On adjacent-vertex-distinguishing total coloring of graphs

In this paper, we present a new concept of the adjacent-vertex-distinguishing total coloring of graphs (briefly, AVDTC of graphs) and, meanwhile, have obtained the adjacent-vertex-distinguishing total chromatic number of some graphs such as cycle, complete graph, complete bipartite graph, fan, wheel and tree.  

关于幂和公式的一般性质

一、前言 当n与k都是正整数时,我们简称sum from m=1 to n ()m~k为“幂和”,并以S_k(n)记之。从古希腊的阿基米德开始,这个问题就吸引着很多数学家的兴趣。十七世纪以前的数学家们仅仅求出了二次和三次幂的求和公式。而雅各·伯努利在《猜度术》中,一举得到了任意次幂的求和公式如下:  

A semismooth equation approach to the solution of nonlinear complementarity problems

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.  

A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems

A new algorithm for the solation of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10 000 variables are presented. Partially supported by Agenzia Spaziale Italiana, Roma, Italy.  

Newton-Thiele's rational interpolants

It is well known that Newton's interpolation polynomial is based on divided differences which produce useful intermediate results and allow one to compute the polynomial recursively. Thiele's interpolating continued fraction is aimed at building a rational function which interpolates the given support points. It is interesting to notice that Newton's interpolation polynomials and Thiele's interpolating continued fractions can be incorporated in tensor‐product‐like manner to yield four kinds of bivariate interpolation schemes. Among them are classical bivariate Newton's interpolation polynomials which are purely linear interpolants, branched continued fractions which are purely nonlinear interpolants and have been studied by Chaffy, Cuyt and Verdonk, Kuchminska, Siemaszko and many other authors, and Thiele-Newton's bivariate interpolating continued fractions which are investigated in another paper by one of the authors. In this paper, emphasis is put on the study of Newton-Thiele's bivariate rational interpolants. By introducing so‐called blending differences which look partially like divided differences and partially like inverse differences, we give a recursive algorithm accompanied with a numerical example. Moreover, we bring out the error estimation and discuss the limiting case. This revised version was published online in June 2006 with corrections to the Cover Date.  

Integration and approximation in arbitrary dimensions

We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ^−p for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{^-1} in these bounds. We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents, and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2. For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces. This revised version was published online in June 2006 with corrections to the Cover Date.  

Dissipativity of Runge-Kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipativeinitial value problems with delays by Runge-Kutta methods. Asufficient condition for the dissipativity of the systems isgiven. The concepts of D(l)-dissipativity and GD(l)-dissipativityare introduced. We investigate the dissipativity propertiesof (k,l)-algebraically stable Runge-Kutta methods with piecewiseconstant or linear interpolation procedures for finite-dimensionaland infinite-dimensional dynamical systems with delays.  

BIVARIATE BLENDING RATIONAL INTERPOLANTS

Both the Newton interpolating polynomials and the Thiele-type interpolating continued fractions based on inverse differences are used to construct a kind of bivariate blending rational interpolants and an error estimation is given.  

A Note on Adjacent Strong Edge Coloring of K（n,m）

In this paper, we prove that the adjacent strong edge chromatic number of a graph K（n,m） is n ＋ 1, with n ≥ 2, m ≥ 1.