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• 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]

• 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]

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]

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]

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]

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]

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]

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t = 0 , we use the non-uniform meshes to capture the rapid change of the solution at t = 0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L 2 -norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O (τ 2 + h x 2 + h y 2 + h z 2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method. • In this paper, we proposed a high-order difference scheme for the nonlocal evolution equation with R-L fractional integral terms based on non-uniform meshes. However, most works mainly study the Caputo fractional derivative terms on non-uniform meshes. As we all known, there are little works in our area. • We extend our scheme to the high dimensional problems and provide the three dimensional example. • In this paper, the influence of different multi-memory kernels $﹨alpha, ﹨beta, ﹨gamma$ on the convergence accuracy of three-dimensional equation is studied. Most studies have focused on a multi-memory kernel $﹨mathcal{L}̂{(﹨alpha)}﹨Delta﹨varpi$, as shown in Equation (1.4). [ABSTRACT FROM AUTHOR]

An edge-colored graph G is called rainbow if all the colors on its edges are distinct. Given graphs G and H , the anti-Ramsey number A R (G , H) is the maximum number k such that there exists an edge-coloring of G with exactly k colors containing no rainbow copy of H. Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this line. We refine the bound of n and determine the precise value of A R (K n , P 4 ∪ t P 2) for all n ≥ 2 t + 4 in complete graphs. • An edge-colored graph G is called rainbow if all the colors on its edges are distinct. Given graphs G and H , the anti-Ramsey number A R (G , H) is the maximum number k such that there exists an edge-coloring of G with exactly k colors containing no rainbow copy of H. • Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. As one of popular graph classes, the anti-Ramsey problem of matchings, which clearly consists of small components, has been well studied. • So far, several results about disconnected graphs, mainly forests, have been obtained in K n , but only for large enough n. For small n , there are almost few results. In this paper, we continue the work in this line. We refine the bound of n and determine the precise value of A R (K n , P 4 ∪ t P 2) for all n ≥ 2 t + 4 in complete graphs K n. [ABSTRACT FROM AUTHOR]

A k -edge coloring ϕ of graph G is injective if any two edges at distance 2 or in the same triangle get different colors. The minimum k in such an edge coloring is the injective edge chromatic number of G , written as χ i ′ (G). We prove in this paper that for any graph G with Δ (G) ≤ 5 , χ i ′ (G) ≤ 18 if m a d (G) < 16 5 , χ i ′ (G) ≤ 19 if m a d (G) < 7 2 and χ i ′ (G) ≤ 20 if m a d (G) < 15 4. • The results obtained in this paper enrich the previous work concerning on the study of injective coloring of sparse graphs. • The discharging method used in this paper is especially useful to solve the problem of injective coloring. • Some important reducible configurations are presented and the discharging rulers are well designed. • It is the first time to study the sufficient conditions for graphs with maximum degree Δ = 5 satisfying χ i ′ (G) ≤ Δ (Δ − 1). [ABSTRACT FROM AUTHOR]

In this paper, we study a class of discrete-time cyber-physical systems (CPSs) with transmission delays and sparse malicious attacks on both input and output transmission channels, and present design methods for secure observer and controller. First, the states and sparse malicious attacks of the original system are reorganized into the states of the new system obtained by an improved augmentation method (rather than a singular augmentation method). Then, based on the new system, the states and attacks can be reconstructed simultaneously by the designed Luenberger-like observer. Furthermore, a novel dynamic output feedback (DOF) controller is proposed using the estimated values of the attacks, and a cone complementarity linearization (CCL) algorithm is designed to obtain the controller gains. The results show that the closed-loop system is stable in the sense of Lyapunov that both the estimation errors and the original system states will converge to a small neighborhood of the origin. In addition, the correlation results without delay are given. Finally, two examples are given to illustrate the effectiveness and superiority of the proposed method. • In addition to the sparse attacks that occur in both input and output channels, this paper also considers the transmission delays in both channels. • This paper introduces a multi-delay dynamic output feedback controller that transforms the inherently unstable original system into one that ultimately achieves Lyapunov stability. • In the case without transmission delay, compared with the results in previous researches, the proposed method of designing the observer and controller separately can reduce the difficulty of solving observer gains and controller gains, and enhance the feasibility. [ABSTRACT FROM AUTHOR]

Let Γ = (V , E) be a graph of order p. Recently, the Sombor index is introduced, defined as S O (Γ) = ∑ v i v j ∈ E (Γ) d Γ (v i) 2 + d Γ (v j) 2 , where d Γ (v i) is the degree of the vertex v i in Γ. Cruz and Rada [4] obtained an upper bound on the Sombor index of unicyclic and bicyclic graphs of order p , but did not characterize the extremal graphs. In the same paper, they mentioned that the maximal graphs over the set of unicyclic and bicyclic graphs with respect to Sombor index, is an interesting problem that remains open. In this paper we completely solve these problems. • Sombor index of a graph Γ is defined as S O (Γ) = ∑ v i v j ∈ E (Γ) d Γ (v i) 2 + d Γ (v j) 2 . • To determine maximal graph over the set of unicyclic graphs with respect to Sombor index. • To characterize the maximal bicyclic graph with respect to Sombor index. [ABSTRACT FROM AUTHOR]

The stability issue and stabilization issue for a class of singular neutral systems are studied in this paper. Firstly, we give some sufficient conditions such that the considered open-looped systems are admissible. Secondly, we give some results which design the feedback controllers and ensure that the resulting close-looped systems are admissible. Finally, the superiority and effectiveness of the obtained methods are illustrated by two numerical examples. The results obtained in this paper can be employed to analyze the singular neutral systems which do not satisfy the restrictive conditions presented in some published papers. • The singular neutral system that does not satisfy the restrictions presented in some papers is considered. • The considered system contains the neutral delay and the state delay. • This paper presents some sufficient conditions which guarantee the considered system to be admissible. • Two different methods are used to design the feedback controllers. • Numerical examples are presented to illustrate the effectiveness and superiority of the obtained results. [ABSTRACT FROM AUTHOR]

This paper addresses the stabilization problem of continuous-time Markovian jump systems (MJSs) by applying an optimization controller. A new stabilizing method based on a mode separation algorithm is proposed, whose separation will be optimized by minimizing the established cost function value. It will be seen that the presented optimal controller will have hybrid decision variables such as continuous and discrete variables. Meanwhile, the related cost function is in contrast to the traditionally linear quadratic functions. The conditions for the existence of the developed controller are established via using rigorous theory analysis, and its detailed computation method is presented too. It can be concluded that traditionally mode-dependent and -independent controllers about MJSs are contained as two special situations. A practical example is offered so as to verify the effectiveness and superiority of the methods proposed in this study. • A new stabilizing controller is proposed to deeply depend on the separation of modes or subsystems. • A novel control cost function is proposed to optimize the mode separation. • It has been proved that the stabilizing controller based on the optimized mode separation can be realized for MJSs, if a precondition is satisfied. • Compared with the similar but very few result [19] , it has showed that the method in this paper is better in having smaller cost function values. • The presented controller can include mode-dependent and -independent controllers as special situations, which is less conservative than mode-independent ones. [ABSTRACT FROM AUTHOR]

A k -limited packing partition (k LP partition) of a graph G is a partition of V (G) into k -limited packing sets. We consider the k LP partitions with minimum cardinality (with emphasis on k = 2). The minimum cardinality is called k LP partition number of G and denoted by χ × k (G). This problem is the dual problem of k -tuple domatic partitioning as well as a generalization of the well-studied 2-distance coloring problem in graphs. We give the exact value of χ × 2 for trees and bound it for general graphs. A section of this paper is devoted to the dual of this problem, where we give a solution to an open problem posed in 1998. We also revisit the total limited packing number in this paper and prove that the problem of computing this parameter is NP-hard even for some special families of graphs. We give some inequalities concerning this parameter and discuss the difference between 2TLP number and 2LP number with emphasis on trees. [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]

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]

• 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]

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]

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 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]