When one compares a difference scheme with another, only a qualitative comparison is usually given. Such a comparison is not enough. For example, Scheme A is more accurate than Scheme B, but it would take more time to use Scheme A than to use Scheme B. In order to determine which scheme is the best, it is necessary to make a quantitative comparison among difference schemes.

The problem discussed in this paper is to determine a nonnegative interpolating polynomial which takes the prescribed nonegative values $y_0,y_1,\cdots,y_n$ at given distinct points $x_0,x_1,\cdots,x_n$: $p(x_i)=y_i),i=0,1,\cdots,n$. This paper shows：(1) $2n$ is the least number of $m$ such that there exists a polynomial $p\in P_m^{+}$, the set of all nonnegative polynomials of degree $\leq m$, satisfying the above equations for any choice of $y_i\geq 0$. (2) the above equations have a unique solution in $P_{2n}^{+}$ if and only if at most one of the $y_i's$ is nonzero.

For each vector norm ‖x‖, a matirx $A$ has its operator norm $‖A‖=\mathop{\rm min}\limits_{x≠0}\frac{‖Ax‖}{‖x‖}$ and a condition number $P(A)=‖A‖ ‖A^{-1}‖$. Let $U$ be the set of the whole of norms defined on $C^n$. It is shown that for a nonsingular matrix $A\in C^{n\times n}$, there is no finite upper bound of $P(A)$ whch ‖·‖ varies on $U$ if $A\neq \alpha I$; on the other hand, it is shown that $\mathop{\rm inf}\limits_{‖·‖\in U} ‖A‖ ‖A^{-1}‖ =ρ(A)ρ(A^{-1})$ and in which case this infimum can or cannot be attained, where $ρ(A)$ denotes the spectral radius of $A$. 

It is proved in this paper that there exists an expansion for the derivative of the linear finite element approximation to a model Dirichlet problem in a polygonal domain with a piecewise uniform triangulation.  

This paper is intended to study the finite difference method for the periodic boundary and initial value problem of a class of system of generalized Zakharov equations.