We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statist...We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.展开更多
In this paper, authors compute the Packing dimension of statistically selfsimilar sets and obtaine the dimension and dimension distribution of statistically self-similar measure.
In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statist...In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statistical contraction operators. Now we will introduce some things about the probability basis and fractal properties of fractals in the last class. The probability basis contains (1) the convergence and measurability of a random recursive setK(ω) as a random element, (2) martingals property. The fractal properties include (3) the character of various similarity, (4) the separability property, (5) the support and zero-one law of distributionP k =P·K ?1, (6) the Hausdorff dimension and Hausdorff exact measure function.展开更多
I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is the...I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.展开更多
Let K be a statistically self-similar set defined by Graf.In this paper,we construct a random measure ρ which is supported by K and study the multifractal decomposition for K with p. Under such a decomposition,we obt...Let K be a statistically self-similar set defined by Graf.In this paper,we construct a random measure ρ which is supported by K and study the multifractal decomposition for K with p. Under such a decomposition,we obtain the expression of the spectrum function f(α).展开更多
Self-similar sets（SSS） are the most important class of Fractals and play an important role in the studies in Fractal. In this note, we introduce self-similar-like sets（SSLS） which generalize self-similar sets,we w...Self-similar sets（SSS） are the most important class of Fractals and play an important role in the studies in Fractal. In this note, we introduce self-similar-like sets（SSLS） which generalize self-similar sets,we will show some properties of SSLS which distinguish essentially that of SSS, and we will give also some applications of SSLS.展开更多
基金the National Natural Science Foundation of China
文摘We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.
文摘In this paper, authors compute the Packing dimension of statistically selfsimilar sets and obtaine the dimension and dimension distribution of statistically self-similar measure.
文摘In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statistical contraction operators. Now we will introduce some things about the probability basis and fractal properties of fractals in the last class. The probability basis contains (1) the convergence and measurability of a random recursive setK(ω) as a random element, (2) martingals property. The fractal properties include (3) the character of various similarity, (4) the separability property, (5) the support and zero-one law of distributionP k =P·K ?1, (6) the Hausdorff dimension and Hausdorff exact measure function.
基金Project supported by the National Natural Science Foundation of China the Doctoral Progamme Foundation of China and the Foundation of Wuhan University.
文摘I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.
基金Research supported by National Natural Science Foundation of China (No.19671066)
文摘Let K be a statistically self-similar set defined by Graf.In this paper,we construct a random measure ρ which is supported by K and study the multifractal decomposition for K with p. Under such a decomposition,we obtain the expression of the spectrum function f(α).
基金Supported by the National Natural Science Foundation of China(No.61071066)
文摘Self-similar sets（SSS） are the most important class of Fractals and play an important role in the studies in Fractal. In this note, we introduce self-similar-like sets（SSLS） which generalize self-similar sets,we will show some properties of SSLS which distinguish essentially that of SSS, and we will give also some applications of SSLS.
