路和圈多重联图的邻点可区别E-全染色

设G（V,E）是一个简单图,k是一个正整数,f是一个V（G）∪E（G）到{1,2,...,k}的映射.如果u,v∈E（G）,则f（u）=f（v）,f（u）=f（uv）,f（v）=f（uv）,C（u）=C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.称f是图G的邻点可区别E-全染色,称最小的数k为图G的邻点可区别E-全色数.讨论了路和圈的多重联图的邻点可区别E-全色数。     

Estimate of a random field based on the trajectory of a point moving in it

We consider the stochastic differential equationd _t =( _t )dt+ _t ( _t )dw _t in Euclidean space, where (x) is a Gaussian random field andw _t is a standard Wiener process. Let f _t ={ _s ,st}. Equations are obtained for the conditional meansm _t (x)=f _t } andB _t (x, y)=M{(x)(y)|f _t }.Translated fromTeariya Sluchaínykh Protsessov, Vol. 14, pp. 7–9, 1986.     

Positive Lyapunov exponents for a class of ergodic orthogonal polynomials on the unit circle

Consider ergodic orthogonal polynomials on the unit circle whose Verblunsky coefficients are given by αn(ω)=λV(Tnω), where T is an expanding map of the circle and V is a C¹ function. Following the formalism of [Jean Bourgain, Wilhelm Schlag, Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Comm. Math. Phys. 215 (2000) 143-175; Victor Chulaevsky, Thomas Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995) 455-466], we show that the Lyapunov exponent γ(z) obeys a nice asymptotic expression for λ>0 small and z∈∂D?{±1}. In particular, this yields sufficient conditions for the Lyapunov exponent to be positive. Moreover, we also prove large deviation estimates and Hölder continuity for the Lyapunov exponent.     

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

We discuss Totik’s extension of the classical Bernstein theorem on polynomial approximation of piecewise analytic functions on a closed interval. The error of the best uniform approximation of such functions on a compact subset of the real line is studied.     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.     

Matlab implementation of a moving grid method based on the equidistribution principle

The objective of this paper is to report on the development of a method of lines (MOL) toolbox within MATLAB, and especially, on the implementation and test of a moving grid algorithm based on the equidistribution principle. This new implementation includes various spatial approximation schemes based on finite differences and slope limiters, the choice between several monitor functions, automatic grid adaptation to the initial condition, and provides a relatively easy tuning for the non-expert user. Several issues, including the sensitivity of the numerical results to the tuning parameters, are discussed. A few test problems characterized by solutions with steep moving fronts, including the Buckley-Leverett equation and an extended Fisher-Kolmogorov equation, are investigated so as to demonstrate the algorithm and software performance.     

Control of the optimum synthesis process of a four-bar linkage whose point on the working member generates the given path

This paper presents the optimum synthesis of a four-bar linkage in which the coupler point performs a path composed of rectilinear segments and a circular arc. The Grashof four-bar linkage whose geometry provides minimum deviations from the given problem for certain parts of the crank cycle is chosen. The motion of the coupler point of the four-bar linkage is controlled within the given values of allowed deviations so that it is always in the prescribed environment of the given point on the observed segment. The synthesis process tends to bring only those path segments that are beyond the boundaries within the prescribed boundary deviations. During the synthesis, allowed deviations change from the initial maximum values to the given minimum ones. Groups of mechanisms realising satisfactory approximation to the desired motion can be obtained by the method of controlled decrease of allowed deviations with the application of the Differential Evolution (DE) algorithm.     

Geometrically nonlinear shell finite element based on the geometry of a planar curve

An engineering approach for constructing a curved triangular finite element of a thin shell is considered. The approach is based on the assumption that the triangle sides are planar nearly circular curves before and after deformation. A geometrically nonlinear formulation of a triangular finite element of a thin Kirchhoff–Love shell is given. The predictive capabilities of the element are tested using benchmark problems of nonlinear deformation of elastic plates and shells.     

Complexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier term

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization(SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 q2 1, the algorithm has O((q1 + 1) nq1+1/2(q1-q2)logn/ε)and O((q1 + 1)3q1-2q2+1/2(q1-q2)n~1/2 logn/ε) complexity results for large- and small-update methods, respectively.     

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.     

A theoretical analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the one dimensional wave equation

This paper details our note [6] and it is an extension of our previous works  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for second order hyperbolic equations with Dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in [14]. We aim in this work (and some forthcoming studies) to get higher order (both in time and space) finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in [14] on low order schemes from which the matrices used to compute the discrete solutions are sparse. We focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in [4]. The implicit finite volume scheme approximating the one dimensional wave equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal, symmetric and definite positive. The finite volume approximate solution of the basic finite volume scheme is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution. The computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [13] without any condition. To reach the above results, a theoretical framework is developed and some numerical examples supporting the theory are presented. Some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions. In particular a new and useful a prior estimate for a suitable discrete problem is developed and proved. The proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems. A new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives.     

A full analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation

The present work is an extension of our previous works ,  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for parabolic equations. We aim in this work (and some forthcoming studies) at getting higher order (both in time and space) finite volume approximations for the exact solution of parabolic equations using the class of spatial generic meshes introduced recently in [13]. We focus in the present contribution on the one dimensional heat equation and its implicit finite volume scheme described in [3]. The implicit finite volume scheme approximating the one dimensional heat equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal. The finite volume approximate solution is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allows us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using the same linear systems involved in the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximations of the second, third, and fourth spatial derivatives of the exact solution. The computation of the approximation of these high-order derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [12] without any restrictive condition on the spatial mesh. A full analysis for the stated theoretical results as well as some numerical examples supporting the theory is presented. The results obtained in the present study are based essentially on two facts. The first fact is the use of the results provided in [3] which state the convergence order of the finite volume approximate solution in several norms. The second fact is the comparison between the stated new higher order approximations and suitable auxiliary finite volume approximations.