We consider those subsets of the self-affine Sierpinski carpets that are the union of an uncountable number of sets each of which consists of the points with their location codes having prescribed group frequencies. It is proved that their Hausdorff dimensions equal to the supremum of the Hausdorff dimensions of the sets in the union. The main advantage is that we treat these subsets in a unified manner and the value of the Hausdorff dimensions do not need to be guessed a priori.

A class of regular subsets of the self-affine set introduced by Gatzouras and Lalley are studied.Their Hausdorff and packing dimensions are explicitly obtained.

The Cauchy problem of Helmholtz equation is severely ill-posed problem.In this paper,we consider the Cauchy problem for the Helmholtz equation where the Cauchy data is given at x = 0 and the solution is sought in the interval 0<x<1.A semi-discrete difference schemes together with a choice of regularization parameter is presented and error estimate is obtained.

The well-known self-affine Sierpinski carpets, first studied by McMullen and Bedford independently, are constructed geometrically by repeating a single action according to a given pattern. In this paper, we extend them by randomly choosing a pattern from a set of patterns with different scales in each step of their construction process. The Hausdorff and box dimensions of the resulting limit sets are determined explicitly and the sufficient conditions for the corresponding Hausdorff measures to be positive finite are also obtained.

In this paper we study Besicovitch type subsets of a class of self-affine set which are defined in terms of the asymptotic behaviour of the frequencies of their digits in their codings. A formula for their Hausdorff dimension is given.

In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.

A class of Cauchy problem of Laplace equation was discussed in following way: the Cauchy data was given at x=1 and the solution was sought in the interval 0x1.Due to the ill-posed feature of the problem,a method for its spectral regularization was given.Meanwhile,the difference between regularized solution and exact solution was analyzed.

In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.

The problem of identifying an unknown source in the heat equation is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In this paper, we proposed a regularization strategy-mollification method to analysis the stability of the problem. Meanwhile, we proposed numerical implement. Numerical example show that the proposed method is effective and stable.

In this paper we study a class of subsets of self-affine carpets which are indexed by fibre-coding, but their limiting frequencies have nonlinear relation. We obtain their Hausdorff dimension spectrum and give the necessary and sufficient conditions of these fractal sets to be s-set.

We consider a class of subsets of the general Sierpinski carpet which is characterized by insisting that the allowed digits in the expansion occur with prescribed mixing group frequencies, determine their Hausdorff dimensions and give the necessary and sufficient conditions for their corresponding Hausdorff measures to be positive and finite.

We consider a random version of the McMullen–Bedford general Sierpinski carpet which is constructed by randomly choosing patterns in each step instead of a single pattern in its original form. Their Hausdorff, packing and box-counting dimensions are determined. A sufficient condition and a necessary condition for the Hausdorff measures in their dimensions to be positive are given. As an application, we discuss the issue on the intersection of the general Sierpinski carpet with its translations.

Under Pitman Closeness(PC) criterion,the mixed regression estimator of regression coefficient is compared with the least square estimator and the condition under which the mixed regression estimator is superior to the least square estimator is achieved for Growth Curve Model.

In this paper we study a class of subsets of the general Sierpinski carpets for which frequencies of the horizontal fibres have linear relation. We calculate the Hausdorff dimension of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.

This paper studied computation of the Hausdorff measure for a class of inhomogeneous self-similar sets in R~3 satisfying the separate condition and Hausdorff dimension less than 1. By means of the upper convex density theorem, some conditions on the contractive ratios were given so that the Hausdorff measures can be exactly determined.

In this paper we study a class of subsets of the general Sierpinski carpets for which the limiting frequency of a horizontal fibre falls into a prescribed closed interval. We obtain the explicit expression for the Hausdorff dimension of these subsets in terms of the parameters of the construction and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.

In this paper we study a class of subset of Sierpinski carpets for which the allowed digits in the expansions fall into each fiber set with a prescribed frequency. We calculate the Hausdorff and packing dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff and packing measures to be finite.

In this paper we study a class of subsets of the general Sierpinski carpets for which the digits in the expansions lie in two specified horizontal fibres with proportional frequencies. We calculate the Hausdorff dimension of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.

In this paper we study the existence of positive global solutions for a semilinear elliptic equation. We use the methods of super-solution and sub-solution to overcome the difficulty of an inhomogeneous term, thus we obtaine the necessary condition which positive global solutions for a semilinear eliptic equation possibly exist, and given the neighborhood which positive global solutions for a semilinear elliptic equation lie in.

In this paper we study the problem of efficiency about unknow mean matrix in singular growth curve model. We derive the estimation of the deviation between the Least squares and the Best linear unbias estimation of the mean matrix. Meanwhile a relative efficiency of LSE is proposed and its upper bound is given.