In this paper, properties of shape preserving for Meyer-Konig and Zeller Operators are considered, of which the monotonicity and convexity preserving as well as the inverse theorem of shape preserving are obtained by analytical method clopend on the special construct of the operators.

In this paper, we will use the 2r-th Ditzian-Totik modulus of smoothness to discuss the direct and inverse theorems of Lp metric approximation by Left-Bernstein-Durrmeyer quasi-interpolant operator Mn[2r-1](f), for functions which are defined in the space Lp[0,1] (1≤p≤+∞).

The inverse theorem of global approximation of derivatives of generalized Bernstein polynomials;

Generalized Baskakov operators and the direct and inverse theorem of its derivatives;

Both direct and inverse theorems of pointwise approximation are derived.

This paper investigates the converse theorem of rectangular projection theorem in descriptive geometry,and furthermore supplements and confirms another converse theorem,"If the projection of two lines intersecting vertically fo rms a rectangle on a certain projecting plane,at least one line of the two lines is parallel to the projecting plane.

The converse theorems of mean value theorem of two and three dimensional biharmonic function are presented and prove

The mean value theorems and converse theorems of bending and vibration of bars are presented and proven.

Heilmann[1J, gives the direct and converse theorems of approxi-mation and the character theorem of derivative.