APPROXIMATE REPRESENTATION OF THE p-NORM DISTRIBUTION

In surveying data processing,we generally suppose that the observational errors distribute normally.In this case the method of least squares can give the minimum variance unbiased estimation of the parameters.The method of least squares does not have the character of robustness,so the use of it will become unsuitable when a few measurements inheriting gross error mix with others.We can use the robust estimating methods that can avoid the influence of gross errors.With this kind of method there is no need to know the exact distribution of the observations.But it will cause other difficulties such as the hypothesis testing for estimated parameters when the sample size is not so big.For non_normally distributed measurements we can suppose they obey the p _norm distribution law.The p _norm distribution is a distributional class,which includes the most frequently used distributions such as the Laplace,Normal and Rectangular ones.This distribution is symmetric and has a kurtosis between 3 and -6/5 when p is larger than 1.Using p _norm distribution to describe the statistical character of the errors,the only assumption is that the error distribution is a symmetric and unimodal curve.This method possesses the property of a kind of self_adapting.But the density function of the p _norm distribution is so complex that it makes the theoretical analysis more difficult.And the troublesome calculation also makes this method not suitable for practice.The research of this paper indicates that the p _norm distribution can be represented by the linear combination of Laplace distribution and normal distribution or by the linear combination of normal distribution and rectangular distribution approximately.Which kind of representation will be taken is according to whether the parameter p is larger than 1 and less than 2 or p is larger than 2.The approximate distribution have the same first four order moments with the exact one.It means that approximate distribution has the same mathematical expectation,variance,skewness and kurtosis with p _norm distribution.Because every density function used in the approximate formulae has a simple form,using the approximate density function to replace the p _norm ones will simplify the problems of p _norm distributed data processing obviously.

Approximate Representation of Bergman Submodules

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L_a^2(B_d) satisfies P_M-sum from i to ( M_(zi)P_MM*_(zi))≤c/(N + 1)P_M for some number c > 0, then there is a sequence {f_j } of multipliers and a positive number c such that c'P_M ≤sum from j to ( M_(fj)M*_(fj))≤ P_M, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p > d.

A general approach to attribute reduction in rough set theory

The concept of a consistent approximation representation space is introduced. Many types of information systems can be treated and unified as consistent approximation representation spaces. At the same time, under the framework of this space, the judgment theorem for determining consistent attribute set is established, from which we can obtain the approach to attribute reductions in information systems. Also, the characterizations of three important types of attribute sets （the core attribute set, the relative necessary attribute set and the unnecessary attribute set） are examined.