Search

• 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]

• The study aimed to quantify the cyclic dominance resembling rock-paper-scissors observed in the optional prisoner's dilemma, considering cooperators, defectors, and loners as the three competing strategies. • The research utilized theoretical and numerical analyses to explore the impact of environmental feedback on the replicator dynamics. • The rate of environmental replenishment and the payoff from draw games were identified as crucial factors influencing gameplay dynamics. • The study revealed that the loner's payoff and the strength of the dilemma played pivotal roles in determining the dominant strategies. This research investigation delves into the concept of cyclic dominance and its interplay with environmental feedback within the framework of the Optional Prisoner's Dilemma (OPD) game. A quantitative model has been meticulously constructed to encompass both the OPD and Rock-Paper-Scissors (RPS) games, wherein the outcomes are contingent upon environmental feedback. The intricate relationship between player populations and their surrounding environment is elucidated through the utilization of replicator dynamics. The empirical findings of this study shed light on the existence of distinct equilibria contingent upon the prevailing environmental circumstances. Under scenarios characterized by resource depletion, an equilibrium state emerges wherein only solitary individuals (referred to as "loners") persist. Conversely, under conditions of resource replenishment, an equilibrium is established where various strategic approaches coexist in equal proportions, akin to the dynamics of the Rock-Paper-Scissors game. Notably, cyclic dominance manifests itself during periods characterized by intermediate environmental conditions. Several pivotal factors have been identified as determinants of these dynamic phenomena, including the rate of environmental replenishment, the strength of the dilemma presented, the payoff for loners, and the reward for a draw. It has been empirically demonstrated that the presence of extreme values in the replenishment rate, irrespective of whether exceptionally high or exceptionally low, serves as an impediment to the emergence of cyclic dominance. This model underscores the substantial influence of environmental feedback on social dilemmas, particularly in the context of the OPD game, ultimately leading to the emergence of competitive cyclic dominance games reminiscent of the Rock-Paper-Scissors game. In doing so, it elucidates the intricate interplay between evolutionary game incentives and the dynamics of the surrounding environment. [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]

Abstract: Using advanced machine learning techniques as an alternative to conventional double-entry volume equations known as classical allometric models, regression models of the inside-bark volume (dependent variable) for standing Eucalyptus globulus trunks (or main stems) have been built as a function of the following three independent variables: age, height and outside-bark diameter at breast height (D). The allometric models of volume, biomass or carbon support the estimation of carbon storage in forests and agroforestry systems. On one hand, this paper presents the construction of allometric models of the inside-bark volume for E. globulus trees. On the other hand, the experimental observed data (age, height, D and inside-bark volume) for 142 trees (E. globulus) were measured and a nonlinear model was built using a data-mining methodology based on multivariate adaptive regression splines (MARS) technique and multilayer perceptron networks (MLP) for regression problems. Coefficients of determination and Furnival’s indices indicate the superiority of the MARS technique over the allometric regression models and the MLP network. The agreement of the MARS model with observed data confirmed the good performance of the same one. Finally, conclusions of this innovative research are exposed. [Copyright &y& Elsevier]

Abstract: We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature. [Copyright &y& Elsevier]

• 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]

Abstract: In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. İzgi in the paper “Voronovskaya type asymptotic approximation by modified Gamma operators” [Applied Mathematics and Computation 217 (2011) 8061–8067]. We show that this theorem is not true. We present a correct form of this theorem and its proof. Furthermore, we obtain an error estimate for modified Gamma operators. [Copyright &y& Elsevier]

Abstract: In this paper, we focus on the proposed algorithms to solve a linear programming problem with the convex combination of the max–min and the max–average composition and the max–star composition, respectively. They have been proposed by Ghodousian and Khorram [A. Ghodousian, E. Khorram, Solving a linear programming problem with the convex combination of the max–min and the max–average fuzzy relation equations, Appl. Math. Comput. 180 (2006) 411–418] and Khorram et al. [E. Khorram, A. Ghodousian, A. Abbasi Molai, Solving linear optimization problems with max–star composition equation constraints, Appl. Math. Comput. 179 (2006) 654–661], respectively. Firstly, we show that the “Tabular method algorithm” in the first paper and the “First procedure” in the second paper may not lead to the optimal solutions of the two models in some cases. Secondly, we generalize the proposed algorithm by Abbasi Molai and Khorram [A. Abbasi Molai, E. Khorram, A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian, Appl. Math. Comput. 190 (2007) 1161–1167] to solve the two models. In fact, it modifies the presented algorithms in the two papers. Finally, some numerical examples are given to illustrate the purposes. [Copyright &y& Elsevier]

Abstract: Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form where f(x) and g(x) are continuous on [a, b], and A i and B i (i =1,2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments. [Copyright &y& Elsevier]

Abstract: One of the desirable attributes of real-life examination timetabling solutions is the maximization of paper spread, which is a measure of the amount of study time that each student has between examinations. In this study, we face with a predefined examination schedule that must be modified in order to maximize paper spread in course of examination. We will show how the integer programming can be employed to achieve this aim. The model presents constraints for the quality of feasible examination timetables and all the requirements found in most academic institutions. The approach is tested on real-world exam timetabling problems. The computational experiments and results will be reported. [Copyright &y& Elsevier]

Abstract: In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the formThe results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper. [Copyright &y& Elsevier]

• 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]

Abstract: In this paper, we establish new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which improve the recent results of Tang and He [X.H. Tang, X. He, Lower bounds for generalized eigenvalues of the quasilinear systems, J. Math. Anal. Appl. 385 (2012) 72–85] when 1< p i <2 for i =1,2,…, n. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations where and . On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations where . The results obtained here complement and correct some known results in . Some examples are given to illustrate the main results. [Copyright &y& Elsevier]

Abstract: In this short note an addition to the paper [A.G. Bratsos, M. Ehrhardt, I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Appl. Math. Comput. 197(1) (2008) 190–205] using the modulus of the terms evaluated from the Adomian decomposition method on p. 194 and their relation to the convergence of the resulting series is presented. Conclusions for the accuracy of the approximated solution are derived. [Copyright &y& Elsevier]

Abstract: This paper considers certain neutral differential equation with the distributed delayWithout imposing a nonnegative restriction on coefficients of the equation, we establish a delay-dependent asymptotic stability condition for the equation by introducing a new Lyapunov function. The condition presented here is different from that of Park [J.H. Park, LMI optimization approach to asymptotic stability of certain neutral delay differential equation with time-varying coefficients, Appl. Math. Comput. 160 (2005) 355–361] in the sense that it is dependent on all delays of the equation. Numerical examples illustrate effectiveness and sharpness of our theoretical result. [Copyright &y& Elsevier]

Abstract: The aim of this note is to show that the optimization algorithm proposed in may not lead to the optimal solution in some cases. In fact, the optimization problem remains open if we do not apply the branch-and-bound method to solve it or compute all the objective values of minimal solutions of the feasible domain. [Copyright &y& Elsevier]

Abstract: A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out. [Copyright &y& Elsevier]

Abstract: In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function. [Copyright &y& Elsevier]

• 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]

Abstract: In this paper we expose new method to analyse time series. The method is essentially applied for the prediction of time series among approximation and modelisation. It is based on wavelet decomposition combined with autoregressive models. An iterative procedure is applied and the performance of the estimator is measured by standardized error. Finally, a comparison with some existing models and methods has been pointed out to prove the performance of our’s. [Copyright &y& Elsevier]

The sole purpose of this short note is to correct false claims made in the paper "A new approach for finding the determinant of matrices" by Omid Rezaifar and Hossein Rezaee (Appl. Math. Comput. 188 (2007) 1445–1454). [ABSTRACT FROM AUTHOR]

The present comment concerns some doubtful mathematical calculation that produced misleading results which are presented in the above paper. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the problem of local asymptotic stability for a competitive Lotka–Volterra system with time-varying delays. By employing a linear matrix inequality (LMI) approach, we not only prove that the local asymptotic stability of the positive equilibrium for the Lotka–Volterra type competitive system will be preserved for suitable delays under a well known condition, but also obtain the maximal allowable length of delays by using Matlab’s Control Systems Toolbox to solve a feasible LMI. Compared with some known results, our estimate on the length of delays is less conservative. [Copyright &y& Elsevier]

Abstract: A nonlinear periodic food-chain predator–prey model is proposed in this paper, which can be seen as the modification of the traditional Lotka–Volterra food-chain system. Sufficient conditions which guarantee the existence of the positive periodic solution of the system are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. [Copyright &y& Elsevier]

Abstract: A multisymplectic integrator for the periodic nonlinear Schrödinger equation is presented in this paper. Its accuracy is proved. By introducing a norm, we investigate its nonlinear stability. We also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method. [Copyright &y& Elsevier]

Abstract: In this note, we will modify several gaps in Gurudwan and Sharma [N. Gurudwan, B.K. Sharma, Convergence theorem for the common solution for a finite family of -strongly accretive operator equations, Appl. Math. Comput. 217 (2011) 6748–6754]. [Copyright &y& Elsevier]

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]

A tree T with exactly one node u of degree larger than two is called a starlike tree and u is called the root node of T. Let S T n be a family of graphs consists of all starlike trees and the path with n + 1 nodes. For a fixed connected graph H with at least one edge, we construct a family of graphs F H (v) , n = { H (v) • T | T ∈ S T n } , where H (v) • T denotes the graph obtained by identifying some node, say v , of H with the root node of T. In this paper, we give an ordering of graphs in F H (v) , n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths. • The spectral radius of the graph powerfully characterizes dynamic processes on networks, such as virus spread and synchronization. • In this paper, we give an ordering of graphs in F H (v) , n by spectral radii coincides with the shortlex ordering of nondecreasing sequences of their branch lengths. • The result generalizes the main result of Oliveira et al. in [10]. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating the stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established, which shows the second-order accuracy. Finally, the numerical experiments are presented confirming the accuracy of the proposed scheme. • In this paper, a new approach to solve numerically neutral Volterra integro-differential equation (NVIDE) has been considered. • The last works related to numerical analysis of NVIDE were concerned with collocation method, Legendre spectral method and etc. But the most important feature of our study is a finite difference scheme has been constructed on a uniform mesh. • Using composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points, we construct the finite difference scheme which have a second-order accuracy. • Therefore we obtain numerical methods which are as high accuracy as possible. [ABSTRACT FROM AUTHOR]

In the existing papers considering the stability analysis of discrete-time systems with distributed delays via the Lyapunov-Krasovskii functional (LKF) method, the delay kernels are normally restricted to be non-negative. In this paper, we aim to remove such a restriction and deal with the stability analysis of systems with arbitrary delay kernels. For this purpose, a kernel-related summation inequality is first constructed. Then, a stability condition is derived based on the proposed inequality and a model transformation. Finally, two numerical examples are presented to show that the proposed stability condition not only has a wider scope of application and is less conservative than the existing ones. [ABSTRACT FROM AUTHOR]

This paper studies the stability of the hybrid neutral stochastic functional differential equations (NSFDEs) with different structures under highly nonlinear conditions. NSFDEs are of wide suitability for simulating random processes with memory, such as tumor evolution mechanism in life science field. The new stochastic system studied in this paper has completely different system structures in different switching subspaces, and the coefficients are highly nonlinear. Moreover, the neutral term is subject to the Markovian switching. This work fills the gap of the stability analysis for differently structured highly nonlinear neutral stochastic functional systems. By using the Lyapunov functional method and the generalized Khasminskii-type conditions, we first establish the existence and uniqueness theorem as well as the asymptotical bounded property of the unique global solution. Then the important stable properties are obtained including H ∞ stability, asymptotic stability and exponential stability in L p. Finally a numerical example is given to illustrate the effectiveness of the results. • This work is on the study of differently structured stochastic functional system. • The neutral term considered in this paper is subject to the Markovian switching. • The generalized Khasminskii's condition is used for the highly nonlinear system. • The asymptotically and exponentially stable results of the system are obtained. [ABSTRACT FROM AUTHOR]

This paper investigates a class of output-mask-based adaptive neural network (NN) tracking control for nonlinear stochastic time-delayed multi-agent systems (STMASs) based on a unified event-triggered approach. The output signal relies on an output mapping acted as a mask, defined as a privacy-protection-like method, so that the internal state of one agent cannot be identified by other distrustful eavesdroppers or attackers. Moreover, the construction of a unified event-triggered control scheme retains the advantages of the saturation threshold triggering strategy, incorporates distributed errors, and increases the flexibility of thresholds. Furthermore, for stochastic time-delay multi-agent systems, the initial value limitation of the conventional first-order filter is removed by a first-order Levant differentiator, and a new estimation term in the fuzzy observer is established to solve the nonlinear fault. The unknown function in pure-feedback form is addressed via combining Butterworth low-pass filter and radial basis function neural networks (RBF NNs). Finally, the boundedness of all signals in the closed-loop systems is demonstrated, and the effectiveness of the proposed algorithm is verified by some simulation results. • This paper proposes a privacy-protection-like scheme for continuous systems to make those signals transmitted in ciphertext. • A unified approach for event-triggered control, namely dynamic saturation threshold event trigger, was established. • The proposed method relaxes two conditions and reduces the conservatism. [ABSTRACT FROM AUTHOR]

A graph G is two-disjoint-cycle-cover [ r 1 , r 2 ] -pancyclic if for any integer ℓ satisfying r 1 ≤ ℓ ≤ r 2 , there exist two vertex-disjoint cycles C 1 and C 2 in G such that | V (C 1) | = ℓ and | V (C 2) | = | V (G) | − ℓ. The k -dimensional data center network DCell with n port switches and order t k , n , denoted by D k , n , plays an important role in big data cloud computing. In this paper, we study the structural properties of D k , n and show that D k , n is two-disjoint-cycle-cover [ 3 , ⌊ t k , n 2 ⌋ ] -pancyclic for n ≥ 7 and n + k ≥ 11 , that is, for any integer ℓ satisfying 3 ≤ ℓ ≤ ⌊ t k , n 2 ⌋ , there exist two vertex-disjoint cycles C 1 and C 2 such that | V (C 1) | = ℓ and | V (C 2) | = t k , n − ℓ in D k , n. This paper generalizes the results about the Hamiltonicity in (Wang et al., 2015 [25]) and the pancyclicity in (Hao and Tian, 2021 [11]). • It is the first study to consider two-disjoint-cycle-cover pancyclicity in data center networks. • The two-disjoint-cycle-cover pancyclicity of data center network DCell is shown. • This result extends and generalizes the Hamiltonicity and bipancyclicity of DCell. [ABSTRACT FROM AUTHOR]

The time-varying formation problem under directed topology is investigated for the uncertain nonlinear multi-agent systems (MASs) via the neural networks-based adaptive feedback control (NNAFC). Compared with the simple adaptive control, the designed control protocol consists of two parts: NNAFC and the event-triggered impulsive estimator. To save communication costs, communications among followers are only carried out at discrete instants determined by the event-triggered impulsive rule. Two kinds of estimators and controllers are designed for two cases with or without communication delay, respectively. Further, NNAFC can be further improved to eliminate errors due to uncertain function approximation for some special cases. Utilizing the Lyapunov stability theory and impulsive control theory, we obtain some sufficient conditions to implement time-varying formation task. Finally, three examples are given to show the feasibility of theoretical analysis. • Two kinds of NNAFC in this paper can compensate for negative factors caused by uncertain terms so that the formation error is smaller. • This paper can apply to time-varying formations of homogeneous or heterogeneous MASs. • The event-triggered (delay) impulsive estimator only communicates at the impulsive instant determined by a rule, which saves bandwidth and reduces the communication cost. • Communication topology has fewer constraints and only requires a directed spanning tree. [ABSTRACT FROM AUTHOR]

A graph G is (d 1 , d 2) -colorable if its vertices can be partitioned into two subsets V 1 and V 2 such that Δ (G [ V 1 ]) ≤ d 1 and Δ (G [ V 2 ]) ≤ d 2. Let G 5 denote the family of planar graphs with girth at least 5. In this paper, we prove that every graph in G 5 without adjacent 5-cycles is (1 , 6) -colorable. • This is a new result towards planar graphs with girth 5 and containing no adjacent 5-cycles. • The skill of discharging method in this paper is a powerful tool to solve the problem of this kind of coloring. • The discharging rulers are well designed. • Establish some reducible configurations by using clever skills. [ABSTRACT FROM AUTHOR]

The exact controllability research framework of complex networks points out that the controllability of the network has a great relationship with the minimum number of driver nodes. It is generally believed that the smaller the minimum number of driver nodes and the lower the cost of external control of the whole network to achieve the ideal state, the better the controllability of networks. In this paper, the minimum driver node problem of a hypernetwork is transformed into the maximum multiplicative problem of the eigenvalue of its 2-section graph. The minimum driver node numbers of two types of typical k -uniform hypertree networks are described, and their bounds are also given. By designing an algorithm, the method of characterizing the driver node set is obtained, and it is found that the selection of the minimum driver nodes of the network tends to the nodes with low hyperdegree. In addition, the paper verifies the minimum driver node set and the theoretical analysis results of controllability by simulation analysis. • Transforming the problem of a hypernetwork to the problem of its 2-section graph. • Giving the minimum driver node number of k-uniform hypertree networks and its bounds. • The driver nodes of the k-uniform linear hypertree networks tend to be the nodes with low hyperdegree. [ABSTRACT FROM AUTHOR]