Fractals and generalized inner products

The generation of fractals by a generalization of the matrix product is described. These matrix products allow visualization of additional structure of the fractals as well as producing fractals of new types.     

Calculus on fractals based upon local fields

The main contents in this note are: 1. introduction: 2. locally compact groups and local fields: 3. calculus on fractals based upon local fields: 4. fractional calculus and fractals: 5. fractal function spaces and PDE on fractals.     

Calculus on fractals based upon local fields In memory of Founding Editor Professor M. T. Cheng with great respect and deep sorrow

The main contents in this note are: 1. introduction: 2. locally compact groups and local fields: 3. calculus on fractals based upon local fields: 4. fractional calculus and fractals: 5. fractal function spaces and PDE on fractals.     

随机自仿射集的Hausdorff维数估计

定义了一类广泛的随机自仿射集,得到了此类集合的Ｈａｕｓｄｏｒｆｆ维数估计．此前的随机自相似（包括Ｇｒａｆ,Ｍａｕｌｄｉｎ与Ｆａｌｃｏｎｅｒ等定义的随机自相似情形）和Ｆａｌｃｏｎｅｒ定义的（严格）自仿射以及作者定义的μ 统计自仿射情形均成为该文结果的特例．     

相应于随机自相似分形的记忆函数和分数次积分

For a physics system which exhibits memory, if memory is preserved only at points of random self-similar fractals, we define random memory functions and give the connection between the expectation of flux and the fractional integral. In particular, when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al. .     

Homogenization of quadratic forms on fractals

We define a notion of homogenization for quadratic energies defined on fractals structures and present, without proofs, a result of homogenization for fractals having special symmetry properties.     

Weak Uncertainty Principles on Fractals

We use the analytic tools such as the energy, and the Laplacians defined by Kigami for a class of post-critically finite (pcf) fractals which includes the Sierpinski gasket (SG), to establish some uncertainty relations for functions defined on these fractals. Although the existence of localized eigenfunctions on some of these fractals precludes an uncertainty principle in the vein of Heisenberg’s inequality, we prove in this article that a function that is localized in space must have high energy, and hence have high frequency components. We also extend our result to functions defined on products of pcf fractals, thereby obtaining an uncertainty principle on a particular type of non-pcf fractal.     

一类Moran测度的Lq-谱

此文考虑了一类Moran分形，在其生成过程中每一阶的压缩比及压缩比的个数可以是不相同的，证明了支撑在此类Moran分形上的Moran测度的L^q-谱的存在性．     

齐次集及其拟Lipschitz等价

本文定义并研究一类齐次分形,该类分形包含所有的(拟)Ahlfors-David正则集和许多非正则的Moran集,这里如果一个分形的Hausdorff维数与packing维数不相等,则称它是非正则的.对于这类齐次分形,本文得出它们的分形维数,并且给出在适当分离条件下两个齐次分形拟Lipschitz等价的充要条件.随后,本文将这些结果应用到非正则的Moran集上.     

Multivalued fractals in <Emphasis Type="Italic">b</Emphasis>-metric spaces

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.     

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.     

Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.     

A stochastic process for fractal design

A special family of stochastic processes which allow the design of fractals is considered. We show that a process with arbitrary but continuous given functions of variance and mean value can be constructed. The paths of the process are fractals. They are computed by a very simple and fast algorithm.     

函数系矩阵生成的分形完备簇的豪斯夺夫维数和盒维数

本文给出了函数系矩阵的定义及运算,用函数系矩阵构造了分形完备簇,并确定了相似分形完备簇在开集条件下的Hausdorff维数和Box维数.     

Heat Kernel Estimates and Homogenization for Asymptotically Lower Dimensional Processes on Some Nested Fractals

We consider the class of diffusions on fractals first constructed in [12] on the Sierpinski and abc gaskets. We give an alternative construction of the diffusion process using Dirichlet forms and extend the class of fractals considered to some nested fractals. We use the Dirichlet form to deduce Nash inequalities which give upper bounds on the short and long time behaviour of the transition density of the diffusion process. For short times, even though the diffusion lives mainly on a lower dimensional subset of the fractal, the heat flows slowly. For the long time scales the diffusion has a homogenization property in that rescalings converge to the Brownian motion on the fractal.     

Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.     

有限义性质下一类广义MORAN集的重分形分解

本文在一类广义Moran集上定义了一种弱分离条件－有限交性质,并证明了重分形分解在广泛的一类分形上仍然成立。本文部分的推广了Cawley和Manlain的结果。     

Scaling of mathematical fractals and box-counting quasi-measure

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.     

Uniqueness of eigenforms on fractals

I give some rather general criteria for the uniqueness of the eigenform on finitely ramified fractals. In particular, I characterize the uniqueness in the case of four vertices with some additional condition, and give some examples of non‐tree symmetric fractals where the eigenform is not unique.     

格点巢分形上的渗流模型

该文研究了一类格点分形图 (格点巢分形 )上的渗流模型 ,证明了该模型没有临界现象 ,进一步给出一个指数衰减律 .同时 ,指出一般有限分岔图上的渗流模型没有临界现象