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Fine-grained B4C–SiC/C composites were fabricated using a ball-milling dispersion process. The oxidation behaviors of both fine-grained B4C–SiC/C composites and coarse-grained B4C–SiC/C composites at temperatures of up to 1400 °C were analyzed by the differential thermal analysis technique, and the surface morphology of the composites after isothermal oxidation at 800, 1200 and 1400 °C was examined by scanning electron microscopy (SEM). The results indicated that fine-grained B4C–SiC/C composites had excellent oxidation resistance with self-healing properties at 1400 °C. A general model and mechanism for self-protection against oxidation of carbon materials were proposed.

In this paper the solvability of linear index-3 DAEs and the stability of the numerical methods are investigated. By choosing suitable projectors, the solutions of DAEs are decomposed into their essential components. Some equivalent systems and consistent initial conditions are given. In a similar way the stability of the BDFs and IRK methods applied to some DAEs is discussed.

Ti doped and Si–Ti doped graphites have been developed. The influence of the dopants on the properties and microstructure of doped graphites was analyzed. Test results reveal that Ti doped graphite has excellent bending strength and high thermal conductivity, with highest values reaching 50.2 MPa and 424 W m −1 K −1 for a Ti concentration of 15 wt% in the raw materials. Si added simultaneously with Ti promotes the growth of graphite crystals resulting in an increased thermal conductivity. A kind of Si–Ti doped graphite has been developed with thermal conductivity as high as 494 W m −1 K −1 by optimizing the compositions. Correlation between the content of dopant and the properties and microstructure of doped graphites was studied, and catalytic graphitization mechanism of dopants is also discussed.

In this paper we introduce a conception on the splitting of a matrix A, which is based on the outer inverse of the matrix A. It will be evidence that splittings early are the special cases. Several necessary and sufficient conditions to endure the convergence of the fixed pointed system generated by the splitting are also induced.

This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved.

In this paper, monotonicity of iterative methods for solving general solvable singularly systems is discussed. The monotonicity results given by Berman, Plemmons, and Semal are generalized to singular systems. It is shown that for an iterative method introduced by a nonnegative splitting of the coefficient matrix there exist some initial guesses such that the iterative sequence converges towards a solution of the system from below or from above. The monotonicity of the block Gauss-Seidel method for solving a p-cyclic system and Markov chain is considered.

In [M. Knorrenschild, SIAM J. Numer. Anal. 29 (1992) 1694-1715], a class of regularization methods for solving autonomous index-1 DAEs (differential-algebraic equations) has been proposed. In this paper, we generalize it to non-autonomous index-1 and higher index DAEs. We will treat a DAEs as an infinitely stiff ODEs (ordinary differential equations). Therefore, the solutions of a DAEs can be considered as the limit of the solutions of the corresponding stiff ODEs in some sense. Some similar results with M. Knorrenschild (1992, loc. cit.) are obtained.

Comparison theorems for spectral radii of two block USSOR methods, two block SSOR methods, two block AOR methods, two block SOR methods with respect to a partition π and the corresponding finer partition π* are given, respectively, whenever A is an M-matrix or an H-matrix.

Linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x′(t) + B(t)x(t) = q(t), which are tractable with a higher index, are discussed. Their essential properties are investigated. Some equivalent systems are given. Using them the paper shows how to state properly initial and boundary conditions for these DAEs. The existence and uniqueness theory of the solution of the initial and boundary value problems for higher index DAEs are proposed.

In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above.

When A∈ R n×n is an M -matrix we prove the following inequality: q(A∘A −1 ) =1 for n=2, > 2 n for n>2, where A∘B is the Hadamard product of two matrices A and B , q(B)=[p(B −1 )] −1 and p(C) denotes the Perron eigenvalue of a nonnegative matrix C . This also gives a positive answer to the conjecture posed by Fiedler and Markham.

In this paper, we discuss convergence of the extrapolated iterative methods for solving singular linear systems. A general principle of extrapolation is presented. The semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coefficient matrix A is a singular M-matrix with ‘property c’ and an irreducible singular M-matrix, respectively. Since the (generalized, block) JOR and AOR methods are respectively the extrapolated methods of the (generalized, block) Jacobi and SOR methods, so the semiconvergence of the (generalized, block) JOR and AOR methods for solving general singular systems are proved. Furthermore, the semiconvergence of the extrapolated power method, the (block) JOR, AOR and SOR methods for solving Markov chains are discussed.

Suppose Ax = b is a system of linear equations where the matrix A is symmetric positive definite and consistently ordered. A bound for the norm of the errors e k = x− xk of the AOR method in terms of the norms of δ k = xk−xk−1 and δk+1 = xk+1−xk and their inner product is derived.

The multi-wall carbon nanotubes (MWNT) were mixed with naphthalene-derived synthetic mesophase pitch, and the pastes were oxidized at 200 oC for 2 hours. Then the oxidized pastes containing different MWNT concentrations were molded into tapes, whose carbonization behaviors were studied. The effects of the MWNT content on the carbonization behaviors and physical properties of the composites were investigated. Results indicated that the sintering of composites was in six stages: thermal swelling, liquid phase, coalescence, semi-coke, coke and carbon. The addition of MWNT severely increases shrinkage in the liquid stage and reduces shrinkages in the coke and carbon stage. The sample shrinkage decreases with increase of concentration of MWNT during the carbonization process. And well dispersion can be obtained at the composite containing 5 wt.% MWNT, and the composite reveals higher mechanical strength than the neat MP-based graphite and the composite containing 20 wt.% MWNT.

Due to their nanometer-scale dimensions, high mechanicalstrength, and very high thermal and electrical conductivity,carbon nanotubes (CNTs) have attracted much attention tobe recognized as an excellent material choice for nano-composites, in which even a very small amount of CNTscan induce significant changes in the material’s properties[1–4].Since carbon and carbon–carbon composites have manyadvantages, such as excellent mechanical behavior at hightemperatures, low reactivity, high heat capacity, and so on,modern advanced technologies make wide use of thesematerials in different fields, including nuclear reactorwalls, rocket nozzles, battery electrodes, and seal andfriction materials [5–7]. However, the conventional tech-niques to prepare carbon and carbon–carbon compositesare complicated and costly due to the time-consumingprocess, such as liquid phase impregnation and chemicalvapor deposition which are indispensable to obtain high-performance carbon and carbon–carbon composites. Mucheffort has been taken to develop new methods to simplifythe processing and shorten the time of preparing carbon–carbon composite. Most significantly, Naphthalene-derivedsynthetic mesophase pitch (MP) have been recognized asan excellent precursor to the high-performance carbon andgraphite due to their high carbon yield [8–11].However, few researchers try to fabricate the MP basedcomposites reinforced with CNTs because of the difficul-ties in homogeneously distributing CNTs in a MP matrixby traditional methods. In this communication, CNTsreinforced oxidized MP based composites were preparedby one-step self-sintering. The influence of CNTs on themechanical and physical properties on obtained CNTs/carbon derived from oxidized MP composites.The CNTs (multi-walled carbon nanotubes, purity99 wt.%, diameter 20–40 nm) synthesized by chemicalvapor deposition were supplied from Wuhan University ofScience and Technology, China. MP was obtained byMitsubishi Gas Chemical Co., Japan. MWNTs were soni-cated in a bath containing toluene, and then the milled MPwas added to the suspension and hot-mixed at 333 K in theother bath containing toluene under moderate stirring for2 h. Subsequently, the system was moved to a plate anddried at 373 K. The dried compound was ground by a ball-mill into grains of particle size in the range of 2.8–8.2

In order to solve least squares problems the block SOR iterative methods are proposed and investigated by some authors. In this paper, we generalize the usual block SOR methods. Other five SOR type iterative methods are defined. Their convergence are discussed, the optimal parameters and optimal spectral radii about the SOR type methods are given.

Suppose Ax = b is a system of linear equations where the matrix A is symmetric positive definite and consistently ordered. Some bounds for ∥e k :∥2 in terms of the norms of and and their inner product for the SOR, SSOR, USSOR, MSOR and AOR methods have been given by many authors. Recently, Li presents a uniform bound for the overrelaxation methods. In this paper, an error bound for the USMAOR method is derived. As special cases of the USMAOR method, bounds of ∥e k :∥2 for the SMAOR, USMSOR, SMSOR, USAOR, SAOR, USSOR, SSOR, MAOR, MSOR, AOR, EAOR, EGS I, EGS II, SOR, JOR, Gauss-Seidel and Jacobi methods are obtained

We consider the generalized AOR method presented by K. R. James. We give some convergence theorems for various kinds of matrices, namely when the coefficient matrix of the linear system is positive definite, an H -, L -, or M -matrix, or strictly or irreducibly diagonally dominant.

The ablation resistance of carbon materials can beeffectively improved by the addition of zirconium dueto the formation of an oxidation-resistant zirconiafilm on the carbon substrate during ablation [1]. How-ever, for ultra-high-temperature structural applications,unsatisfactory values of strength are still obstacles. It iswell established that introducing other elements intographite is a quite effective means to enhance themechanical performances of graphite materials [2]. Bor-on atoms can substitute for carbon atoms withoutchanging the network of carbon [3]. Therefore, boronas a reinforcement aid for carbon materials has attractedattention from researchers. It was found that a low con-tent boron improved the mechanical properties [4].In the present letter, Zr–B-doped graphitised carbonswere prepared. The influences of the content of boronon the strength and ablation resistance of zirconium-doped graphitised carbons were also investigated.Melting coal tar pitch with softening point 180 Cwas mixed with B

For solving the large dynamical systems of initial value problems that can be described by a system of mixed implicit differential equations, the waveform relaxation (WR) methods are proposed and investigated by some mathematicians. In this paper, in order to compute the WR methods on parallel processors and to avoid the synchronization costs, the chaotic waveform relaxation (WR) methods are presented and their convergence is investigated. As for the special cases, the nonlinear ODEs, linear ODEs and semi-explicit DAEs are discussed, respectively.

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type II response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.

This paper presents the multi-parameters overrelaxation (MPOR) method, whose special cases involve many classical iterative methods, for solving linear systems of equations. The convergence conditions with A being a Hermitian matrix, an L-matrix and an H-matrix are derived. The results here imply many well-known statements for the classical iterative methods.

In this paper we relax the some given models of parallel chaotic iterations and give several algorithms of relaxed parallel chaotic iterative methods for solving large nonsingular systems of equations Ax = b. Under some different assumptions of coefficient matrix A and its multisplittings we obtain corresponding sufficient conditions of convergence for some relaxed parameters.

This paper investigates the convergence of parallel multisplitting methods for solving linear systems Ax = b, where A is an H-matrix. Sufficient conditions of convergence for general parallel methods are derived. A class of parallel generalized AOR methods, parallel block AOR methods and parallel AOR methods are proposed, and many convergence conditions are obtained. In case A is an M-matrix, we compare the asymptotic convergence rates of the parallel methods.

In this paper, we propose the meshless approach for the numerical solution of time dependent partial differential algebraic equations (PDAEs) in terms of finite difference scheme generated from radial basis functions (RBF‐FD). Using the method, we can circumvent the influence from an index jump of PDAEs in some degree. Numerical example shows that the method has some advantages over some known methods, such as less computation and more accurate for PDAEs.

In recent years, some high—order iterative methods for finding multiple roots of nonlinear equations are presented. However, almost none of these methods give the information for the convergence radius. Recently, H. M. Ren and I. K. Argyros give an estimate of the convergence radius of the modified Newton’s method for multiple roots under Holder and center—Holder continuous conditions. In this paper, we estimate the convergence radius of Osada’s method for multiple roots under the same conditions. Also, some examples are provided to show applications of our theorems.

In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.

Summary The 3rd generation coherence algorithm, which was proved to be an excellent tool to detect faults and has good antinoise ability, was restricted by its large computational cost on constructing covariance matrix and eigen decomposition. We aim at this problem in this paper and propose an efficient realizing method which was based on recursion and power algorithm to accelerate the 3rd generation algorithm. The results on field data show that our method can greatly reduce calculation.

Let A and B be two n × n matrices with spectra λ ( A )={ λ 1 ,…, λ n } and λ ( B )={ μ 1 ,…, μ n }. Suppose that the nonsingular matrix Q satisfies Q −1 AQ =diag( J 1 ,…, J p ), where each submatrix J i , i=1,…,p , is a Jordan block. Then there exists a permutation π of {1,…, n } such that ∑ j=1 n μ π(j) −λ j 2 ⩽ n (1+ n−p ) max ∥Q −1 (B−A)Q∥ F , ∥Q −1 (B−A)Q∥ F m and for j =1,…, n , μ π(j) −λ j ⩽ n (1+ n−p ) max n ∥Q −1 (B−A)Q∥ 2 , n ∥Q −1 (B−A)Q∥ 2 m , where m is the order of the largest Jordan block of A and ∥ ∥ F and ∥ ∥ 2 denote, respectively, the Frobenius norm and the spectral norm.

In order to solve a linear system Ax = b , Hadjidimos et al. (1992) defined a class of modified AOR (MAOR) method, whose special case implies the MSOR method. In this paper, some sufficient and/or necessary conditions for convergence of the MAOR and MSOR methods will be achieved, when A is a two-cyclic matrix and when A is a Hermitian positive-definite matrix, an H -, L - or M -matrix, and a strictly or irreducibly diagonally dominant matrix. The convergence results on the MSOR method are better than some known theorems. The optimum parameters and the optimum spectral radii of the MAOR and MSOR methods are obtained, which also answers the open problem given by Hadjidimos et al.

The comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors. In this article, some new comparison theorems for two nonnegative splittings (a splitting A = M − N is nonnegative if M −1 exist and M −1 N is nonnegative) are derived. They extend the known results in the literature. In addition, we also point out three incorrect conditions in a paper by Beauwens. Furthermore, we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates. This also answers the open question which additional and appropriate conditions should be imposed on Miller-Neumann splittings to obtain strict inequality. Finally, some applications to a class of generalized AOR, SOR, and JOR iterative methods whose special cases imply block (also point) AOR, SOR, and JOR iterative methods for solving linear systems are discussed.

For an n × n matrix A = (aij) ⩾ 0, we prove the Perron root r(A) satisfies the inequality r(A) ⩾ ∏ 1 ⩽i (a ij a ji ) x i x j ∏ i= 1 n (a ii +q) x i 2 ∑ i= 1 n x i 2 −q for any x=(x1,…,xn)T⩾0 with ∑ i= 1 n x i = 1 and for any q⩾–min{aii}. Using this, a necessary condition for a real n × n matrix A with aii>0 and aij⩽0, i≠j, i, j = 1,…,n, to be an M-matrix is presented, viz. ∏ 1 ⩽i a ij a ji a ii a jj x i x j 1 +q)q − ∑ i= 1 n x i 2 ∑ i= 1 n x i 2 for any x=(x1,…,xn)T⩾0 with ∑ i= 1 n x i = 1 and for any q>0.

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.

A new kind of trigonometrically fitted embedded pair of explicit ARKN methods for the numerical integration of perturbed oscillators is presented in this paper. This new pair is based on the trigonometrically fitted ARKN method of order five derived by Yang and Wu in [H.L. Yang, X.Y. Wu, Trigonometrically-fitted ARKN methods for perturbed oscillators, Appl. Numer. Math. 9 (2008) 1375–1395]. We analyze the stability properties, phase-lag (dispersion) and dissipation of the higher-order method of the new pair. Numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature.