In this paper, the relationship between the s-dimensional Hausdorff measures and the g-measures in Rd is discussed, where g is a gauge function which is equivalent to ts and 0 < s≤d. It shows that if s=d, then Hg ...In this paper, the relationship between the s-dimensional Hausdorff measures and the g-measures in Rd is discussed, where g is a gauge function which is equivalent to ts and 0 < s≤d. It shows that if s=d, then Hg = c1Hd, Cg = c2Cd and Pg = c3Pd on Rd, where constants c1, c2 and c3 are determined by where Wg, Cg and Pg are the g-Hausdorff, g-central Hausdorff and g-packing measures on Rd respectively. In the case 0<s<d, some examples are given to show that the above conclusion may fail. However, there is always some s-set F (?) Rd such that Hg|F=C1HS|F, Cg|F = c2Cs|F and Pg|F = c3Ps|F, where the constants c1, c2 and c3 depend not only on g and s, but also on F. A criterion is presented for judging whether an s-set has the above properties.展开更多
Let （Ω, F, P）=（[0, 1], [0, 1], μ）<sup>N</sup> （μ is the Lebesque measure, N={1, 2,…}）.{X<sub>n</sub>}<sub>n=1</sub><sup>∞</sup> are independent random variabl...Let （Ω, F, P）=（[0, 1], [0, 1], μ）<sup>N</sup> （μ is the Lebesque measure, N={1, 2,…}）.{X<sub>n</sub>}<sub>n=1</sub><sup>∞</sup> are independent random variables on （Ω, F, P） with X<sub>n</sub>（ω）=ω<sub>n</sub>, where ω=（ω<sub>1</sub>, ω<sub>2</sub>,…）. The {X<sub>n</sub>}<sub>n=1</sub><sup>∞</sup> are almost surely distinct. Thus to almost all sample points ω there is a random partial order 【 of the integers given展开更多
The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each l...The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.展开更多
Let 0<λ_1,λ_2<1 and 1-λ_1-λ_2≥max｛λ_1,λ_2｝.Let ~K(λ_1,λ_2) be the attractor of the iterated function system ｛φ_1,φ_2｝on the line,where φ_1(x)=λ_1x and φ_2(x)=1-λ_2+λ_2x,x∈R.~K(λ_1,λ_2) is ...Let 0<λ_1,λ_2<1 and 1-λ_1-λ_2≥max｛λ_1,λ_2｝.Let ~K(λ_1,λ_2) be the attractor of the iterated function system ｛φ_1,φ_2｝on the line,where φ_1(x)=λ_1x and φ_2(x)=1-λ_2+λ_2x,x∈R.~K(λ_1,λ_2) is called a non-symmetry Cantor set. In this paper,it is proved that the exact Hausdorff centred measure of K(λ_1,λ_2) equals 2s(1-λ)s,where λ=max｛λ_1,λ_2｝ and s is the Hausdorff dimension of K(λ_1,λ_2).展开更多
In this paper we discuss tLhe existence results of the integral solutions to nonlinear evolution inclusion: u' (t) ∈ Au(t) +F(t,u(t)), where A is m-dissipative and F is a set valued map in separable Banach spaces...In this paper we discuss tLhe existence results of the integral solutions to nonlinear evolution inclusion: u' (t) ∈ Au(t) +F(t,u(t)), where A is m-dissipative and F is a set valued map in separable Banach spaces, and extend the relative results in references.展开更多
First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction ...First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction operators.展开更多
In this article, the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions. Several examples are given at the end.
Let X= (Ω, ■, ■_t, X_t,, θ_t, p~x) be a self-similar Markov process on (0,∞) with non-decreasing path. The exact Hausdorff and Packing measure functions of the image X([0,t] ) are obtained.
文摘In this paper, the relationship between the s-dimensional Hausdorff measures and the g-measures in Rd is discussed, where g is a gauge function which is equivalent to ts and 0 < s≤d. It shows that if s=d, then Hg = c1Hd, Cg = c2Cd and Pg = c3Pd on Rd, where constants c1, c2 and c3 are determined by where Wg, Cg and Pg are the g-Hausdorff, g-central Hausdorff and g-packing measures on Rd respectively. In the case 0<s<d, some examples are given to show that the above conclusion may fail. However, there is always some s-set F (?) Rd such that Hg|F=C1HS|F, Cg|F = c2Cs|F and Pg|F = c3Ps|F, where the constants c1, c2 and c3 depend not only on g and s, but also on F. A criterion is presented for judging whether an s-set has the above properties.
文摘Let （Ω, F, P）=（[0, 1], [0, 1], μ）<sup>N</sup> （μ is the Lebesque measure, N={1, 2,…}）.{X<sub>n</sub>}<sub>n=1</sub><sup>∞</sup> are independent random variables on （Ω, F, P） with X<sub>n</sub>（ω）=ω<sub>n</sub>, where ω=（ω<sub>1</sub>, ω<sub>2</sub>,…）. The {X<sub>n</sub>}<sub>n=1</sub><sup>∞</sup> are almost surely distinct. Thus to almost all sample points ω there is a random partial order 【 of the integers given
基金This research is partly supported by NNSF of China (60204001) the Youth Chengguang Project of Science and Technology of Wuhan City (20025001002)
文摘The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.
文摘Let 0<λ_1,λ_2<1 and 1-λ_1-λ_2≥max｛λ_1,λ_2｝.Let ~K(λ_1,λ_2) be the attractor of the iterated function system ｛φ_1,φ_2｝on the line,where φ_1(x)=λ_1x and φ_2(x)=1-λ_2+λ_2x,x∈R.~K(λ_1,λ_2) is called a non-symmetry Cantor set. In this paper,it is proved that the exact Hausdorff centred measure of K(λ_1,λ_2) equals 2s(1-λ)s,where λ=max｛λ_1,λ_2｝ and s is the Hausdorff dimension of K(λ_1,λ_2).
文摘In this paper we discuss tLhe existence results of the integral solutions to nonlinear evolution inclusion: u' (t) ∈ Au(t) +F(t,u(t)), where A is m-dissipative and F is a set valued map in separable Banach spaces, and extend the relative results in references.
基金Supported by the National Natural Science Foundation of China (10371092)the Foundation of Wuhan University
文摘First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction operators.
基金supported by the National Natural Science Foundation of China(10371092)Foundation of Ningbo University(8Y0600036).
文摘In this article, the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions. Several examples are given at the end.
基金the National Natural Science Foundation of China
文摘Let X= (Ω, ■, ■_t, X_t,, θ_t, p~x) be a self-similar Markov process on (0,∞) with non-decreasing path. The exact Hausdorff and Packing measure functions of the image X([0,t] ) are obtained.
