# Showing papers in "International Journal of Computer Mathematics in 2017"

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TL;DR: It is proved that the iterative sequence generated by the alternating direction method converges to a critical point of the problem, provided that the penalty parameter is greater than 2L, where L is the Lipschitz constant of the gradient of one of the involved functions.

Abstract: The efficiency of the classic alternating direction method of multipliers has been exhibited by various applications for large-scale separable optimization problems, both for convex objective functions and for nonconvex objective functions. While there are a lot of convergence analysis for the convex case, the nonconvex case is still an open problem and the research for this case is in its infancy. In this paper, we give a partial answer on this problem. Specially, under the assumption that the associated function satisfies the Kurdyka–Łojasiewicz inequality, we prove that the iterative sequence generated by the alternating direction method converges to a critical point of the problem, provided that the penalty parameter is greater than 2L, where L is the Lipschitz constant of the gradient of one of the involved functions. Under some further conditions on the problem's data, we also analyse the convergence rate of the algorithm.

80 citations

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TL;DR: 2-extra connectivity of is 4n−4 and 2-extra edge-connectivity of is 6 n−4 for .

Abstract: The balanced hypercube , as a new variation of the hypercube, possesses many attractive properties such that the hypercube dose not have. Given a connected graph G and a non-negative integer g, the g-extra connectivity resp. g-extra edge-connectivity of G, denoted by resp. , is the minimal cardinality of a set of vertices resp. edges of G, if exists, whose deletion disconnects G and each remaining component contains more than g vertices. In this paper, we show that the 2-extra connectivity of is 4n−4 and 2-extra edge-connectivity of is 6n−4 for . Also, we determine 3-extra connectivity of for .

51 citations

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TL;DR: This paper gives that the 1-good-neighbour diagnosability of under the PMC model and MM model is except the bubble-sort graph under MM model, where, and the1- good-neIGHbour diagnOSability ofUnder theMM model is 4.

Abstract: Diagnosability is an important metric for measuring the reliability of multiprocessor systems. In 2012, Peng et al. proposed a new measure for fault tolerance of the system, which is called g-good-neighbour diagnosability that restrains every fault-free node containing at least g fault-free neighbours. As a favourable topology structure of interconnection networks, the Cayley graph generated by the transposition tree has many good properties. In this paper, we give that the 1-good-neighbour diagnosability of under the PMC model and MM model is except the bubble-sort graph under MM model, where , and the 1-good-neighbour diagnosability of under the MM model is 4.

51 citations

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TL;DR: An improved rough k-means clustering based on weighted distance measure with Gaussian function is proposed in this paper and the validity of this algorithm is demonstrated by simulation and experimental analysis.

Abstract: Rough k-means clustering algorithm and its extensions are introduced and successfully applied to real-life data where clusters do not necessarily have crisp boundaries. Experiments with the rough k-means clustering algorithm have shown that it provides a reasonable set of lower and upper bounds for a given dataset. However, the same weight was used for all the data objects in a lower or upper approximate set when computing the new centre for each cluster while the different impacts of the objects in a same approximation were ignored. An improved rough k-means clustering based on weighted distance measure with Gaussian function is proposed in this paper. The validity of this algorithm is demonstrated by simulation and experimental analysis.

46 citations

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TL;DR: This paper studies the numerical solutions of singularly perturbed parabolic convection–diffusion problems with a delay in time, and proves that the proposed scheme is -uniform convergence of first-order in time and first- order up to a logarithmic factor in space.

Abstract: This paper studies the numerical solutions of singularly perturbed parabolic convection–diffusion problems with a delay in time. We divide the domain using a piecewise uniform adaptive mesh in the spatial direction and a uniform mesh in the temporal direction. Further, we discretize the time derivative by the backward-Euler scheme and the spatial derivatives by the upwind finite difference scheme. We obtain the maximum principle and carry out the stability analysis. Then we prove that the proposed scheme is -uniform convergence of first-order in time and first-order up to a logarithmic factor in space. Numerical results are carried out to verify the theoretical results.

43 citations

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TL;DR: An improved approximate Chinese remainder theorem (CRT) is presented with the aim of performing efficient residue-to-binary conversion for general RNS moduli sets and a method is proposed to substitute fractional calculations by similar computations based on integer numbers to have a hardware amenable algorithm.

Abstract: The residue number system (RNS) is an unconventional number system which can lead to parallel and fault-tolerant arithmetic operations. However, the complexity of residue-to-binary conversion for large number of moduli reduces the overall RNS performance, and makes it inefficient for nowadays high-performance computation systems. In this paper, we present an improved approximate Chinese remainder theorem (CRT) with the aim of performing efficient residue-to-binary conversion for general RNS moduli sets. To achieve this aim, the required number of fraction bits for accurate residue-to-binary conversion is derived. Besides, a method is proposed to substitute fractional calculations by similar computations based on integer numbers to have a hardware amenable algorithm. The proposed approach results in high-speed and low-area residue-to-binary converters for general RNS moduli sets. Therefore, with this conversion method, high dynamic range residue number systems suitable for cryptography and digital ...

41 citations

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TL;DR: An F-transform-based Picard-like numerical scheme is proposed in order to solve a class of delay differential equations and leads to a non-recursive approximate solution by means of operational matrices and vectors of known quantities.

Abstract: Fuzzy transforms (or F-transforms for short) are an approximation technique recently introduced. The main application is referred to image and data compression. There are really few works devoted t...

36 citations

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TL;DR: The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively.

Abstract: In this paper, the fractional delay differential equation FDDE is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.

35 citations

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TL;DR: The existence, uniqueness of the weak solution, and the numerical stability of the scheme are proved in great detail and the optimal error estimate computed by -norm showed both in time and space is derived by introducing a fractional orthogonal projection.

Abstract: In this paper, we consider a fully discrete finite element method FEM to solve the two-dimensional nonlinear Fisher' equation with Riesz fractional derivatives in space. This method is chiefly performed by using Crank–Nicolson discretization in conjunction with a linearized approach in time and FEM in space. The existence, uniqueness of the weak solution, and the numerical stability of the scheme are proved in great detail. The optimal error estimate computed by -norm showed both in time and space is derived by introducing a fractional orthogonal projection. Moreover, several numerical examples are conducted on unstructured triangular meshes by a properly designed algorithm.

35 citations

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TL;DR: A numerical scheme to compute approximate solutions of two dimensional unsteady convection–diffusion equation using collocation of modified bi-cubic B-spline functions for dependent variable u and for its derivatives w.r.t. space variables x and y is presented.

Abstract: This research study presents a numerical scheme to compute approximate solutions of two dimensional unsteady convection–diffusion equation. We used collocation of modified bi-cubic B-spline functions for dependent variable u and for its derivatives w.r.t. space variables x and y. Strong stability preserving Runge–Kutta method SSP-RK54 has been used for solving system of first-order ordinary differential equations obtained from the collocation form of the partial differential equation. We did not linearize the nonlinear terms by using any transformation or linearization method. The number of computations and the required storage space is very less for the proposed scheme. Four examples have been taken as described in available literature to demonstrate the effect and utility of the proposed scheme. These numerical experiments show that the obtained results are not only quite satisfactory w.r.t. the exact solutions but also competent with the solutions available in earlier research studies. Computational complexity of the proposed scheme has been discussed and shown that it is , where p is total number of nodes. The proposed scheme is easy to implement and the size of required computational work is very small. Moreover, using this scheme, we can compute approximate solutions not only at the mesh points but at any other point of the solution domain as well.

33 citations

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TL;DR: Fast numerical methods for solving space-fractional diffusion equations are studied in two stages and a method with fourth-order accuracy in both space and time can be achieved.

Abstract: In this paper, fast numerical methods for solving space-fractional diffusion equations are studied in two stages. Firstly, a fast direct solver for an implicit finite difference scheme proposed by Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 2015, pp. 787–805], which is fourth-order accurate in space and second-order accurate in time, is developed based on a circulant-and-skew-circulant CS representation of Toeplitz matrix inversion. Secondly, boundary value method with spatial discretization of Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 2015, pp. 787–805] is adopted to produce a numerical solution with higher order accuracy in time. Particularly, a method with fourth-order accuracy in both space and time can be achieved. GMRES method is employed for solving the discretized linear system with two preconditioners. Based on the CS representation of Toeplitz matrix inversion, the two preconditioners can be applied efficiently, and the convergence rate of the preconditioned GMRES method is proven to be fast. Numerical examples are given to support the theoretical analysis.

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TL;DR: This paper establishes higher order numeric solutions for the IVP of the singular Lane–Emden-type equation, including the Emden–Fowler equation using the multi-stage modified decomposition method.

Abstract: In this paper, we establish higher order numeric solutions for the IVP of the singular Lane–Emden-type equation, including the Emden–Fowler equation. We use the multi-stage modified decomposition method to effectively treat these types of equations and develop numeric solutions that are effective in the large. The step-size and the order in our numeric solutions are two parameters that may be arbitrarily specified. Fast algorithms of the Adomian polynomials guarantee the efficiency of our approach, and a higher order numeric solution can be readily generated at will. The proposed method overcomes the singular behaviour at the origin and exhibits approximations of high accuracy with a large effective region of convergence. Several numerical examples are examined to demonstrate the reliability of our new approach. In these examples, we have demonstrated that our numeric solutions are consistent by halving the step-size, i.e. the numeric solutions of different step-sizes nearly coincide.

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TL;DR: A new filled function which is continuous and differentiable without any parameter to tune is proposed which has three advantages: firstly, it is not easier to produce extra local minima, secondly, more efficient local search algorithms using gradient information can be applied and thirdly, a continuous andDifferentiable filled function can be optimized more easily.

Abstract: Many real world problems can be modelled as optimization problems. However, the traditional algorithms for these problems often encounter the problem of being trapped in local minima. The filled function method is an effective approach to tackle this kind of problems. However the existing filled functions have the disadvantages of discontinuity, non-differentiability or sensitivity to parameters which limit their efficiency. In this paper, we proposed a new filled function which is continuous and differentiable without any parameter to tune. Compared to discontinuous or non-differentiable filled functions, the continuous and differentiable filled function mainly has three advantages: firstly, it is not easier to produce extra local minima, secondly, more efficient local search algorithms using gradient information can be applied and thirdly, a continuous and differentiable filled function can be optimized more easily. Based on the new proposed filled function, a new algorithm was designed for unco...

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TL;DR: The proposed dynamic system can achieve superior convergence performance and is called the finite-time convergent dynamic system, and the upper bound of the convergence time is derived analytically with the error bound being zero theoretically.

Abstract: A new dynamic system is proposed and investigated for solving online simultaneous linear equations. Compared with the gradient-based dynamic system and the recently proposed Zhang dynamic system, the proposed dynamic system can achieve superior convergence performance (i.e. finite-time convergence) and thus is called the finite-time convergent dynamic system. In addition, the upper bound of the convergence time is derived analytically with the error bound being zero theoretically. Simulation results further indicate that the proposed dynamic system is much more efficient than the existing dynamic systems.

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TL;DR: It is shown that determining the outer-independent Roman domination number in graphs is NP-complete for bipartite graphs and lower and upper bounds are presented on .

Abstract: A Roman dominating function (RDF) on a graph G is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. A function f:V(G)→{0,1,2} is an outer-independent Roman dominating function (OIRDF) on G if f is an RDF and V0 is an independent set. The outer-independent Roman domination number γoiR(G) is the minimum weight of an OIRDF on G. In this paper, we initiate the study of the outer-independent Roman domination number in graphs. We first show that determining the number γoiR(G) is NP-complete for bipartite graphs. Then we present lower and upper bounds on γoiR(G). Moreover, we characterize graphs with small or large outer-independent Roman domination number.

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TL;DR: The exponential lag synchronization for a class of Cohen–Grossberg neural networks with discrete time-delays and distributed delays is investigated via periodically intermittent control through mathematical induction method and analysis technique.

Abstract: In this paper, the exponential lag synchronization for a class of Cohen–Grossberg neural networks with discrete time-delays and distributed delays is investigated via periodically intermittent control. Some simple and useful criteria are derived by using mathematical induction method and the analysis technique which are different from the methods employed in correspondingly previous works. Finally, two examples and their numerical simulations are given to demonstrate the effectiveness of the proposed control schemes.

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TL;DR: Sinc collocation method is considered to obtain the numerical solution of pantograph Volterra delay-integro-differential equation (VDIDE) and this numerical method reduces the VDIDE to an explicit system of algebraic equation.

Abstract: In this article, Sinc collocation method is considered to obtain the numerical solution of pantograph Volterra delay-integro-differential equation VDIDE. This numerical method reduces the VDIDE to an explicit system of algebraic equation. Convergence analysis is given and shows that Sinc solution produces an error of order , where k>0 is a constant. Moreover, Sinc method is applied to the test examples to illustrate accuracy and implementation of the method.

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TL;DR: Experimental results show that either smaller maximum node degree or larger power-law exponent is conducive to the containment of virus spreading, the first time the effect of network topology on virus spreading is investigated in this context.

Abstract: This paper is intended to investigate the effect of network topology on the spread of computer viruses in the presence of removable storage media. For that purpose, a novel network-based computer virus spreading model is proposed. Both theoretically and experimentally, it is shown that, under proper conditions, viruses on a scale-free network would tend to extinction. Experimental results show that either smaller maximum node degree or larger power-law exponent is conducive to the containment of virus spreading. To our knowledge, this is the first time the effect of network topology on virus spreading is investigated in this context.

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TL;DR: Two characteristic block-centred finite difference schemes are introduced and analysed to solve the nonlinear convection-dominated diffusion equation and it is shown that the discrete and errors are , where h corresponds to a finer grid and H corresponding to a coarser grid.

Abstract: In this article, two characteristic block-centred finite difference schemes are introduced and analysed to solve the nonlinear convection-dominated diffusion equation. One scheme is a linear difference approximation which shows that the discrete and errors are , while the other is a two-grid scheme which demonstrates that the discrete and errors are , where h corresponds to a finer grid and H corresponds to a coarser grid. Error estimates with both schemes are established on a non-uniform rectangular grid. Finally, numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis and validate the efficiency of the two-grid method.

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Muğla University

^{1}TL;DR: A numerical solution method which is based on Taylor Matrix Method to give approximate solution of the Bagley–Torvik equation, which is transformed into a system of algebraic equations.

Abstract: In this paper, we present a numerical solution method which is based on Taylor Matrix Method to give approximate solution of the Bagley–Torvik equation. Given method is transformed the Bagley–Torvik equation into a system of algebraic equations. This algebraic equations are solved through by assistance of Maple 13. Then, we have coefficients of the generalized Taylor series. So, we obtain the approximate solution with terms of the generalized Taylor series. Further some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm.

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TL;DR: A unified difference-spectral method for stably solving time–space fractional sub-diffusions and super-diffusion equations is developed by combining the spectral Galerkin method in space and the fractional trapezoid formula in time.

Abstract: In this paper we develop a unified difference-spectral method for stably solving time–space fractional sub-diffusion and super-diffusion equations. Based on the equivalence between Volterra integral equations and fractional ordinary differential equations with initial conditions, this proposed method is constructed by combining the spectral Galerkin method in space and the fractional trapezoid formula in time. Numerical experiments are carried out to verify the effectiveness of the method, and demonstrate that the unified method can achieve spectral accuracy in space and second-order accuracy in time for solving two kinds of time–space fractional diffusion equations.

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TL;DR: This paper considers a GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns, and obtains the stationary queue length distribution at both arrival and arbitrary epochs.

Abstract: In this paper, we consider a GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns. When the server is working in any normal service phases, it may suffer disastrous interruptions, causing all present customers to leave the system. At an exponential failure instant, the system becomes defective, and goes directly to repair phase. During the repair period, instead of stopping service completely, the system is equipped with a substitute server which continues to provide service to the arriving customers. After an exponential repair time, the substitute server stops service and the system moves to normal operative phase i with probability qi,i=1,2,…,N. Using the matrix analytic approach and semi-Markov process, we obtain the stationary queue length distribution at both arrival and arbitrary epochs. We also provide the elaborate analysis of some performance measures and sojourn time distribution of an arbitrary customer. In addition, some numerical examples are presented.

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TL;DR: This paper proposes the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models and focuses on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions.

Abstract: Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 2012, pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.

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TL;DR: Two recursive relations to count the number of subtrees of trees are presented and all trees of n vertices are characterized, which implies that for given families of graphs, determining the graphs with maximum number ofsubrees within the family becomes important.

Abstract: The number of subtrees in a graph is related to the reliability of a network with possible vertex failure and edge failure. It is known that networks with larger number of subtrees would be more reliable. Therefore, for given families of graphs, determining the graphs with maximum number of subtrees within the family becomes important. For a tree T of n vertices, let denote the number of all subtrees in T. Yan and Yeh gave a linear-time algorithm to count the number of subtrees in T and determined the tree of diameter d and order n with the maximum number of subtrees. In this paper, we present two recursive relations to count the number of subtrees of trees. We also characterize all trees of n vertices with .

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TL;DR: This work presents exact values of the metric dimension of several networks, which can be obtained as categorial products of graphs.

Abstract: A set of vertices W is a resolving set of a graph G if every two vertices of G have distinct representations of distances with respect to the set W. The number of vertices in a smallest resolving set is called the metric dimension. This invariant has extensive applications in robotics, since the metric dimension can represent the minimum number of landmarks, which uniquely determine the position of a robot moving in a graph space. Finding the metric dimension of a graph is a non-deterministic polynomial-time hard problem. We present exact values of the metric dimension of several networks, which can be obtained as categorial products of graphs.

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TL;DR: A high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer is introduced and it is shown that the method is -uniformly convergent of second-order accurate in time and in the spatial direction.

Abstract: In this article, we aim to introduce a high-order uniformly convergent method to solve singularly perturbed delay parabolic convection diffusion problems exhibiting a regular boundary layer. The domain is discretized by a uniform mesh in the time direction and a piecewise-uniform Shishkin mesh for the spatial direction. We use the Crank–Nicolson method for the time derivative and we develop a fourth-order compact difference method to solve the set of ordinary differential equations at each time level. The stability analysis and the truncation error are discussed. Parameter-uniform error estimates are derived and it is shown that the method is e-uniformly convergent of second-order accurate in time, and in the spatial direction it is of second-order outside region of boundary layer, and of almost fourth-order inside the layer region. Numerical examples are presented to verify the theoretical results and to confirm the efficiency and high accuracy of the proposed method.

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TL;DR: It is proved that the optimal range for the limit on the number of consecutive zero bits in each digit and average Hamming weight of n-bit integers in the compact SD representation are to and n/3, respectively.

Abstract: This paper presents and evaluates a compact Signed-Digit SD encoding algorithm which is an efficient class of the large integer encoding algorithms. The compact SD algorithm results in a series of digits which show the number of consecutive zero bits. Using the proof-of-concept code, we proved that the optimal range for the limit on the number of consecutive zero bits in each digit and average Hamming weight of n-bit integers in the compact SD representation are to and n/3, respectively. In the modular multiplication algorithm, the compact SD is applied to the multiplier. The compact SD modular multiplication processes consecutive zero bits and followed nonzero digits in one clock cycle. Implementation results on Xilinx Virtex 5 FPGA show that the compact SD modular multiplication outperforms previous modified modular multiplication algorithms in terms of throughput and area × time complexity.

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TL;DR: In this paper, two optimal Newton-Secant-like iterative methods for solving nonlinear equations were proposed, which support the Kung and Traub conjecture and possess a high computational efficiency.

Abstract: We construct two optimal Newton–Secant like iterative methods for solving nonlinear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.

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TL;DR: Improved delay-dependent stability and stabilization criteria for guaranteeing the asymptotic stability of the system are proposed with the framework of linear matrix inequalities.

Abstract: In this paper, the problems of stability and stabilization for linear systems with time-varying delays and norm-bounded parameter uncertainties are considered. By constructing augmented Lyapunov functionals and utilizing auxiliary function-based integral inequalities, improved delay-dependent stability and stabilization criteria for guaranteeing the asymptotic stability of the system are proposed with the framework of linear matrix inequalities. Four numerical examples are included to show that the proposed results can reduce the conservatism of stability and stabilization criteria by comparing maximum delay bounds.

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TL;DR: This paper shows that the 3-extraconnectivity of the k-ary n-cubes is , where and .

Abstract: Given a graph G and a non-negative integer g, the g-extraconnectivity of G is the cardinality of a minimum set of vertices in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with more than g vertices. The 2-extraconnectivity of k-ary n-cubes is gotten by Hsieh and Chang [Extraconnectivity of k-ary n-cube networks. Theoret. Comput. Sci. 443 2012 63–69] for . This paper shows that the 3-extraconnectivity of the k-ary n-cubes is , where and .