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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.

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

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.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (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.

This paper was an invited contribution for The 9th International Symposium on Numerical Analysis of Fluid Flow and Heat Transfer – Numerical Fluids 2014, Rhodes, Greece. © 2015 The Authors. Published by Elsevier Inc. 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. Rolls-Rolls plc; EPSRC grants: EP/L000261/1; EP/I017771/1; EP/I010440/1; EP/I017747/1; EP/H001395/1; EP/G027633/.

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 ? -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.

In the paper the second order half-linear delay differential equation ( r ( t ) | u ' ( t ) | α - 1 u ' ( t ) ) ' + p ( t ) | u ( ? ( t ) ) | α - 1 u ( ? ( 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 ? 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.

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.

In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. Izgi 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.

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.

The authors appreciate Professor Cheng-De Zheng pointing errors in definition of LMI variable [1–8] and would like to correct an error in definition of an LMI variable of the paper [1–8]. In [1–8], Theorems or Corollaries present stability or stabilization criteria for the system with uncertainties or nonlinear perturbations. In [1–8], the LMI variables H in [1–3,5– 8], Hi ði 1⁄4 1;2Þ in [4], and P2 in Theorem 1 of [6] are defined as a positive matrix. However, the aforementioned LMI variables in [1–6] should be positive scalars. Then, all the Theorems and Proofs in [1–8] are still valid without any further changes. Note that the numerical results for systems without uncertainties ([1–3,5,6]) or nonlinear perturbations ([4]) do not change even though the H in [1–3,5–8], Hiði 1⁄4 1;2Þ in [4], and P2 in Theorem 1 of [6] are corrected to a positive scalar. In case that the system has uncertainties or nonlinear perturbations, the numerical results become to change a little. In [1], Example 2 considers the case with uncertainties. The corrected versions for Table 2 and 3 in [1] are given in Tables 1 and 2, respectively. Example 3 in [2] considers the system with uncertainties. The corrected versions for Tables 2–4 in [2] are shown in Tables 3–5, respectively. In [3], Example 2 considers the case with uncertainties. The corrected versions for Tables 2 and 3 in [3] are given in Tables 6 and 7, respectively. In [4], Example 1 and 2 consider the case with nonlinear perturbations. The corrected maximum allowable bound under the same condition in Example 1 is corrected to 2.8687 from 3.4039. The corrected result still gives larger delay bound than one of previous work given in Ref. [6] of paper [4]. And the corrected versions for Table 1 in [4] are given in Table 8. In [5], Example 1, 3, 4, 5, and 6 consider the case with uncertainties. The corrected versions for Tables 1–4, 6 and 7 in [5] are given in Tables 9–14, respectively. In [6], Example 1 and 3 consider the case with uncertainties. The corrected versions for Tables 1, 2 and 4 in [6] are given in Tables 15–17, respectively. In [7], the corrected results are given in Table 18. In [8], the corrected maximum allowable bound under a = 0, hL = 0 and unknown hD changes from 0.7245 to 0.6918, while is the result still larger than the ones of the previous works mentioned in [8]. Also, the corrected results for Tables 1 and 2 [8] are given in Tables 19 and 20, respectively.

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.

In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations(r(t)@j(x(t))|Z^'(t)|^@a^-^1Z^'(t))^'+q(t)f(x(@s(t)))=0,t>=t"0>0, where Z(t)=x(t)+p(t)x(t-@t) and @a>0,0==t"0>0, where @!"t"""0^~dtr(t)

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 (1) d 4 y d x 4 + f ( x ) y = g ( x ) , a ⩽ x ⩽ b , (2) y ( a ) = A 1 , y ( b ) = A 2 , y ′ ( a ) = B 1 , y ′ ( b ) = B 2 , 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.

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 Schrodinger 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.

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.

In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u [ τ ( t ) ] | α - 1 u [ τ ( t ) ] = 0 . The 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.

The aim of this note is to show that the optimization algorithm proposed in [6] 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.

This paper considers certain neutral differential equation with the distributed delayddt[x(t)+c(t)x([email protected])]+p(t)x(t)+q(t)x([email protected])+r(t)@!"t"-"@d^tx(s)ds=0.Without 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.

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.

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.

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 ϕ -strongly accretive operator equations, Appl. Math. Comput. 217 (2011) 6748–6754].

The recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self-adjoint differential equations. The present brief article contains clarifying comments.

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

Ensemble clustering has attracted much attention for its robustness, stability, and accuracy in academic and industry communities. In order to yield base clusterings with high quality and diversity simultaneously in ensemble clustering, many efforts have been done by exploiting different clustering models and data information. However, these methods neglect correlation between different base clusterings during the process of base clusterings generation, which is important to obtain a quality and diverse clustering decision. To overcome this deficiency, a sequential ensemble clusterings generation algorithm for mixed data is developed in this paper based on information entropy. The first high quality base clustering is yield by maximizing the entropy-based criterion. Afterward, a sequential paradigm is utilized to incrementally find more base clusterings, in which the diversity between a new base clustering and the former base partitions is measured by the normalized mutual information. Extensive experiments conducted on various data sets have demonstrated the superiority of our proposal as compared to several existing base clusterings generation algorithms.

In this short paper, we introduce and analyze a generalized preconditioned parameterized inexact Uzawa method for solving singular saddle point problems, this method is a generalization of the PPIU method (see, Gao and Kong, 2010), then we give the semi-convergent conditions of this method, which are more practical for the applications.

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u,?v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in Y. Zhang, Y. Hu, 2016 38. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus-Gaddum-type results for Wp(G). Our lower bound on W p ( G ) + W p ( G ? ) is always better than the previous lower bound given by Zhang and Hu.

Recently, some results on the trigonometric approximation of Lip(ω(t), p)-class have been given by Srivastava and Singh (2014). In this note, we point out that one of the assumption conditions in their theorems is invalid in some cases. Thus, all conclusions in the paper based on the condition lose the foundation. Moreover, we obtain the similar results for a class of more extensive functions after deleting the mentioned condition, and weakening another condition in their theorems.

Cognitive processing by intelligent systems involves the deletion of information in favor of higher level abstractions. This process can be addressed through the physics of computation but a formal model that explains this process has not been proposed yet. In this short paper, we propose a model that through physical constraints only generates optimal solution to the collapse of n objects into n sets. A numerical simulation of the model results in a logarithmic function of information loss and condensation that perfectly fits our knowledge of cognitive processes.

In this short paper we study the counterpart, within the context of a general class of fluids, of two famous unsteady flows originally studied by Stokes, within the context of Navier-Stokes fluid, namely Stokes' first and second problems. The class of fluids considered, stress power-law fluids, are capable of stress thinning or stress thickening and can describe phenomena that the classical power-law fluids are incapable of modeling. Within the context of the problems considered, we are able to find solutions wherein stress boundary layers develop.

In this short paper, we present time scales version of mean value theorems for integrals in the single variable case.

Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. This method suffers from a number of problems. The major problem is choosing initial and final step length of dividers. In this short paper we would like to demonstrate that the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity.

Investigation of the asymptotic properties of solutions to systems of discrete equations is a topic of permanent interest. Numerous papers analyze the so-called prescribed behaviour of solutions giving sufficient conditions for the existence of at least one solution having a given asymptotic behaviour. To a smaller extent the problem is considered of determining the initial data generating such solutions. The present paper finds its niche being concerned with the case of a triangular system of linear discrete equations. The existence of solutions with a prescribed asymptotic behaviour is proved and algorithms suggested for computing stepwise the initial data defining such solutions and eventually suggesting these. Illustrative examples (supported with a MATLAB program) are considered and some open problems are formulated as well.

In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N, the Nth method of this family has the order of convergence 2 N + 1 . In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family.

This paper has attended to obtain the approximate analytical expressions of periodic solutions in the delayed non-smooth Internet TCP-RED congestion control system employing the semi-analytical method named as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique. For the first time, the methodology presented herein provides the accurate approximate analytical expressions of periodic solutions, including the non-impacting periodic solutions and impacting periodic solutions, in the non-smooth dynamical system with time-delays. They agree very well with the results of numerical simulations by MATLAB. It implies that the proposed method in this paper is accurate and effective. Furthermore, rich dynamics of this delayed non-smooth system is discovered in the present paper. Three kinds of bi-stability, i.e. coexistence of asymptotically stable equilibrium and equilibrium, equilibrium and periodic solution, periodic solution and periodic solution, have been found in the system with the variation of the delay. And the grazing solution, the impacting solution traversing both the lower threshold Tmin and the larger threshold Tmax, and a route to chaos, i.e., intermittent to chaos, have been also obtained by varying the delay employing numerical method.

In modern control theory, it is more applicable for digital computers using a sampled-data controller than a continuous-time controller. Meanwhile, input saturation and disturbance inevitably exist in control systems. Thus, how to propose an applicable sampled-data controller for nonlinear systems considering input saturation together with external disturbances is essential for applications. To deal with this control problem, a novel auxiliary control signal is designed in this paper. Then, by using Lyapunov method, the control design process is simplified compared with the traditional backstepping technique, and only one Lyapunov function is needed. In addition, the state and disturbance observers are proposed with the new auxiliary control signal to track the unavailable states and external disturbances. Finally, we propose a novel controller reflecting the sampled-data output-feedback for the nonlinear systems to ensure the system stability. The effectiveness of proposed control method in this paper is confirmed through two sets of numerical simulation results.

On the basis of the previous work by Tang and Zhang [37], in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series.

The solution of Bernoulli differential equation can be described as a generalized Logistic curve function. Weibull cumulative distribution function is also an analytic solution of one variable coefficient differential equation. But, however because of the exact analytical solution problem of the equation, up to now, the developing coefficient of all the grey forecasting models is still defined as a constant. So, the aim of this paper is just to construct a novel grey differential equation model by combining NGBM(1,1) and Weibull cumulative distribution function. The proposed model(WBGM(1,1)) has the advantages of NGBM(1,1) and Weibull cumulative distribution. Where there are double shape parameters, the developing coefficient of the grey forecasting model is extended to be a variant. Property analysis of WBGM(1,1) shows that the fitting accuracy is higher and the applicable confines are wider. Finally, this paper gives an optimization method for the parameters of WBGM(1,1). A classic example and a practical case are studied for confirming the effectiveness of WBGM(1,1). The case study is the prediction for the number of invention patents of integrated circuit(IC) filed in China from 2007 to 2017. Results of the example and case study are compared to other forecasting models, including GM(1,1), NGBM(1,1), Holt exponential smoothing and ARIMA. Results show that WBGM(1,1) is a more general and more efficient model in grey prediction theory.

This paper focuses on the behavior of a linear quadratic Gaussian (LQG) controller for discrete-time systems utilizing industrial communication protocols. A new design of LQG problem is investigated for constrained networked control systems by using a new class of a quadratic cost function with a communication cost. This paper is considered two industrial communication protocols namely: the transportation control protocol (TCP), in which the sender should receive an acknowledgment regarding the delivery of the sending packets and the user datagram protocol (UDP) where the acknowledgment signal is absence. In the latter one, the driver (actuation) does not afford acknowledgment to inform the controller/estimator regarding the reception of the action packets. In such setting, the UDP-like protocol can react nonlinearly, where the absence of the acknowledgment creates a nonlinear optimization control law. In this sense, a LQG with standard Kalman filtering is failed because the separation principle can not reached. Under this setting, a suboptimal methodology can be successfully implemented to compensate the absence of communication and acknowledgment packets. The prescribed performance of networked system is sustained in the presence of communication constraints. The probabilities of the data signal dropouts are expressed in terms of random Bernoulli processes. Then, simulation results are provided to demonstrate the validity of the proposed schemes and to compare them with others from literature.

This paper is focused on the stability problem of a class of stochastic switched genetic regulatory networks, where the stable property of every subsystem is not imposed. By employing the stabilization effects of switching behaviors and stochastic differential equation theory, a sufficient condition for globally asymptotic stability in mean is derived. Furthermore, inspired by the idea of switching interval segmentation, an easily verifiable criterion is established by means of multiple discretized Lyapunov functions. Then, we extend the attained results to the case with time delays via the multiple discretized Lyapunov-Krasovskii functionals approach. Finally, the results obtained in the paper are illustrated by a numerical example.

This paper considers the problems on the existence and uniqueness, the pth(p ≥ 1)-moment and the almost sure stability with a general decay for the global solution of stochastic delay differential equations with Markovian switching, when the drift term and the diffusion term satisfy the locally Lipschitz condition and the monotonicity condition. By using the Lyapunov function approach, the Barbalat Lemma and the nonnegative semi-martingale convergence theorem, some sufficient conditions are proposed to guarantee the existence and uniqueness as well as the stability with a general decay for the global solution of such equations. It is mentioned that, in this paper, the time-varying delay is a bounded measurable function. The derived stability results are more general, which not only include the exponential stability but also the polynomial stability as well as the logarithmic one. At last, two examples are given to show the effectiveness of the theoretical results obtained.

This paper debated the exponential input-to-state stability (EITSS) of the solution for a kind of quaternion-valued neural networks (QVNNs) with time delay. It fills the blank of QVNN in the aspect of input-to-state stability (ITSS). In virtue of the quaternion multiplication is not suitable for commutative law, QVNN is ordinarily resolved into four real-valued neural networks (RVNNs). Making use of a novel Lyapunov–Krasovskii function and some inequalities, we obtain a little sufficient conditions to assure the considered system is EITSS. Finally, by means of two examples, it is certified that the calculation results in this paper are fine.

This paper designs the robust fault tolerant controller for unmanned marine vehicle (UMV) systems with thruster faults and external disturbances via sliding mode output feedback. A comprehensive and unified fault model including thruster partial, complete and stuck faults is built for the first time. Based on input matrix full-rank decomposition technique and H∞ technique, a sufficient condition of sliding mode in the form of linear matrix inequality (LMI) is given. Then taking advantage of adaptive mechanism, a nonlinear discontinuous control term and an output feedback controller are aimed to reduce the oscillation amplitudes of the yaw velocity error and the yaw angle. Compared with the existing methods, the general faults including time-varying stuck fault can be dealt with. Eventually, the comparative simulation results have demonstrated the effectiveness and feasibility of the presented strategy in this paper.

Let Γ be a simple connected graph with vertex set V(Γ). The eccentric distance sum (EDS for short) of Γ is defined as ξ d ( Γ ) = ∑ v ∈ V ( Γ ) e Γ ( v ) D Γ ( v ) , where ɛΓ(v) is the eccentricity of a vertex v and DΓ(v) is the sum of all distances from v to other vertices. In the paper Li and Wu [11], the strict upper bound on ξd(Γ) among the k-connected graphs Γ with an integer k even were given, and proposed an open question: the corresponding problem on k-connected graphs with k odd. In this paper, first, we divide cubic transitive graphs into two cases: super-connectedness and non-super-connectedness, and characterized non-super-connected cubic transitive graphs, filling the gaps in this field. Then, by using the characterization, we show the upper bound on EDS among (3-connected) cubic transitive graphs of order n and characterize the extremal graphs: the ladders when n ≡ 0 ( mod 4 ) and the ladders or the Mobius ladders when n ≡ 2 ( mod 4 ) . Finally, we conclude the paper with conjectures about the upper bound on EDS of k-connected graphs with odd integer k ≥ 3.