Finite Mean Family With Multi-parameter In Higher Space

In paper[l],mean value theory has been taken axiomatize treatment,and mean family theory has been established. Countable mean family and continuous mean family theory also have been given in paper[4].In this paper, the results of [1,4] are generalized, and mean value theory are discussed further. In this basis, we talk over mean function and mean family of n variables in m dimensions.Three chapters are included in this paper. The first part is introduction in which some excellent results are given. For example, LeachandSholander mean of two variables E(r, s; x, y) and its monotonicity, Some results of paper[l] and [4] are also given. In Chapter l,mean function and mean family of n variables in one dimension are given. On the base of mean function of n variables of paper [1], weproof that is mean function and discuss its monotonicity andcontinuity. In particular, Stolarsky mean are generalized from two variables to n variables. Moreover,we make ωi become weight function to diversity the kinds of mean function. For the mean family, we discuss mean family with single parameter or multi-parameter . At last, we also talk about the convex inequalities, the reformation of Minkowski inequality and the convex inequality about Dresher mean with double-parameter are given. In Chapter 2, the concepts of mean mapping and mean family of m dimensions are given, and concerning on whether they satisfy the convex hull property or the square hull property, they are generalized for two classes, and then we also find and testify their concrete examples. In Chapter 3, the applications of mean family theory in defining new norm and in Computer Aided Geometric Design are given. We define a new norm with the non-degenerate weight and discuss the curve and surface equations of Bezier, B Spline, NU RBS.