objective is to minimize the total cost of all agents. In a game setting, agents have the option to strategically report their locations or types. Therefore, a mechanism which ensures that agents do not have any incentive to misreport their private information is considered strategy-proof. We design strategy-proof mechanisms with approximation ratios for both continuous and discrete facility locations, which are designed with respect to various restrictions on misreporting. Furthermore, we establish lower bounds on approximation ratios. • We study a novel model of two facility location games with mixed types of agents. • The goal is to design strategy-proof mechanisms to minimize the total cost. • We present upper and lower bounds on the approximation ratio of strategy-proof mechanisms. [ABSTRACT FROM AUTHOR]

A recent result shows that there exists d 0 > 0 such that an idealized mass-spring model of electrostatically actuated relativistic Micro-Electro-Mechanical System (MEMS) has at least two, at least one, or no periodic solution, according to d > d 0 , d = d 0 , or d < d 0 , respectively. Here d represents the initial distance between two parallel capacitor plates. In this paper, it is further proved that the model has exactly two classes of periodic solutions. The first class of periodic solutions uniformly tends to 0 at the rate d − 1 2 as d tends to +∞, while the second one uniformly tends to +∞ at the rate d. • Prove a relativistic MEMS model has exactly two classes of periodic solutions. • Prove the first class of periodic solutions uniformly tends to 0, while the second one uniformly tends to +∞. • Provide the convergence and divergence rate, of periodic solutions, with respect to the parameter. [ABSTRACT FROM AUTHOR]

The scattering number of a strong digraph D is defined by s (D) = m a x { ω (D − R) − | R | } , where R and ω (D − R) respectively represent a separator of D and the number of strong components of D − R. This concept is a natural extension of the scattering number of undirected graphs which has a strong background of applications in measuring the vulnerability of networks. In this paper, we get sharp upper and lower bounds of the scattering number of general strong digraphs. We also determine the precise values for scattering numbers of some Cartesian product digraphs. • We introduce and study the concept of scattering number of digraphs for the first time. • We get sharp upper and lower bounds of the scattering number of general strong digraphs. • We obtain the precise values for scattering numbers of some Cartesian product digraphs. [ABSTRACT FROM AUTHOR]

The backward substitution method is a newly developed meshless method that has been used for the simulation of many problems in science and engineering with high accuracy and efficiency. In this paper, we explore the feasibility of employing the backward substitution method for simulating two-dimensional incompressible flows. Two non-increment pressure correction projection methods are considered to decompose the original velocity and pressure coupling system into two boundary value problems of intermediate velocity and pressure. Then, two boundary value problems are solved by the backward substitution method in each time iteration step. Five numerical examples are provided to demonstrate the accuracy, computational efficiency and convergence of the method. Comparisons with some existing meshless methods verify the method's advantages and potential applications to engineering. • The backward substitution method is first raised for the simulation of incompressible flows. • Investigated effects of the time step and the number of collocation points on the accuracy. • Verification benchmark comprising several fluid flows problems. • Two projection methods are considered to verify the algorithm. • The algorithm is easily extended to three-dimensional incompressible flows. [ABSTRACT FROM AUTHOR]

In this paper, for leader-follower Lipschitz nonlinear multi-agent systems (MASs) affected by external disturbances, observer-based distributed controller design problems are discussed in order to realize the MASs consensus. Firstly, to deal with unmeasured system states and the unknown disturbances, for each follower, a joint unknown input observer is proposed. To do so, through an interval observer, a mathematical relationship between the state and the disturbance is set up in advance. Secondarily, by using the estimations of the state and the disturbance, a local feedback controller is designed aiming at stabilizing the linear part of each follower. Thirdly, an odd distributed impulsive control method is proposed which is utilized to deal with the nonlinearity. Then the MASs consensus goal is realized by combining the local feedback controller and the distributed impulsive controller. At last, a simulation is discussed so as to display the effectivity of the raised scheme. • For a nonlinear system with external disturbances and unknown states, a joint unknown input observer(JUIO) is proposed. • With resort to the JUIO, a feedback controller is designed, which can stabilize the linear part and compensate disturbances. • Aiming at addressing the nonlinearity, an impulsive controller is designed. Consequently, the MASs consensus is realized. [ABSTRACT FROM AUTHOR]

Among the numerous applications of the Ginzburg-Landau theory to the analysis of significant reaction-diffusion systems, modeling of the behavior of promising polar dielectric materials should be especially highlighted. The paper is devoted to the theoretical and numerical analysis of the Landau-Khalatnikov model describing the dynamics of 2D domain pattern formation in ferroelectrics. The unique solvability of the initial-boundary value problem for the system of 2D cubic-quintic Landau-Khalatnikov equations is proved. The proof is based on the derivation of new a priori estimates for the solution of the system of nonlinear parabolic equations. A series of computational experiments are conducted to examine both spontaneous and polar-induced domain pattern formation in biaxial ferroelectrics. Finite-element simulations allow us to visualize different types of ferroelectric domain structures depending on the varying boundary conditions. • The Landau-Khalatnikov model describing the dynamics of 2D domain pattern formation in ferroelectrics is studied. • A priori estimates for the solution of the initial-boundary value problem for the 2D Landau-Khalatnikov model are derived. • The unique solvability of the initial-boundary value problem for the 2D Landau-Khalatnikov model is proved. • Computational experiments to examine both spontaneous and polar-induced ferroelectric domain patterns are conducted. [ABSTRACT FROM AUTHOR]

In this paper, a class of non-negativity-preserving and maximum-principle-satisfying finite difference methods have been derived by Vieta theorem for one-dimensional and two-dimensional Fisher's equation. By using the positivity and boundedness of numerical and exact solutions, it is shown that numerical solutions obtained by current methods converge to exact solutions with orders of O (Δ t + (Δ t / h x) 2 + h x 2) for one-dimensional case and O (Δ t + (Δ t / h x) 2 + (Δ t / h y) 2 + h x 2 + h y 2) for two-dimensional case in the maximum norm, respectively. Here, Δ t , h x and h y are meshsizes in t -, x - and y -directions, respectively. Finally, numerical results verify that the proposed method can inherit the monotonicity, boundedness and non-negativity of the continuous problems. • New Du Fort-Frankel methods are devised for Fisher's equation. • They are non-negativity-preserving and maximum-principle-satisfying schemes. • Errors in maximum norm are given for them. • They are very good at long-term simulations. • They are very easy to be implemented. [ABSTRACT FROM AUTHOR]

This paper presents a biconjugate gradient stabilized (BiCGSTAB) method and a biconjugate residual stabilized (BiCRSTAB) method for solving the Sylvester tensor equation, respectively. Preconditioned BiCGSTAB and BiCRSTAB algorithms are also developed to solve the Sylvester tensor equation. The convergence of each proposed iterative algorithm is proved. Several numerical examples are shown to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a social contagion model with group interactions on heterogeneous network, in which the group interactions are represented by incorporating higher-order terms. The dynamics of the proposed model are analyzed, revealing the effect of group interactions on the system dynamics. We derive a global asymptotically stability condition of the zero equilibrium point. If the parameters enhancement factor and transform probability meet the condition, the social contagion will eventually disappear. The bifurcation behavior arising from group interactions is investigated, when the enhancement factor exceeds a threshold, the system undergoes a backward bifurcation. This implies that R 0 < 1 does not guarantee the disappearance of social contagion, we also need to control the initial values of social contagion at a lower level. The optimal control strategies for the model are provided. Moreover, the numerical simulations validate the accuracy of the theoretical analysis. It is worth noting that the group interactions lead to the emergence of a bistable phenomenon, in which the social contagion will either fade away or persist eventually, depending on the initial values. • The simplicial SIS social contagion model with delay is proposed on heterogeneous networks. • The dynamics of the model are analyzed, revealing the effect of group interactions on the system dynamics. • An optimal control strategy is proposed for the model, incorporating group interactions. [ABSTRACT FROM AUTHOR]

This paper addresses the dynamic event-triggered fuzzy control problem of nonlinear singular semi-Markovian jump systems. Firstly, the interval type-2 fuzzy model is utilized to approximate nonlinear singular semi-Markovian jump systems. Secondly, by employing the state decomposition approach, the interval type-2 fuzzy singular system is decomposed into a differential equation and an algebraic equation. Thirdly, a new periodic dynamic event-triggered scheme with the adaptive variable is proposed to further relieve the communication pressure while ensuring the system performance. The hidden semi-Markov model is used to describe the phenomenon that the modes of fuzzy controller and the modes of original system are asynchronously operational. By designing the asynchronous event-triggered fuzzy controller based on quantized decomposed state vectors and constructing suitable Lyapunov-Krasovskii functional, sufficient conditions are derived to guarantee that the closed-loop fuzzy singular system is stochastically admissible. Finally, two simulation examples are provided to verify the validity and advantages of obtained results. • The interval type-2 fuzzy singular semi-Markovian jump systems are investigated. • A new periodic dynamic event-triggered scheme is proposed. • The fuzzy event-triggered controller based on decomposed state vectors is designed. • Coupled nonlinear terms are handled by applying slack matrix variables. [ABSTRACT FROM AUTHOR]

This Editorial paper provides an overview of a selection of contributions from researchers working in the fields of Subdivision, Applied Geometric and Algebraic Methods, Isogeometric Analysis, Refinability and Geometric Deep Learning, with special emphasis on numerical computation, advanced algorithms, their analysis, and their applications. [ABSTRACT FROM AUTHOR]

• Improved the value and judgment conditions of the psychological force. • The modified social force model eliminates the abnormal phenomenon of evacuees' speed oscillation near fixed obstacles that exists in the original model. • The effectiveness of the modified social force model was validated by comparing the simulation results with the actual experimental results. The social force model is one of the most common models used in the study of pedestrian dynamics. In the social force model, there are oscillation in the movement of pedestrians. To address the oscillation phenomenon, this paper modifies the way of setting psychological forces and judgment conditions based on previous research. By comparing the simulation results with the real experimental results, the evacuation time results are basically the same, which validates the effectiveness of the modified social force model. The simulation results show that the modified social force model eliminates the phenomenon of evacuee velocity oscillations near fixed obstacles in the original model and fixes the phenomenon of abnormal behavior of evacuees during collisions. In addition, we simulate different numbers of evacuees (i.e.,25,50,100,200) and the results show that the modified social force model successfully replicates the phenomenon of arch congestion during evacuation. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of improving Kalman filtering for a class of multi-layer cyber physical systems (CPS) in the face of multiple consecutive replay attacks. CPS is vulnerable in an open network environment, and the replay attack model is described based on the number and duration of successful message transmissions. For all possible occurrences of replay attacks, a detection and compensation method is obtained with the help of the residual monitoring mechanism and an attack detection function. At the same time, in order to identify the attack has occurred in which, the register is integrated into the system. Furthermore, a novel Kalman filter with a compensation mechanism is designed to estimate the ideal states of CPS subject to replay attacks, and the Kalman filter gain is optimized based on the detection results. Finally, a simulation example is given to verify the effectiveness of the proposed improving Kalman filter. • The relationship between the physical layer output signal and the cyber layer control signal is pointed out. • The detection identifies the occurrence of attacks and the attacked layer by combining integrated register and Kalman filter. • For a multi-layer CPS, to achieve better filtering effects, the proposed Kalman filter is adjusted based on detection results. [ABSTRACT FROM AUTHOR]

This paper investigates a high-order disturbance observer-based safe tracking control scheme for a class of uncertain multiple-input and multiple-output systems under time-varying full state constraints and disturbances. To achieve the safe tracking objective, a boundary protection algorithm is introduced to generate new safe desired signals which are within corresponding state constraints. An improved second-order dynamic surface control technology is developed to deal with the piecewise differentiability of safe desired signals and the phenomenon of repeatedly differentiation, simultaneously. To handle the negative effects of system uncertainties and obtain better estimation effect of high order time-varying disturbances, the radial basis function neural networks and high-order disturbance observer methods are developed. The safety and performance of the closed-loop nonlinear system under the proposed control scheme have been rigorous proved and discussed by Lyapunov stability analysis. Finally, a two-link manipulator model has been given as an example, and the numerical simulations are given to express the availability of the proposed controller. • Different with the high-gain principle of common constrained control methods, a boundary protection algorithm (BPA) has been proposed to calculate new safe desired signals, which can solve the full-state constrained control problem with the time-varying and unpredicted state constraints. • A novel auxiliary system and a second-order filter (SOF) technique have been developed to solve the problem caused by BPA. Comparing with conventional filter in dynamic surface control [13], this SOF can reach the unbiased estimation of the time derivative of the input signal. • The HODO-based safe tracking control scheme is able to realize the safety and convergence of the higher-order, under-actuated nonlinear system under high frequency disturbances proved by rigorous Lyapunov analysis, and the restricted conditions have been strictly discussed. [ABSTRACT FROM AUTHOR]

Abstract: In abstract of the paper [A. Rafiq, A note on “A family of methods for solving nonlinear equations”, Appl. Math. Comput. 195 (2008) 819–821] we can find the following sentences. We cite: Ujević et al. introduced a family of methods for solving nonlinear equations. However the main Algorithm 1 put forward by Ujević et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujević et al. – the end of the citation. Here it is shown that all of the mentioned assertions are not true. In other words, the Algorithm 1 is correct (up to an obvious misprint, which is not mentioned in the above paper) and there are no major bugs in the paper by Ujević et al. In fact, these observations, which will be given in this note, show that the main aim of the paper by Rafiq is wrong. [Copyright &y& Elsevier]

In the framework of partial unknown states, this paper studies the exponential stability problem of nonlinear systems. A novel event-triggered mechanism (ETM) just involving partial known states is designed to guarantee the exponential stability of considered system, where forced impulse with great freedom degree is introduced in ETM. What's more, the information of known states of the system is fetched and applied to supplement the unknown part at impulse instants. With the help of matrix analysis technique, some linear matrix inequalities (LMI) based sufficient conditions are presented from the perspective of the quantity of known and unknown states. Besides, our proposed conditions establish the relationship of event-triggered parameters, impulse interval, and impulse strength. Finally, expository examples and their numerical modeling are appeared to prove the availability of main results. • In this paper, we not only allow systems to have incomplete states, but also relax restrictions on unmeasurable states. • The event triggered mechanism just involves partial known states. • The forced impulse possesses a greater freedom degree. • Our proposed results improve and extend some recent publications. [ABSTRACT FROM AUTHOR]

As a typical class of Cyber-Physical Systems (CPSs), microgrids (MGs) are susceptible to malicious cyber attacks and disturbances that can compromise their security and reliability. To address this issue, an event-triggered distributed resilient control method is proposed in this paper to mitigate the impacts of disturbances, FDI and DoS attacks on MGs. The proposed approach employs a piecewise control strategy and utilizes adaptive method to achieve the secondary control objective of MGs. In order to conserve computing and communication resources, an event-triggered mechanism is adopted and its parameter selection is associated with attacks. This design ensures system stability while also eliminating Zeno behavior. By applying Lyapunov stability theory, it can be proven that the algorithm proposed in this paper is capable of achieving the desired control objective. Finally, the effectiveness and superiority of the developed method are demonstrated through simulation examples. • We propose a control algorithm to solve the problems caused by simultaneous disturbances, FDI, and DoS attacks on MGs. • The employed adaptive control tackles unknown-upper-bound FDI attacks and disturbances on any number of DGs. • To ensure system stability, an attack-related event-triggered mechanism is employed in this study. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce Δ ω -Laguerre based Appell polynomials. The first main aim of the paper is to investigate their main properties including explicit representation, determinantal form, recurrence relation, lowering operators (L O) , integro-partial raising operator (I P R O) and integro-partial difference equation (I P D E). The second aim is to give the corresponding results, which are known or new, to the usual Laguerre based Appell polynomials which can be obtain immediately from the limiting case ω → 0. Furthermore, we introduce Δ ω -Laguerre-Carlitz Bernoulli, Δ ω -Laguerre-Carlitz Euler, Δ ω -Laguerre-Miller-Lee, Δ ω -Laguerre-Boole polynomials and Δ ω -Laguerre-Bernoulli polynomials of the second kind as particular cases of Δ ω -Laguerre based Appell polynomials. The corresponding main results have also been exhibited for these new families of polynomials. [ABSTRACT FROM AUTHOR]

• This paper is the first to add nodes with different dimensions to the structurally balanced network(including directed and undirected network), which is different from the case where only the same dimensional nodes are considered. This means that the model proposed in this paper is more universal. • The topology of the above network is structural balance, which is used to describe the connections between nodes, and weights of connections include either positive (cooperative) and negative (antagonistic). In this paper, we mainly focus on the design of stabilization schemes for a structurally balanced network with communication delays and nonlinear similar nodes in this paper. • A decentralized control scheme is designed to realize asymptotic stabilization of the above network. Compared to the controllers proposed in other papers, the method proposed in this paper not only solves the nodes' stabilization problem with different dimensions, but also has better simulation results for the same dimension of nodes. This paper dedicates to the issue of stabilization for structurally balanced coupled complex networks with distinct nodes, in which the time delay is arbitrary and finite. We put forward a network model that contains similar nodes of different state dimensions. Proper decentralized controllers are proposed for this model with some necessary assumptions. Besides, the control scheme is also effective even though the nodes' state dimensions are identical. Finally, we prove the effectiveness of the theoretical results by giving two numerical examples. [ABSTRACT FROM AUTHOR]

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k -th order Taylor coefficient is O (k) whereas by using the existing AD methods it amounts to O (k 2). • The paper describes the QTSM method, and proves that it outperforms in terms of execution time the classic TSM method. • The capability of QTSM for more general nonlinear ODEs than formally polynomal ones, is discussed. • The paper illustrates, in a simulation example, that TSM correctly calculates the solution whereas lower order methods fails. • The paper illustrates the performance of QTSM vs TSM for the Kepler, and the Van der Pol ODEs. [ABSTRACT FROM AUTHOR]

• Two-channel stochastic network attacks for discrete-time nonlinear switched systems on sensor networks are introduced. These attacks are detected in both the communication channel between the model and the sensor and among the sensors themselves. The attacks are modeled using an independent Bernoulli process with uncertain probabilities. • To address the distributed filter design problem for switched systems on sensor networks, the paper proposes a novel event-triggered communication mechanism. This mechanism involves a combination of switching signals and sensor nodes, enabling efficient communication and filtering. • The various asynchronous modes that can arise between the target switched systems and the switched distributed filter on each sensor node, considering the impact of different event-triggered communication mechanism, are taken into account. To capture these asynchronous modes, a compressed filtering error system is constructed, incorporating all switching signals on the sensor networks. A piecewise vector modal function is developed to represent the synchronous and asynchronous modes between the distributed filter and switched systems on all sensor nodes. • Some novel sufficient conditions in the presence of asynchronous modes, twochannel network attacks, and the event-triggered communication mechanism are established. These conditions ensure that the filtering error system is finite-time bounded and satisfies the H ∞ performance index. This paper addresses the problem of distributed event-triggered finite-time H ∞ filtering for switched systems on sensor networks, subject to two-channel network attacks and asynchronous modes. To tackle the issue posed by the switched event-triggered communication mechanism and network attacks, a distributed filter with switching signal is introduced. Additionally, considering the mismatched behavior between switched systems and distributed filters caused by the event-triggered communication mechanism and network attacks, a piecewise vector modal function which considers all switching conditions is proposed for handling the problem of the sensor network's filtering error system containing multiple switching signals. By utilizing the multiple Lyapunov functional method and the average dwell time switching mechanism, sufficient conditions are derived to guarantee that the filtering error system with synchronous and asynchronous modes is both finite-time bounded and satisfies the H ∞ performance index. Finally, an example is presented to demonstrate the validity of the results. [ABSTRACT FROM AUTHOR]

• A cooperative game is introduced to model the interaction of multiple DoS attacks. Subsequently, an option scheme is adopted based on the Pareto front to obtain the optimal power allocation that makes all attackers expect consistently. Importantly, this paper uses the approach of averaging the optimum power allocation that both effectively guarantees the strength of the attack and reasonably distributes the power of the attackers. • The interactions among attackers and users are analyzed using Stackelberg games to model the optimal defense/response strategy. In addition, the optimal control strategy for NMAS under the strongest attack is developed to ensure that the cost function is minimized. The double-layered game is interlinked through the data packets reception rate. • Sufficient criteria that can achieve stability and consistency in the mean-squared sense for NMAS are pre sented under multiple DoS attacks. In this paper, the stability and consistency are researched for networked multiagent system (NMAS) with double-layered game under multiple denial-of-service (DoS) attacks. Concretely, a cooperative game is applied to model the coordination among DoS attackers in the cyber-layer. Meanwhile, a choice scheme based on the Pareto front is introduced to obtain an optimum power distribution that the DoS attackers expect to be consistent. A Stackelberg game frame is developed to simulate the interactions among DoS attackers and agents. Sufficient conditions that can achieve stability and consistency for NMAS are proposed. Eventually, the significance of the scheme is verified through simulation outcomes. [ABSTRACT FROM AUTHOR]

• We are very grateful to the editor and the reviewer for evaluating our manuscript and for the comments. In accordance with the comments, we have carefully and thoroughly revised and rewritten our manuscript. To our best knowledge, this work is the first paper to consider the stationary distribution of the Milstein scheme for stochastic differential delay equations. We reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Then the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, we have demonstrated the reliability of the theoretical results through abundant numerical experiments. This paper focuses on the stationary distribution of the Milstein scheme for stochastic differential delay equations. The numerical segment process is constructed, which is proved to be a time homogeneous Markov process. We show that this numerical segment process admits a unique numerical stationary distribution. Then we reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Moreover, the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, abundant numerical experiments confirm the reliability of theoretical findings. [ABSTRACT FROM AUTHOR]

• Let Γ be (di)graph. A family P of sub(di)graphs of Γ is called a path cover (resp. a path partition) of Γ if ∪ p ϵ P V (P) = V (Γ) (resp. ∪ p ϵ P V (P) = V (Γ)) and every element of P is a path or a directed path. The minimum cardinality of a path cover (resp. a path partition) of Γ is denoted by pc(Γ) (resp. pp(Γ)). • The authors recently gave Ramsey-type results for the path cover/partition number of graphs. • We continue the research about them focusing on digraphs, and find a relationship between the path cover/partition number and forbidden structures in digraphs. • In this paper, we find forbidden structure conditions assuring us that pc(D) (or pp(D)) is bounded by a constant for every weakly connected digraph D. Recently, the authors [Electron. J. Combin. 29 (2022), Paper No. 4.8] gave Ramsey-type results for the path cover/partition number of graphs. In this paper, we continue the research about them focusing on digraphs, and find a relationship between the path cover/partition number and forbidden structures in digraphs. Let D be a weakly connected digraph. A family P of subdigraphs of D is called a path cover (resp. a path partition) of D if ⋃ P ∈ P V (P) = V (D) (resp. ⋃ ˙ P ∈ P V (P) = V (D)) and every element of P is a directed path. The minimum cardinality of a path cover (resp. a path partition) of D is denoted by pc (D) (resp. pp (D)). In this paper, we find forbidden structure conditions assuring us that pc (D) (or pp (D)) is bounded by a constant. [ABSTRACT FROM AUTHOR]

• A fast finite-time criterion for deterministic systems is established. • We establish a fast finite-time control criterion for stochastic systems by revamping the fast finite time method. • The control strategy can effectively improve the control speed compared to the existing results in this paper. This paper elucidates the finite-time stability in probability problem for a class of p-norm stochastic nonlinear systems. Among them, from the point of view of convergence speed, finite-time control does have optimal properties. Consequently, a fast finite-time criterion for deterministic systems is established. However, although there exist a significant amount of results on fast finite time stability, there is no article involves p-norm stochastic nonlinear systems case available so far. There are random disturbance terms that made the traditional fast finite-time control methods inapplicable. To overcome this problem, we establish a fast finite-time control criterion for stochastic systems by revamping the fast finite time method of deterministic systems. Then, with the aid of sign function and adding a power integrator technology, a non-Lipschitz continuous state feedback controller is established successfully. Next, the control strategy can effectively improve the control speed compared to the existing results in this paper. Finally, numerical and dynamic suspension simulations are offered to augment our theoretical analysis results, and comparison plots with conventional control strategies are shown. [ABSTRACT FROM AUTHOR]

• The introduction is shorten and a historical note has been added in the appendix. • Section 2 is the critical part of the paper and has been deeply reformulated. In this paper, we propose a new paradigm for finite differences numerical methods, based on compact schemes to provide high order accurate approximations of a smooth solution. The method involves its derivatives approximations at the grid points and the construction of structural equations deriving from the kernels of a matrix that gathers the variables belonging to a small stencil. Numerical schemes involve combinations of physical equations and the structural relations. We have analysed the spectral resolution of the most common structural equations and performed numerical tests to address both the stability and accuracy issues for popular linear and non-linear problems. Several benchmarks are presented that ensure that the developed technology can cope with several problems that may involve non-linearity. [ABSTRACT FROM AUTHOR]

• A stabilization method for probabilistic Boolean networks using Q -learning is studied. • The reward design method for deriving a reduced-order controller is proposed. • The reward design method for deriving both a pinning-order controller and pinning nodes is proposed. • The effectiveness of the proposed method is presented by numerical examples. In this paper, we study a stabilization method for probabilistic Boolean networks (PBNs) using Q -learning, which is one of the typical methods in reinforcement learning. A PBN is a class of discrete-time stochastic logical systems in which update functions are randomly chosen from the set of the candidates of Boolean functions. In the existing methods using reinforcement learning, a design method of structured controllers has not been studied. In this paper, we propose reward design methods to derive reduced-order controllers and pinning controllers. The key idea is to adjust the structure of controllers by the reward for the structure of the Q -table. The advantage of the proposed method is that implementation is easy, because the proposed method can be embedded in the existing Q -learning-based stabilization algorithm. In design of pinning controllers, we can calculate not only controllers but also pinning nodes in which the control input is assigned. We demonstrate through numerical examples that the controller obtained from our proposed method is simpler than that from the existing method. [ABSTRACT FROM AUTHOR]

• Distinct from most existing output/state constraints problem [13, 20, 21, 22, 32, 33, 34, 35, 36], function constraints on full states in this paper are asymmetric time-varying, and they are dependent of both state and time. This will increase the difficulty of controller design due to their partial derivatives. Moreover, our proposed control scheme has wide applications, that is, symmetric constraints, constant constraints or time-varying constraints are the special cases of function constraints. • Without growth conditions on nonlinear functions, FLS is introduced to estimate some complex terms generated by the derivation of constraint boundary, the auxiliary variable and uncertain nonlinear functions which cannot be directly handled in the constructed controller. • Different from [13, 14, 19, 26, 28, 29, 33], all the control coefficients and their boundaries are not available with respect to state-function constraints. As a consequence, the Nussbaum function involved in [38, 39, 40, 41, 42] is borrowed in the process of controller design. • By incorporating BLF method into the backstepping idea, a novel adaptive fuzzy controller is devised such that the tracking error ultimately changes near the origin and all signals of the closed-loop system are bounded. Meanwhile, full states remain in the asymmetric function-constraint set. The adaptive control problem is studied for nonlinear systems with asymmetric full state-function constraints, unknown control gains, input delay and external disturbances in this paper. Firstly, the effect of input time delay is removed by using Laplace transform and Pade approximation. Then, Nussbaum function and fuzzy logic system (FLS) are recommended to dispose unknown control gains and unknown nonlinear functions, respectively. By incorporating barrier Lyapunov function (BLF) into the backstepping idea, a new adaptive fuzzy controller is presented such that the tracking error finally fluctuates near the origin and all signals of the closed-loop system are bounded. Meanwhile, the asymmetric full state-function constraints do not violate. Ultimately, a numerical example is used to explain the feasibility of the devised control scheme. [ABSTRACT FROM AUTHOR]

• This paper firstly proposes improved Milne-Hamming method to solve high-order uncertain differential equations. • Compare with existing numerical methods, this method further improves the accuracy of solving high-order uncertain differential equations. • This method is firstly implemented to acquire the expected value, extreme value, and integral of the solution. Uncertain systems can be modeled using high-order uncertain differential equations, however, which are often difficult to resolve via analytical methods, so what's necessary is how to devise a numerical method. Based on the Milne-Hamming formula, this paper presented a new numerical solution—improved Milne-Hamming method. It showed that improved Milne-Hamming method is better than other numerical methods through numerical experiments. In addition, the analysis of its convergence, stability, and time complexity was given. After this, this method was implemented to acquire the expected value, extreme value, and integral of the solution. Finally, a concise conclusion was obtained. [ABSTRACT FROM AUTHOR]

• In this paper, H ∞ finite-time stability and boundedness are first applied to discrete-time MJSPSs with partially known probability information. Compared with other existing literature, it guarantees the boundedness in the finite-time region of MJSPSs, but not the state convergence in the infinite interval. • The HMM is adopted to solve the difficulty where system modes are inconsistent with controller modes, and a situation where obtaining probabilistic information is not easy is analyzed based on the HMM. It considers that partially known probability information exists in the transition probability matrix of Markov chain, or in the observation probability matrix, or in both. It is more comprehensive and practical than the case of the completely known probability information. • The conditions for the existence of the finite-time asynchronous H ∞ controller are presented, and the design method of the controller is proposed by solving the linear matrix inequalities independent of SPP. Finally, two examples are given to verify the effectiveness and practicability of the proposed method. This paper investigates the finite-time asynchronous H ∞ control problem for Markov jump singularly perturbed systems. The hidden Markov model is introduced to describe the asynchrony between system modes and controller modes, where partially known probability information exists in the transition probability matrix, in the observation probability matrix or in both. On this basis, a controller which ensure the boundedness in the finite-time and satisfy the H ∞ performance of Markov jump singularly perturbed systems is designed. Whereafter, the designed controller gain is obtained by solving linear matrix inequalities which are independent of singularly perturbed parameters. Finally, the validity and practicability of the proposed method are verified by two examples. [ABSTRACT FROM AUTHOR]

• A balanced space-time Sinc-collocation method for viscoelastic heat. • An effective single exponential Sinc method to deal with the initial singularity. • The error estimate is proved rigorously. • Space-time spectral-order convergence order simultaneously. The purpose of this paper is to investigate a space-time Sinc-collocation method for solving the fourth-order nonlocal heat model arising in viscoelasticity, which is a class of partial integro-differential equations (PIDEs) whose solution typically exhibits the weak singularities at the initial time. Most of existing approximate methods for solving such kind PIDEs are unbalanced, since most work have used a low order scheme for integrating the temporal variable and a high order numerical framework for discretization of space variables. The current paper contributes to a space-time spectral-order Sinc-collocation method which is balanced in both time and space variables. In order to deal with the initial singularity of solution in the time direction, the Sinc-collocation method with single exponential (SE) transformation is used for the first time, which is an effective method to deal with the singularity of the equation. Also, we fully analyse the error bounds of the method which show the exponential convergence rate in space and time. Meanwhile, we consider several test problems for examining the suggested scheme. The experiment results highlight a good precision of the new Sinc-colloction method and the correctness of the theoretical prediction for this kind nonlocal heat problems. [ABSTRACT FROM AUTHOR]

• This paper considers the estimation problem for two-dimensional uncertain systems with redundant channels and randomly occurring deception attacks. The redundant channel transmission is applied to improve the transmission reliability, where the channel packet losses are characterized by a set of random variables. The random deception attacks are also considered to characterize the insecure network environment. • By employing the set-membership method with mathematical induction technique, sufficient conditions are derived guaranteeing existence of the expected set-membership estimator, which would limit the state estimate error within a predetermined ellipsoidal area. • The criteria obtained are in the form of recursive matrix inequalities that are convenient for online computation. In addition, an optimization algorithm is also provided to ensure the locally optimal estimation performance. This paper considers the estimation issue of two-dimensional uncertain systems with redundant channels and randomly occurring deception attacks. The redundant channel transmission is applied to improve transmission reliability, with channel packet losses characterized by a set of mutually uncorrelated random variables that follow the Bernoulli distribution. To simulate the random deception attacks, a Bernoulli distribution-based sequence is employed. The purpose of this study is to build a set-membership estimator for the considered system with redundant channels in the insecure network environment that limits the state estimate error to a predetermined ellipsoidal area. By employing the set-membership method with the mathematical induction technique, sufficient criteria are derived to guarantee the expected estimated performance. In addition, an optimization algorithm is further provided to ensure locally optimal estimation performance. Finally, a numerical simulation of the suggested set-membership estimation technique is presented to demonstrate its usefulness. [ABSTRACT FROM AUTHOR]

• The technique bypasses technically challenging derivation of stochastic representative solutions underlying Monte Carlo solution of initial boundary value problems. • In contrast to MC solutions, the technique estimates non-problem-specific impulse-response functions, i.e., Green's functions, which can be used to solve IBVP's subject to an infinite array of boundary and internal forcing conditions. • A random walker swarm is launched from a chosen response (or impulse) point, well-developed RWboundary interaction models can be implemented, and a simple estimator is used to estimate the transition density - and corresponding Green's function - between the response/impulse point and the impulse/response point. • Deleterious depletion of RW's at absorbing Dirichlet boundaries is solved using a RW respawning algorithm, while spatial variance in estimated GF's, produced by our naĩve transition density estimator, is minimized using a simple area-averaging technique. • The proposed technique is tested against three problems having known Green's functions. • The paper is presented in recipe fashion, and is aimed at non-specialists. A stochastic method is described for estimating Green's functions (GF's), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF's, within any geometry, the technique solves the central challenge in obtaining Green's function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF's that can be used: a) to obtain (infinite) sets of solutions, subject to any combination of (random and deterministic) boundary, initial, and internal forcing, b) as high fidelity direct models in inverse problems, and c) as high quality process models in thermal and mass transport design, optimization, and process control problems. The technique exploits an equivalence between the adjoint problem governing the transport problem Green's function, G (x , t | x ′ , t ′) , and the backward Kolmogorov problem governing the transition density, p (x , t | x ′ , t ′) , of the stochastic process used in Green's function construction. We address nonspecialists and report four contributions. First, a recipe is outlined for diagnosing when stochastic Green's function estimation can be used, and for subsequently estimating the transition density and associated Green's function. Second, a naive estimator for the transition density is proposed and tested. Third, Green's function estimation error produced by random walker absorption at Dirichlet boundaries is suppressed using a simple random walker splitting technique. Last, spatial discontinuity in estimated GF's, produced by the naive estimator, is suppressed using a simple area averaging method. The paper provides guidance on choosing key numerical parameters, and the technique is tested against two simple unsteady, linear heat conduction problems, and an unsteady groundwater dispersion problem, each having known, exact GF's. [ABSTRACT FROM AUTHOR]

• The paper investigates some mathematical properties of the Photo Response Non Uniformity Pattern noise that can be used for the identification of the source device of an image. • The paper proves that the 2D decreasing rearrangement of PRNU image provides specific and device-dependent geometric structures that can be properly coded by the Radon transform. • The empirical distribution of the Radon transform of PRNU 2D decreasing rearrangement is a feature robust to image manipulations and independent of image size. Source camera identification represents a delicate, crucial but challenging task in digital forensics, especially when an image has to be used as a proof in a court of law. This paper investigates some properties of the Photo Response Non Uniformity (PRNU) pattern noise that represents the fingerprint of any acquisition sensor. The main goal is to define specific and distinctive features for this noise source that enable the identification of the acquisition sensor by simply analysing a few images. These features are required to be independent of image size, modifications, storage mode, etc. The discrimination power of the decreasing rearrangement of a function, combined with the Radon transform, has been investigated in this paper. Preliminary tests show that a proper rearrangement of PRNU image provides specific and device-dependent geometric structures that can be properly coded through the Radon transform. In particular, the empirical distribution of the Radon Transform of rearranged Flat Field images alone is capable to correctly characterize each device with high accuracy, showing robusteness to some standard image modifications, such as quantization and blurring; in addition, it guarantees independence of image size. [ABSTRACT FROM AUTHOR]

• The principal purpose of this paper is the development in numerical treatments for the resonance problem of multi-phase systems by constructing a well-balanced finite volume scheme for two-dimensional shallow water equations with variable topography. • This scheme is constructed based on exact solution of local planar Riemann problems at each grid cells. • The well-balancedness of this scheme is shown in the sense that it can preserve exactly the lake at rest solution over both smooth and non-smooth bottom. • Several numerical experiments related to steady and time-dependent problems are conducted to show the potential of this approach. • The remain challenges are the ineffectiveness of this scheme towards time-dependent solutions throughout a long process and the instability of errors caused by the relaxation while coding. • This work provides an impetus to enhance the performance of this scheme for time-dependent solutions, and motivates further studies in numerical treatments of the resonance problem for multi-phase flow models as well. We consider in this paper a finite volume scheme based on local planar Riemann solutions for the two-dimensional shallow water equations with bathymetry. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The scheme to be presented is a development of the preliminary works that will be cited below. Our foremost purpose is to extend those results to two-dimensional formalism while preserving the physical and mathematical properties, including the well-balancedness. The proposed scheme is applied to some specific families of solutions, especially lake at rest and partially well-balanced solution. The numerical results show that this approach can give a good accuracy, except for resonant cases. Furthermore, it is proved that our finite volume scheme can preserve the C -property in the sense that it can capture exactly the lake at rest solution. [ABSTRACT FROM AUTHOR]

• The algorithm proposed in this paper is simpler and easier to understand compared to the algorithms in references [3] and [8]. • The paper focuses on the power domination problem on trees and presents a linear-time algorithm to solve it using the labeling method. In this paper, the power domination problem on trees is studied and we give a linear-time algorithm to it by using the labeling method. Our algorithm is simpler and easier to understand than those in [3,7]. [ABSTRACT FROM AUTHOR]

• 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]