多重胞元和规则多重分形

以超级分形纤维和双重分形纤维的研究结果为基础,达成了如下目标:首先,归纳、抽象出了多重胞元概念;其次,基于多重胞元概念,证实:具有严格自相似性的规则多重分形,不仅是可构造的,而且其构造模式具有普遍性;再者,通过分析构造模式,发现:任何规则多重分形,都可以在多重胞元意义下,被精确地等价成具有多重精细结构的广义单重分形．而基于这种等价性,单重分形维数公式就能够推广至规则多重分形维数公式,单重分形几何就能够推广至规则多重分形几何;最后,借助规则多重分形,构造了几种黄金分形．     

Designing Fractal Nanostructured Biointerfaces for Biomedical Applications

Fractal structures in nature offer a unique “fractal contact mode” that guarantees the efficient working of an organism with an optimized style. Fractal nanostructured biointerfaces have shown great potential for the ultrasensitive detection of disease‐relevant biomarkers from small biomolecules on the nanoscale to cancer cells on the microscale. This review will present the advantages of fractal nanostructures, the basic concept of designing fractal nanostructured biointerfaces, and their biomedical applications for the ultrasensitive detection of various disease‐relevant biomarkers, such microRNA, cancer antigen 125, and breast cancer cells, from unpurified cell lysates and the blood of patients.     

On singularity and fine spectral structure of random continued fractions

We study properties of the distribution of a random variable of the continued fraction form where are independent and not necessarily identically distributed random variables. We prove the singularity of and study the fine spectral structure of such measures.     

Uniqueness of eigenforms on fractals – II

I give an explicitly verifiable necessary and sufficient condition for the uniqueness of the eigenform on finitely ramified fractals, once an eigenform is known. This improves the results of my previous paper [16], where I gave some necessary and some sufficient conditions, and with a relatively mild additional requirement on the known eigenform. The result of this paper can be interpreted as a uniqueness result for self‐similar energies on finitely ramified fractals.     

相伴于随机自相似分形的随机测度的强测度的收敛性

构造了随机自相似分形及其上的记忆函数,并得出了有关结论,在此基础上,我们可以定义一个随机概率测度dΦn(τ)=Kn(τ)dτ,Φn(τ)弱收敛于Φ,进一步可得到强测度序列Ψn(·)=EΦn(·),则{Ψn}弱收敛于Ψ=EΦ.     

液态In-80wt%Sn合金结构的分形分析

在X射线衍射实验的基础上，把分形理论用于分析In-80wt%Sn在300℃时的液态结构，发现液态结构是一种多度域分形结构，并提出液态结构多度域分形模型.通过分形分析，液态合金的整体结构被清晰地呈现出来.推测多度域两分形之间过渡区的大小和液态合金性质有关，过渡区的变化可能反映了液态合金性质的变化. 关键词： 液态合金结构 分形理论 多度域分形     

Monte Carlo study of the XY-model on Sierpinski gasket

We have performed a Monte Carlo study of the classical XY-model on two-dimensional Sierpinski gaskets (SGs) of several cluster sizes. From the dependence of the helicity modulus on the cluster size we conclude that there is no phase transition in this system at a finite temperature. This is in agreement with previous findings for the harmonic approximation to the XY-model on SG and is analogous to the absence of finite-temperature phase transition for the Ising model on fractals with a finite order of ramification.     

Asymptotic stability of discrete cocycles

We introduce the notion of asymptotic stability of sequences of multifunctions associated with discrete cocycles. Some sufficient conditions for existence of attracting sets are given. The use of the topological (Kuratowski's) limits, as less complicated as commonly used Hausdorff metric, let us to weaken many standard assumptions. We show that in considered case existence of attractor is a property of a cocycle mapping itself and does not depend on properties of a parameter nor a state space. The obtained results generalize earlier on iterated function systems and can be applied for non-autonomous as well as random dynamical systems.