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• Understanding how strategic interactions influence trait evolution remains a challenge in evolutionary games. • The evolutionary RPS game investigates the evolution of three strategies under a fixed payoff matrix. • Exploring the impact of a dynamic instead of a fixed payoff matrix could change the dynamics of the game. • The adaptive RPS game, incorporating a dynamic payoff matrix, investigates how strategies change adaptively through trait evolution. The rock-paper-scissors (RPS) game is a classic model for exploring the performance of how multiple strategies interact and evolve over time. The classic RPS game assumes a fixed benefit and cost for each strategy against another one when two players meet, while its evolutionary game considers the frequency dynamics of the three strategies with each's fitness influenced by its net payoff. This may not reflect the complexity of real-world scenarios as strategies can co-adapt with each other. We introduce an adaptive RPS game that captures the dynamics of strategy densities with trait-mediated payoffs, and the adaptive dynamics of coevolving traits via incremental mutations leading to adaptively evolving payoffs. Results show that the adaptive RPS game approaches a steady state of strategy density, if any, faster than the evolutionary RPS game. The stable coexistence of all strategies in the evolutionary game can be easily destabilized in the adaptive game. Strategies that are allowed to adaptively change their traits also performed better and achieved greater strategy densities than those fixed strategies in the adaptive game. The coevolving strategies in the adaptive RPS game exhibit complex and diverse attractors in the trait space, sensitive to both initial conditions and model parameters, but exhibiting positive payoffs with greater benefits than costs. These findings highlight how adaptive games enhance strategy performance by sacrificing eco-evolutionary dynamic stability. [ABSTRACT FROM AUTHOR]

In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]

In this paper, we first point out a gap in Yang (2012). Next, strong convergence theorem for the common solution for a finite family of φ -strongly accretive operator equations in Banach spaces is established without any bounded assumption, and we also give an example to show universality of the result of this paper, which improve and extend some recent results. [ABSTRACT FROM AUTHOR]

In the paper the second order half-linear delay differential equation ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u ( τ ( t ) ) | α - 1 u ( τ ( t ) ) = 0 , α > 1 is studied under the condition ∫ ∞ r - 1 / α ( t ) d t < ∞ . The oscillation criterion proved by Sun and Meng (2006, Theorem 2.2) is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions. Among others, using elementary arguments we show that an arbitrary positive nondecreasing function ρ from this criterion can be safely and with no loss of generality replaced by a constant function. An extension to other related equations, such as neutral equations is also provided. [ABSTRACT FROM AUTHOR]

The future use of eddy resolving simulations (ERS) such as Large Eddy Simulation (LES), Direct Numerical Simulation (DNS) and related approaches in aerospace is explored. The turbulence modeling requirements with respect to aeroengines and aircraft is contrasted. For the latter, higher Reynolds numbers are more prevalent and this especially gives rise to the need for the hybridization of ERS methods with Reynolds Averaged Navier–Stokes (RANS) approaches. Zones where future use of pure ERS methods is now possible and those where hybridizations with RANS will be needed is outlined. The major focus is the aeroengine for which the component scales are much smaller. This gives rise to generally more benign Reynolds numbers. The use of eddy resolving methods in a wide range of zones in an aeroengine is discussed and the potential benefits and also cost drawbacks with such approaches noted. The tension when using such computationally intensive calculations in an area where the coupling of components and even the airframe and engine is becoming increasingly important is explored. Also, the numerical methods and meshing requirements are considered and the implications of ERS methods for future numerical algorithms. It is postulated that such simulations are ready now for niche uses in industry. However, to perform the scale of simulations that industry requires, to meet pressing environmental needs, challenges remain. For example, there is the need to develop optimal numerical methods that both map to the accuracy requirements for ERS and also future computer architectures. [ABSTRACT FROM AUTHOR]

• The convergence analysis and the techniques used in Lemma 3.3 by X. Wang et al. in [1] are based essentially on Sobolev's inequality in one dimension and they cannot be extended to two dimensions. • All results based on Lemma 3.3 are false, in particular Theorem 3.3, Theorem 4.1, Theorem 5.1 and Theorem 5.2. • Regarding the conservation of mass and energy, the initial terms Q 0 and E 0 depends on U 0 and on U 1 and the authors consider only Eq. (4) without taking into account Eq. (7). Therfore, there is an error in Theorem 3.1 and Theorem 3.2. • In Theorem 4.1, the authors did not show that u1 is uniquely determined by Eq. (7). In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments. [ABSTRACT FROM AUTHOR]

• We build a pairwise (PW) nonlinear fermi dynamics to study the rock-paper-scissors game's under evolutionary dynamics. • The model also investigates the effect of environmental noise and demographic noise. • We perform analytical analysis and numerical simulation to show the impact of linear and nonlinear to explore the noise effect. • Both linear and non-linear dynamics show comparable stability equilibria. The difference between conventional replicator dynamics and pairwise (PW) nonlinear Fermi dynamics can be discerned by studying the evolutionary dynamics of the interactions between the symmetric cyclic structure in the rock–paper–scissors game and inter- and intraspecific competitions. Often, conventional replicator models presume that the payoff difference among species is a linear function (a linear benefit). This study introduces a PW contrast under the properties of the well-known Fermi rule, where species play against one another in pairs. To model a PW nonlinear evolutionary environment (a nonlinear benefit) within this framework, both analytical and numerical approaches are applied. It is determined that the dynamics of the linear and nonlinear benefits can present the same stability conditions at equilibrium. Moreover, it is also demonstrated that, even in an identical equilibrium condition for both dynamics, the numerical result run by a deterministic approach presents a faster stability state for nonlinear benefit dynamics. This study also suggests that introducing mutation as demographic noise can effectively disrupt the phase regions and show the different relationships between linear and nonlinear dynamics. The symmetric bidirectional mutation among all the species reduced to the stable limit cycle by an arbitrary small mutation rate is also explored. Due to the environmental noise, however, linear and nonlinear exhibit the same steady state. Nevertheless, non-linearity illustrates more stable and faster stability situations. Our result suggests that environmental and demographic noise on the evolutionary dynamic framework can serve as a mechanism for supporting PW nonlinear dynamics in multi-species games. [ABSTRACT FROM AUTHOR]

• kinetic description of a binary game • quasi-invariant limit • numerical simulations This article studies the kinetic dynamics of the rock-paper-scissors binary game. We first prove existence and uniqueness of the solution of the kinetic equation and subsequently we prove the rigorous derivation of the quasi-invariant limit for two meaningful choices of the domain of definition of the independent variables. We notice that the domain of definition of the problem plays a crucial role and heavily influences the behavior of the solution. The rigorous proof of the relaxation limit does not need the use of entropy estimates for ensuring compactness. [ABSTRACT FROM AUTHOR]

In this paper, our intention is to investigate the blow-up theory for generalized auto-convolution Volterra integral equations (AVIEs). To accomplish this, we will consider certain conditions on the main equation. This will establish a framework for our analysis, ensuring that the solution of the equation exists uniquely and is positive. Firstly, we analyze the existence and uniqueness of a local solution for a more general class of AVIEs (including the proposed equation in this paper) under certain hypotheses. Subsequently, we demonstrate the conditions under which this local solution blows up at a finite time. In other words, the solution becomes unbounded at that time. Furthermore, we establish that this blow-up solution can be extended to an arbitrary interval on the non-negative real line, thus referred to as a global solution. These results are also discussed for a special case of generalized AVIEs in which the kernel functions are taken as positive constants. • Proof of existence and uniqueness of a local solution for a more general case of AVIEs in two cases (see Section 3). • Discussing continuation of local solution for a large interval (see the last two paragraphs of Section 3). • Proving blowing up and globality properties of local solution. [ABSTRACT FROM AUTHOR]

In this paper, a new direct method based on system solutions is proposed to give global exponential synchronization analysis of multi-link memristive neural networks. The network dynamics are affected by time-varying distribution, leakage and transmission delays, simultaneously. Based on the definition of synchronization, sufficient conditions to ensure the synchronization of multi-link memristive neural networks are investigated, and thereby, a new controller is proposed. Compared with other controllers, the controller design method proposed in this paper is relatively simple, and avoids the construction of Lyapunov–Krasovskii functionals, which greatly reduces the workload. Finally, numerical simulations are given to check the effectiveness of this method. • Global exponential synchronization of multi-link memristive neural networks is studied. • The time-varying distribution, leakage and transmission delays are involved. • A direct method based on system solutions is proposed to give synchronization conditions. • Compared with the usual methods, the direct method reduces computational complexity. • The obtained synchronization conditions can be easily checked by using MATLAB. [ABSTRACT FROM AUTHOR]

The maintenance of biological diversity has perpetually remained a central focus in the field of ecology. In the pursuit of enhanced survival rates, species have begun to explore cooperation with one another. However, the consequences of such collaboration remain largely unexplored. To delve into this matter, we introduce intraspecific cooperation within the framework of the classic rock-paper-scissors (RPS) game. In this model, the competition rate is intricately tied to interactions among individuals of the same species. A greater population of individuals from the same species tends to lead to an increased predation rate and a decreased prey rate. Through extensive simulations, we observe that (i) in the case of homogeneous intraspecific cooperation (all three species have intraspecific cooperation), increased cooperation between predators tends to increase the likelihood of species coexistence. In contrast, high levels of cooperation between prey appeared to decrease the favorability of species coexistence. Measurements of the characteristic length of spiral structures revealed that the characteristic length of spirals became longer when the intensity of prey cooperation increased. (ii) In the case of heterogeneous intraspecific cooperation (two species or only one species with intraspecific cooperation), neither an increase in the intensity of intraspecific cooperation of the predator nor an increase in the intensity of intraspecific cooperation of the prey is favorable for species coexistence. Our work underscores the critical role of intraspecific cooperation in maintaining biodiversity. • In rock-paper-scissors games, intraspecific cooperation promotes and jeopardizes biodiversity. • Under homogeneous intraspecific cooperation, predator cooperation promotes coexistence. • Under heterogeneous intraspecific cooperation, both predator and prey cooperation jeopardize coexistence. [ABSTRACT FROM AUTHOR]

A fan, denoted by H 2 k , is the graph obtained by joining a vertex to a path with 2 k vertices. A graph is H 2 k -free if it does not contain a subgraph H 2 k. Yuan (2021) [13] proved the Turán number and the extremal graphs of H 2 k -free graphs. In this paper, we characterize spectral extremal graphs of H 2 k -free graphs for sufficiently large n and k ≥ 3. • Spectral Turán-type problem is a frontier research field in spectral graph theory. In this paper, we characterize spectral extremal graphs of H 2 k -free graphs for sufficiently large n and k ≥ 3. • The problem investigated in this paper is very interesting and important in spectral graph theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose the frozen form of nonstationary iterated Tikhonov regularization to solve the inverse problems and study its convergence analysis in the Banach space setting, where the iterates of proposed method are defined through minimization problems. These minimization problems involve uniformly convex penalty terms that may be non-smooth, which can be utilized in the reconstruction of several notable features (e.g., discontinuities and sparsity) of the sought solutions. We terminate our method through a discrepancy principle and show its regularizing nature. Moreover, we show the convergence of the iterates of the method with respect to Bregman distance as well as their strong convergence. In order to show the applicability of the proposed method, we show that it is applicable on two ill-posed inverse problems. Finally, by assuming that a stability estimate holds within a certain set, we derive the convergence rates of our method. • In this paper, the frozen form of nonstationary iterated Tikhonov regularization to solve the inverse problems is proposed. • The convergence analysis is studied in Banach spaces for both the noisy as well as non-noisy data. • The convergence rates of the method are derived through stability estimates. • Two concrete examples are discussed to showcase the validity of our results. [ABSTRACT FROM AUTHOR]

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems. • Numerical Gaussian processes for solving delay PDEs models governed by variable-order fractional order operators and nonlinear operators. • Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. • Runge-Kutta methods are applied to enhance the prediction accuracy of these problems. • Seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with higher odd-order boundary value problems. We first prove that the operators associated with the problems are symmetric and the corresponding eigenvalues are real, and we give the resolvent operators related to the differential operators. Then we obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem. Moreover, the differential expressions of the eigenvalues as regards these parameters are given. In particular, we give the Frechet derivatives of the eigenvalues for the leading coefficient function q 0 and coefficient functions q 1 , ⋯ , q n. • This paper is concerned with dependence of eigenvalues of higher odd-order boundary value problems. • We give a brief overview of the studies related to the problems, and show that the eigenvalues of the problem depend not only continuously but also smoothly on all parameters of the problem. • By some derivation methods and techniques, we give the differential expressions of the eigenvalues with respect to the coefficient functions q 0 , q 1 , ⋯ , q n , which are some interesting and concise results. And these results have not been done by the predecessors. [ABSTRACT FROM AUTHOR]

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nyström methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose and introduces new stability properties of them. • Weighted averaged quadrature rules are analyzed and their stability and convergence are proved in weighted function spaces. • A Nyström type method based on averaged rules is developed for the numerical solution of second-kind Fredholm equations. • New iterative methods are given for the solution of systems of equations associated with weighted averaged Nyström's methods. [ABSTRACT FROM AUTHOR]

In this paper, the zero-sum game problem for linear discrete-time Markov jump systems is solved by two novel model-free reinforcement Q -learning algorithms, on-policy Q -learning and off-policy Q -learning. Firstly, under the framework of the zero-sum game, the game-coupled algebraic Riccati equation is derived. On this basis, subsystem transformation technology is employed to decouple the jumping modes. Then, a model-free on-policy Q -learning algorithm is introduced in the zero-sum game architecture to obtain the optimal control gain by measured system data. However, the probing noise will produce biases in on-policy algorithm. Thus, an off-policy Q -learning algorithm is proposed to eliminate the effect of probing noise. Subsequently, convergence is discussed for the proposed methods. Finally, an inverted pendulum system is employed to verify the validity of the proposed methods. • The zero-sum game problem for linear discrete-time Markov jump systems is addressed with the reinforcement learning method. • A novel model-free parallel off-policy Q-learning method is developed, which is not affected by probing noise. • The subsystem transformation technology is employed decoupling the jumping modes to solve the game-coupled algebraic Riccati equation. • The convergence of the proposed methods, on-policy and off-policy, for Markov jump systems are both discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2- 1 σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L 2 -norm and H 1 -norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H = h 1 2 and H = h r 2 r + 2 , respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods. • We establish an efficient two-grid method for the nonlinear time-fractional variable coefficient diffusion-wave equations. • We provide stability and convergence results for both fully discrete scheme and two-grid scheme, respectively. • The L2- 1 σ formula is employed to discretize Caputo fractional derivative to achieve second-order numerical accuracy. • Two-grid methods achieve optimal convergence order when coarse and fine grids satisfy H = h 1 2 and H = h r 2 r + 2 , respectively. • Considering the initial singularity and historical memory, the nonuniform and fast two-grid algorithms are presented. [ABSTRACT FROM AUTHOR]

In this paper, set stability of Boolean networks (BNs) is studied based on semi-tensor product of matrices. First, an index-vector is introduced to verify whether a BN can be set stable from an initial set to a destination set. Then, an algorithm is proposed to obtain the minimal set of pinning nodes to achieve set stability, if a BN is not set stable. Moreover, in order to reduce the computational complexity, a hybrid pinning control technique is proposed, where the designed controllers are not entirely dependent on all the nodes of the networks. Further, the issue of synchronization is discussed by using the proposed hybrid pinning control methods. Finally, simulations are presented to illustrate the effectiveness of the obtained results in this paper. • Based on an index-vector, a new necessary and sufficient condition has been obtained for set stability of Boolean networks from an initial state set to a destination state set. • Compared to the existing methods, fewer pinning nodes have been found to achieve global stability for a Boolean network by using the proposed method in this paper. • By using hybrid pinning control, pinning controllers have been simplified and the computational complexity has been reduced. [ABSTRACT FROM AUTHOR]

This paper studies customers' equilibrium joining behavior in a multi-server queue with threshold policy, and then puts forward effective schemes to control customers' behaviors and to optimize social welfare. The stream of customers consists of two parts: customers in class 1 and customers in class 2, and they wait for service in order until the system is empty. Then the servers take a vacation and they would be reactivated as soon as the queue length reaches a threshold N. The two types of customers are heterogeneous in service revenue and delay sensitivity, i.e., they have different revenue-cost structures. Besides, customers in class 1 can access to the delay information on servers' status and queue length as they arrive, but customers in class 2 can not. Distinguishing the two scenarios of whether there is a joining threshold for customers in class 1 in vacation state, we get the two types of customers' equilibrium joint joining strategies, respectively, and observe the interaction between the behaviors of the two types of customers. Moreover, we also get their socially optimal joint joining strategy for comparisons and for making control measures, and discuss the influence of the fraction of customers in class 1 on social welfare and system throughput. We find that in any scenario, the equilibrium social welfare is always unimodal with respect to the fraction of customers in class 1, which indicates that the higher fraction of customers in class 1 is not necessarily beneficial. However, on the premise of effectively controlling customer behavior, this high fraction is still necessary to optimize social welfare and increase system throughput, although the optimal social welfare is also unimodal. • This paper discusses the joining behavior of heterogeneous customers in a multi-server single-queue system with threshold policy. • This paper classifies two types of customers based on both delay information and revenue-cost structure heterogeneities in a multi-server queue. • This work simultaneously discusses and compares the equilibrium and socially optimal joining behaviors of the two types of customers. [ABSTRACT FROM AUTHOR]

An oriented circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (PST for short) and multiple state transfer (MST for short) on integral oriented circulant graphs. Specifically, a characterization of PST and MST on integral oriented circulant graphs is provided. As an application, we also obtain a closed-form expression for the number of integral oriented circulant graphs having PST and MST. • Use Ramanujan's sum to characterize the eigenvalues of integral oriented circulant graphs and give the complete formula. • The paper provides a characterization of PST and MST on integral oriented circulant graphs. • Closed-form expression for the number of graphs with PST and MST on integral oriented circulant graphs. [ABSTRACT FROM AUTHOR]

In this paper, by means of sliding mode control (SMC), the problem resulted by local input matrices is investigated for fuzzy markovian jumping system. The stability of the considered fuzzy Markovian system with multiple sub-input matrices can be fulfilled using the method in this paper. First, the state equation of the considered system is transformed according to the input matrices. Second, according to the state equation, sliding mode surface is constructed. This surface contains several sub-surfaces. It can deal with this kind of fuzzy Markovian system with multiple sub-input matrices and uncertainties. And another characteristic is that there is not the process of reaching the sliding mode surface; It can settle the problem resulted by Markovian jumping and sliding mode method together. Third, by use of the Cramer's rule, a criterion is provided to judge the existence of sliding mode dynamics equation. A controller containing several sub-controllers components is designed. And these sub-controllers should keep the considered system on the several sub-surfaces and not leave it. At last, simulations are provided to illustrate the validity of the method in this paper. [ABSTRACT FROM AUTHOR]

• To improve the efficiency and stability, by combining the DSMLS approximation and a stabilization factor based on the solution space, an improved stabilized element-free Galerkin (ISEFG) method is proposed for the Stokes problems. • The ISEFG method can lessen the dimensionality and complexity of matrices calculation in solving the shape functions, thereby improving efficiency and accuracy. • The numerical results show that the ISEFG method of this paper can acquire higher accuracy and consumes less CPU time than the EFG method based on the MLS approximation for Stokes problems. Combining the dimensional splitting moving least squares (DSMLS) approximation and the variational weak form, this paper developed an improved stabilized element-free Galerkin (ISEFG) method for Stokes problems. In the ISEFG method, the DSMLS approximation is adopted to construct the shape function, and the stabilization factor is established based on the solution space of velocity and pressure. The Galerkin weak form and integral coordinate transformation are taken to achieve the final discrete equations of the problems. Following the ideas of the dimensional splitting method, the DSMLS method approximates the functions from the direction of dimension splitting and the dimension-splitting subdivision surfaces. Then the ISEFG method can reduce the dimensionality and complexity of matrix operations in solving the shape function, thereby improving the efficiency and accuracy. This paper introduces several numerical examples to demonstrate the effectiveness of the stabilized meshless method. The numerical examples show that the ISEFG method based on the DSMLS approximation can find stable solutions of the velocities and pressure without physical oscillation. The method presented in this paper offers higher accuracy and consumes less CPU time than the EFG method based on the MLS approximation. [ABSTRACT FROM AUTHOR]

In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n × n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem. • The eigen-problems of reduced biquaternion matrices are further studied based on the complex representation form. • Propose new algebraic algorithms for finding the eigenvalues and the eigenvectors of a reduced biquaternion matrix. • An n × n reduced biquaternion matrix has infinite eigenvalues. • There are multiple eigenvalues corresponding to the same eigenvector of a reduced biquaternion matrix. • The proposed method is more comprehensive and can find more eigenvalues of a reduced biquaternion matrix. [ABSTRACT FROM AUTHOR]

Parallel and distributed systems play a significant role in high-performance computing, prompting us to investigate qualitative and quantitative metrics to indicate the fault tolerance and vulnerability of systems. Consider the setup where there are large-scale link malfunctions that disconnect the network and result in various components, each with multiple processors. In this paper, we propose and study the h -extra r -component edge-connectivity of a connected graph G , which is denoted by c λ r h (G) and has not been addressed before. Let h and r be two positive integers with r ≥ 2. An edge subset F ⊆ E (G) is said to be an h -extra r -component edge-cut of G , if any, G − F has at least r components and every component of G − F has at least h vertices. The cardinality of the minimum h -extra r -component edge-cut of G is the h -extra r -component edge-connectivity of G. Let h be a positive integer with the decomposition h = ∑ i = 0 t 2 k i , where k i > k i + 1 , 0 ≤ i ≤ t − 1. In this paper, we derive a lower bound for the exact value of h -extra 3-component edge-connectivity of BC networks B n and demonstrate that it is tight for one member of B n , hypercube Q n , in the interval 1 ≤ h ≤ 2 ⌊ n 2 ⌋ − 1 − 1 , n ≥ 4. This lower bound is also tight for the exact value of 2 c -extra 3-component edge-connectivity of B n with 0 ≤ c ≤ n − 2 , n ≥ 4. Specifically, c λ 3 h (Q n) = 2 n h − ∑ i = 0 t k i 2 k i + 1 − ∑ i = 0 t i 2 k i + 2 − h for 1 ≤ h ≤ 2 ⌊ n 2 ⌋ − 1 − 1 , n ≥ 4 and c λ 3 2 c (B n) = (2 n − 2 c − 1) 2 c for 0 ≤ c ≤ n − 2 , n ≥ 4. Exact value of h -extra 3-component edge-connectivity of n -dimensional folded hypercube F Q n , c λ 3 h (F Q n) is 2 (n + 1) h − ∑ i = 0 t k i 2 k i + 1 − ∑ i = 0 t i 2 k i + 2 − h for 1 ≤ h ≤ 2 ⌈ n 2 ⌉ − 1 − 1 , n ≥ 4 , and that of 2 c -extra 3-component edge-connectivity of F Q n , c λ 3 2 c (F Q n) , is (2 n − 2 c + 1) 2 c for 0 ≤ c ≤ n − 2 , n ≥ 4. • For a connected graph G , this paper proposes a novel conditional edge-connectivity called h -extra r -component edge-connectivity c λ r h (G). • The folded hypercube F Q n and BC networks B n are two of the most intriguing interconnection networks. • Provide a lower bound for c λ 3 h (B n) and show that it is tight for one member of B n , hypercube Q n in the interval 1 ≤ h ≤ 2 ⌊ n 2 ⌋ − 1 − 1 , n ≥ 4. • This lower bound is also tight for the exact value of c λ 3 2 c (B n) with 0 ≤ c ≤ n − 2 , n ≥ 4. • Determining the exact values of c λ 3 h (F Q n) for 1 ≤ h ≤ 2 ⌈ n 2 ⌉ − 1 − 1 , n ≥ 4 and h = 2 c , 0 ≤ c ≤ n − 2 , n ≥ 4 , respectively. [ABSTRACT FROM AUTHOR]

Let D be a digraph with vertex set V (D) and arc set A (D). An antidirected spanning closed trail of D is a spanning closed trail in which consecutive arcs have opposite directions and each arc of D occurs at most once. If all vertices in this antidirected spanning closed trail are distinct, then the antidirected spanning closed trail is called an antidirected hamiltonian cycle. Grünbaum in 1971 conjectured that every even tournament T n with n ≥ 10 has an antidirected hamiltonian cycle. In 1983, Petrović proved that any even tournament with at least 16 vertices has an antidirected hamiltonian cycle, which was the best result supporting this conjecture by far. In this paper, we investigate the existence of antidirected spanning closed trails in tournaments. Moreover, we show that every tournament T n with n ≥ 6 has an antidirected spanning closed trail, and prove that it is optimal. • The existence of antidirected spanning closed trails in tournaments is studied in this paper. • A result that for any integer n ≥ 6 , every tournament T n contains an antidirected spanning closed trail is proved. • The result in this paper is optimal and counterexamples are given for the remaining cases. [ABSTRACT FROM AUTHOR]

The Iterative Filtering method is a technique aimed at the decomposition of non-stationary and non-linear signals into simple oscillatory components. This method, proposed a decade ago as an alternative technique to the Empirical Mode Decomposition, has been used extensively in many applied fields of research and studied, from a mathematical point of view, in several papers published in the last few years. However, even if its convergence and stability are now established both in the continuous and discrete setting, it is still an open problem to understand up to what extent this approach can separate two close-by frequencies contained in a signal. In this paper, first we recall previously discovered theoretical results about Iterative Filtering. Afterward, we prove a few new theorems regarding the ability of this method in separating two nearby frequencies both in the case of continuously and discrete sampled signals. Among them, we prove a theorem which allows to construct filters which captures, up to machine precision, a specific frequency. We run numerical tests to confirm our findings and to compare the performance of Iterative Filtering with the one of Empirical Mode Decomposition and Synchrosqueezing methods. All the results presented confirm the ability of the technique under investigation in addressing the fundamental "one or two frequencies" question. [ABSTRACT FROM AUTHOR]

The rapid informatization and digitalization of the society heavily rely on the extensive use of parallel and distributed, networked computer systems. It is important for large-scale parallel and distributed systems to be able to detect and tolerate faulty vertices in the network. A network's fault status can often be characterized with the network's connectivity and diagnosability. The connectivity/diagnosability can be defined under various conditions. This paper is concerned with the connectivity/diagnosability under the " g -good-neighbor condition", which can more accurately measure a network's fault status. In this paper, we propose a new class of recursive networks, named recursive match networks (RMNs), which contain the well-known BCube and BC networks. We determine the RMNs' g -good-neighbor connectivity and g -good-neighbor conditional diagnosability under the classic MM* and PMC diagnostic models for g ≥ 0. Since the RMN is a more general network covering the BCube and BC networks, our results can be directly applied to these two networks. • We proposed a new class of networks, named recursive match networks (RMNs), which contain the BCube, BC networks, and some other networks. • We obtained the g -good-neighbor connectivity and g -good-neighbor conditional diagnosability of RMNs, under the MM* and PMC models. • We got that the g -good-neighbor connectivity/diagnosability of the RMN are both nearly g + 1 times of its original, unconditional ones. • The results of this paper can be directly applied to the BCube, BC networks, and any other networks that fall within the definition of the RMN. [ABSTRACT FROM AUTHOR]

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k + 1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H 1 and standard L 2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings. • A special variational form possessing two built-in parameters is proposed for the one-dimensional nonlinear convection-diffusion problem. • The WG scheme, developed in this paper, is very flexible because different stencils can be obtained by tuning the built-in weighted parameters. • The optimal error estimates in both standard L 2 and discrete H 1 norms are derived for our numerical method. [ABSTRACT FROM AUTHOR]

This paper takes into consideration the H ∞ filtering problem based on L 2 -gain analyses for switched descriptor systems. To start with, the state jumps that have been ignored for the existing results in the literature are precisely characterized for the filtering error system at the switching moments utilizing a mode-dependent coordinates transformation for the first time. Next, by utilizing multiple Lyapunov functions scheme, sufficient conditions are came up with ensuring the exponential stability as well as weighted H ∞ performance for the filtering error system under average dwell time. In addition, the corresponding filter parameters are gained by solving a series of LMIs. Finally, a numerical example and the ladder step-up SC converter model certify that the presented H ∞ filtering design scheme is effective. • The state jumps are precisely described for the filtering error system at the switching moments for the first time. • This paper studies a class of switched descriptor systems with the mode-dependent derivative term matrices. • Sufficient conditions are got to ensure the exponential stability while guaranteeing weighted H ∞ performance. • The established H ∞ filtering scheme is applied to the ladder step-up SC converter model. [ABSTRACT FROM AUTHOR]

This paper researches the stability issue of generalized neural networks (GNN) with time-varying delay. For the delay, its derivative has an upper bound or is unknown. Firstly, the augmented Lyapunov-Krasovskii functional (LKF) is constructed based on the state vectors of the third order integral inequalities. Then, by introducing two sets of state vectors, the LKF derivative is presented as the quadratic and quintic polynomials of the delay, respectively. Next, the new quadratic and quintic polynomial negative definite conditions (NDCs) are proposed to set up the linear matrix inequalities (LMIs). In addition, based on the same LKF and third order integral inequalities, this paper proves that the introduction of extra state vectors increases the conservatism of the derived stability conditions. Eventually, the advantages of the provided conditions are illustrated by several numerical examples. • Based on the third order FMBIIs, the novel augmented LKF is constructed. • The LKF derivative is presented as the quadratic and quintic polynomials, respectively. • For the quadratic and quintic polynomials, the novel NDCs are provided. • This paper proves that the introduction of extra state vectors increases the conservativeness of the stability conditions. [ABSTRACT FROM AUTHOR]

In this paper, an efficient nonconforming finite element method (FEM) is developed for solving the fourth order nonlinear extended Fisher-Kolmogorov equation. We firstly construct a backward Euler fully discrete scheme with a non- C 0 nonconforming double set parameter rectangular Morley element, and prove that this scheme is uniquely solvable and preserves the discrete energy dissipation law. Then based on some a priori bounds of the discrete solution, the superclose estimate in energy norm is obtained unconditionally, and the global superconvergence result is deduced with the help of the interpolated postprocessing technique. Moreover, a two-grid method (TGM) is presented, which can maintain the superconvergence result and save about half of the computing cost. Finally, numerical results are provided to show that the proposed two-grid nonconforming FEMs have a good performance. Here we mention that the above analysis is also valid for the Crank-Nicolson fully discrete scheme. • Two-grid fully nonconforming finite element method is proposed for fourth order nonlinear EFK equation. • The unconditional superclose and global superconvergence results are obtained. • The above analysis is valid for the fully discrete scheme with Crank-Nicolson type temporal approximation. • Rigorous proofs and detailed numerical results are provided. [ABSTRACT FROM AUTHOR]

Tikhonov regularization involves minimizing the combination of a data discrepancy term and a regularizing term, and is the standard approach for solving inverse problems. The use of non-convex regularizers, such as those defined by trained neural networks, has been shown to be effective in many cases. However, finding global minimizers in non-convex situations can be challenging, making existing theory inapplicable. A recent development in regularization theory relaxes this requirement by providing convergence based on critical points instead of strict minimizers. This paper investigates convergence rates for the regularization with critical points using Bregman distances. Furthermore, we show that when implementing near-minimization through an iterative algorithm, a finite number of iterations is sufficient without affecting convergence rates. • We relax traditional assumptions and allow convergence based on critical points rather than strict minimizers. • When using an iterative near-minimization algorithm, a finite number of iterations is sufficient for convergence rates. • The departure from conventional approaches opens up new ways of solving inverse problems. [ABSTRACT FROM AUTHOR]

WENO scheme is popular for solving hyperbolic conservation laws. However for the traditional WENO schemes, they often increase the order of accuracy by enlarging the stencil, thus lack the compactness and do not obtain the highest resolution and efficiency. In order to overcome this problem, we consider to improve the traditional WENO scheme by using subcell strategy, which basic idea is that: first, in order to obtain higher resolution, each control volume is divided into some subcells further; second, in order to save the computational cost, the WENO reconstruction is still implemented on the control volume; finally, in order to keep the compactness, the whole subcells in control volume and whole or part of subcells in neighbor ones are used to construct the final WENO reconstruction. To the best of our knowledge, the corresponding research is still limited. In this paper, we first propose the framework of WENO scheme with subcell strategy, in which various WENO techniques can be used for the spatial reconstruction in theory; then two traditional WENO schemes, i.e., XWENO (X=DK and AO), are selected for the improvement. The resulted XWENO-SC (X=DK and AO) schemes not only inherit the high order of accuracy and ENO property of original WENO schemes, but also show their significant advantages in the compactness, resolution and efficiency. • Our XWENO-SC (X=DK and AO) schemes inherit the high order of accuracy and ENO property of original WENO schemes. • Our XWENO-SC(X=DK and AO) schemes have significant advantages in the compactness, resolution and efficiency. • Our XWENO-SC (X=DK and AO) schemes can be extended to triangular meshes similarly. [ABSTRACT FROM AUTHOR]

Solving Hammerstein - Fredholm integral equations on unbounded domains is challenging due to its domain of integration. Therefore, this paper discusses a numerical solution to the proposed problem using the Galerkin method, with low computational complexity and high convergence rates. In order to reduce the computational complexity, Laguerre polynomials are used as basis functions and to analyze the better convergence rates Kumar-Sloan (KS) approach has been applied. Further, the multi-Galerkin method with KS approach is used to improve the convergence results of Galerkin method. In addition, we have shown that the derivatives of the approximation solution converge to the corresponding derivatives of the exact solution with the same convergence rates as we attained for approximate solutions. Finally, numerical examples lend credibility to the presented theoretical framework. [ABSTRACT FROM AUTHOR]

In this paper, we propose first- and second-order exponential scalar auxiliary variable (ESAV) schemes for solving the phase field crystal equation with the periodic boundary condition. Specifically, the scalar auxiliary variable (SAV) in this work is constructed based on an exponential function, which differs from the square root form commonly used in the traditional SAV method. This feature allows the proposed schemes that only need to solve one linear system with constant coefficients at each time step. To construct the first-order ESAV scheme, we utilize the backward Euler method. Combining the ESAV method with the leapfrog approach, the second-order ESAV scheme is developed. Theoretically, the proposed schemes are proven to be uniquely solvable and unconditionally energy stable. Furthermore, the error estimates of the second-order ESAV scheme is rigorously proved. The numerical results are provided to demonstrate the effectiveness of the proposed schemes. • The scalar auxiliary variable in this study is constructed based on an exponential function. • The proposed schemes only need to solve one linear system with constant coefficients at each time step. • We use the leapfrog approach to develop the second-order scheme. • The error estimates of the second-order scheme is rigorously proved. [ABSTRACT FROM AUTHOR]

• We study zero-determinant strategies in arbitrary 2 × 2 asymmetric games. • We identify the range of multiplicative factors for different types of games. • We apply our results to asymmetric PD game, the ultimatum game, and the trust game. Recently, a new class of memory-one strategies, zero-determinant (ZD), has been proposed in the context of repeated Prisoner's Dilemma game. A player using a ZD strategy can unilaterally enforce a linear relation between the two players' payoffs. In this paper, we study ZD strategies in arbitrary 2 × 2 asymmetric games. We show that the existence of three typical subclasses of ZD strategies, equalizer, extortionate, and generous strategies, depends crucially on the payoff structure of the games. More interestingly, we find that extortionate and generous strategies can be expressed in a unified form of ZD strategies, so-called "multiplicative" ZD strategies, where this type of strategy can establish a multiplicative relation between the net payoffs of two players. Then, we identify the range of multiplicative factors when two players have different payoff functions or identities and discuss how this type of strategy implies extortion or generosity in different asymmetric games. Finally, we apply the above results to some specific asymmetric games, including the asymmetric PD game, the ultimatum game, the trust game as well as the generalized asymmetric social dilemma games, and analyze the effects of ZD strategies on cooperation and payoff control in these games. [ABSTRACT FROM AUTHOR]

This study explores the Sugeno exponential function, which is the solution to a first order differential equation with respect to nonadditive measures, specifically distorted Lebesgue measures. We define k -distorted semigroup property of the Sugeno exponential function, introduce a new addition operation on a set of distortion functions, and discuss some related results. Furthermore, m -Bernoulli inequality, a more general inequality than the well-known Bernoulli inequality on the real line, is established for the Sugeno exponential function. Additionally, the above concept is extended to a system of differential equations with respect to the distorted Lebesgue measure which gives rise to the study of a matrix m -exponential function. Finally, we present an appropriate m -distorted logarithm function and describe its behavior when applied to various functions, such as the sum, product, quotient, etc., while maintaining basic algebraic structures. The results are illustrated throughout the paper with a variety of examples. • The study analyzes the Sugeno exponential function in a distorted Lebesgue measure's differential equation. • It introduces k -distorted semigroup, broadens k -Bernoulli inequality, explores matrix m -exponential in differential equations. • It introduces k -distorted logarithm analyses in sums/products preserves algebraic structures. • Numerous examples illustrate the findings. [ABSTRACT FROM AUTHOR]

This paper studies the prescribed performance bipartite containment control design issue for nonlinear multiagent systems (MASs) under denial-of-service (DoS) attacks. First, an attack compensator is built for each agent in order to obtain the system output when the DoS attacks appear. Then, a fuzzy estimator is developed to estimate the unknown states based on the attack compensator. Second, a prescribed performance control (PPC) approach is utilized to ensure the bipartite containment error within a preassigned boundary, and the restriction related to the initial condition is removed. Finally, a set of switching laws is designed by combining the average dwell time method with the limited conditions of DoS attacks. Under this framework, an appealing feature of the designed controller is that the output signals of followers converge to a convex hull formed by leaders and all the signals in the closed-loop system remain bounded. A simulation example is given to demonstrate the effectiveness of the control strategy. • A prescribed performance bipartite containment consensus scheme is first proposed for uncertain nonlinear MASs against DoS attacks. • To deal with the challenge that the state signals become unavailable during DoS attacks intervals, a novel fuzzy state estimator in form of an attack compensator is designed to accurately estimate the unavailable state signals. • A novel prescribed performance security control algorithm is developed to remove the condition of the initial errors by introducing a class of performance functions. [ABSTRACT FROM AUTHOR]

For a graph G , the Wiener index W (G) is the sum of distances between all pairs of vertices, which is one of the most popular graph invariants. In this paper, we characterize a class of the unicyclic graphs with given order n and maximum degree Δ which minimize Wiener index, and then identify a class of the unicyclic graphs with g = 3 and Δ ⩾ 4 which minimize Wiener index. [ABSTRACT FROM AUTHOR]

Koopman operator with numerical approximation method for modelling nonlinear systems has become a popular data-driven approach in the past five years. However, when the system contains time-varying parameters, the data-driven Koopman operator-based model produces deviations between the nominal model and the true one. It affects the control performance when it serves as the nominal model in Model Predictive Control (MPC). To solve this issue, the Koopman operator-based model learned by using a multi-step prediction autoencoder and the strategy of online updating model are proposed. In the first step, a neural network with encoder-decoder structure is trained to search for the optimal lifted functions of the corresponding nonlinear system with fixed parameters. In the second step, an online updating strategy is presented to update the Koopman operator-based model using the collected data, when the nominal model is applied in the closed loop operation of MPC. In order to reduce the additional time consumption on updating the nominal model, the matrix inversion lemma is applied and it leads the strategy of updating to be presented in a recursive form. In general, this paper proposed an iterative MPC algorithm for online updating the Koopman operator-based model in a recursive form based on the optimal lifted function using autoencoder. Numerical simulations on three nonlinear cases with time-varying parameters show good performances on tracking control using the proposed Koopman operator-based model using autoencoder and online updating strategy. • A method based on autoencoder to learn the lifted functions was presented to obtain the approximation of Koopman operator. • The learned nominal model shows good prediction on evolution behaviors of nonlinear systems. • Adaptive model predictive control (MPC) was proposed to address the model mismatch on parameter-varying nonlinear systems. • The proposed Koopman operator-based Adaptive MPC can achieve high computational efficiency and good control performances. [ABSTRACT FROM AUTHOR]

A graph G = (V , E) is arbitrarily partitionable if for any sequence (n 1 , ... , n k) that satisfies n 1 + ... + n k = | G | it is possible to divide V into disjoint subsets V = V 1 ∪ ... ∪ V k such that | V i | = n i , i = 1 , ... , k and the subgraphs induced by all V i are connected. In this paper we inspect an on-line version of this concept and show that for graphs of order n , 8 ≤ n ≤ 14 , and size greater than ( n − 3 2 ) + 6 these two concepts are equivalent. Although our result concerns only finitely many graphs, together with a recent theorem of Kalinowski [5] it implies that arbitrarily partitionable graphs of any order n and size greater than ( n − 3 2 ) + 6 are also on-line arbitrarily partitionable. For the proof of our main result, we show some lemmas providing sufficient conditions for a graph to be traceable or Hamiltonian-connected, and they are of interest on their own. [ABSTRACT FROM AUTHOR]

In this paper, we present a fast algorithm for solving the space-fractional Gross-Pitaevskii equation while preserving the law of mass conservation. First we discretize this equation by using a second-order weighted and shifted Grünward difference operator and obtain a system of semilinear differential equations with linear and nonlinear parts. Afterwards, we employ a Strang splitting method to solve this semi-discretization scheme. To further reduce computational time, we propose a two-level Strang splitting method from the linear part. This method significantly reduces computational complexity to O (n log ⁡ n) by implementing the fast Fourier transform. Importantly, our proposed method ensures the unconditional preservation of mass conservation and achieves second-order convergence. At last, we demonstrate the validity of our approach through numerical experiments and graphical results presented. • A fast second-order method for high dimensional space-fractional Gross-Pitaevskii equations. • Split the Toeplitz matrix into the circulant and skew-circulant matrices. • Discrete mass conservation preserved unconditionally and global error estimated. [ABSTRACT FROM AUTHOR]

In this paper we consider various vertex versions of interval edge colorings. We distinguish two types of interval vertex colorings − an open and a closed interval vertex coloring, which are defined in such a way that colors used on open or closed neighborhood of each vertex form an integer interval, respectively. These colorings need not to be necessarily proper, and as coloring of all vertices with only one color is both open and closed, a natural goal is to maximize the total number of colors used for these colorings. We discuss some properties of the corresponding chromatic numbers and conditions for the existence of a proper (open and closed) interval vertex colorings. In the case of open interval coloring we show that only bipartite graphs admit a proper open interval vertex coloring. We also describe a technique for obtaining a closed interval vertex coloring of a tree with the maximum possible number of colors, where the resulting coloring is in fact proper. Finally, we propose some open problems concerning interval vertex colorings in general. [ABSTRACT FROM AUTHOR]

This paper addresses finite-region asynchronous H ∞ filtering for a class of two-dimensional Markov jump systems (2-D MJSs). A mathematical model is established using the Roesser model, and asynchrony is accounted for using a hidden Markov model (HMM). The modes jumping between the target system and the designed filter are determined by the given conditional probability matrix. Sufficient conditions are derived using suitable Lyapunov function and linear matrix inequalities (LMIs) to ensure stable filtering performance. The practical applicability of the approach is illustrated by two examples. Overall, this study offers a method to tackle filtering challenges in 2-D Markov jump systems, incorporating HMM, Lyapunov functions, and LMIs to effectively solve the finite-region asynchronous H ∞ filtering problem. • An HMM is used to describe the asynchronous phenomenon between the system and the filter. • Lyapunov functional and recursive formulas effectively analyze the FRB of filtering error system. • The Darboux equation and the heat exchange process both validate the effectiveness of the filter. [ABSTRACT FROM AUTHOR]

This research paper investigates the convergence and stability of two diagonal drift-implicit second-order stochastic Runge-Kutta methods for weak approximation of systems containing three-dimensional drift and noise terms in Itô stochastic differential equations. The first method is based on the approach presented by Debrabant and Rößler (2008) [5] , while the second method utilizes a Butcher table that, to the best of our knowledge, has not been used in previous research. We compare the convergence and stability of both methods and analyze their respective stability regions. The results show that the method using the newly introduced Butcher table is not only reliable but also highly efficient. • We propose a numerical method for SDEs that utilizes a new Butcher table. • We demonstrate the advantages of the new method over a more standard numerical approach. • We obtain stability regions for different values of the SDE parameters. • The new scheme achieves a substantial improvement in accuracy across various step sizes. [ABSTRACT FROM AUTHOR]

In this paper, an innovative numerical methodology is introduced for solving multiple types of linear one-dimensional Fredholm integro-differential equations using least squares support vector machine (LS-SVM) algorithm, even in singular cases. The proposed method combines the original LS-SVM model with the composite Simpson method, which transforms the original solution problem into an optimization problem with equality constraints. Then the original problem is reduced to seeking the LS-SVM model parameters, which further reduces the calculation cost. An explicit expression for the numerical solution of the Fredholm integro-differential equations is given. Finally, the constructed LS-SVM model is evaluated through numerical experiments, and the obtained solutions demonstrate the robustness and validity of the constructed model. • Introduction to traditional numerical methods for solving integro-differential equations. • The concept of least squares support vector machines and its application are discussed. • Combining LS-SVM method with Simpson's orthogonal formula to transform the problem into linear equations. • Numerical experiments verify the effectiveness and superiority of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of asymmetric beta-binomial generalized autoregressive conditional heteroscedastic (GARCH) models for bounded integer-valued time series, which can capture the asymmetric impact of positive and negative observations. We study the stationarity conditions of the process and derive the moment and covariance functions. Furthermore, we estimate the unknown parameters using the conditional maximum likelihood (CML) method. The asymptotic properties of the estimators are discussed, as well as their finite-sample performance. Finally, we illustrate the model to real time series data in the field of meteorology. • A new asymmetric beta-binomial GARCH model for bounded Z-valued time series. • Asymmetric impact of positive and negative observations. • CML estimation with asymptotic properties. • A real time series data in the field of meteorology. [ABSTRACT FROM AUTHOR]

The solutions of Hamilton-Jacobi (HJ) equations may contain discontinuous derivatives, which brings numerical difficulties in capturing sharply these derivatives. This paper presents a third-order weighted compact nonlinear scheme (WCNS) and a third-order perturbed WCNS (WCNS-P) for solving HJ equations. A linear combination of hybrid cell-edge and cell-node values of the function is applied to approximate the spatial derivatives of the function. For the WCNS, the upwind-biased nonlinear weighted interpolation is used for the unknown cell-edge values, while for the WCNS-P perturbations with a free parameter is introduced to the upwind-biased linear interpolation. The free parameter can be optimized to reduce numerical errors in smooth regions. To enhance the numerical stability, a monotone polynomial interpolation method is designed to switch the WCNS-P to the WCNS. Several numerical experiments are performed to test the order of accuracy and the discontinuity-capturing ability of the two schemes. • Two types of weighted compact nonlinear schemes are constructed to solve Hamilton-Jacobi equations. • A switching condition of two schemes is designed to avoid numerical oscillations near discontinuities. • The two schemes can achieve third-order accuracy and have good resolution near corner singularities. • The two schemes have good performance in the noise reduction. [ABSTRACT FROM AUTHOR]