The Tangent Measure Distribution of Self-Conformal Fractals

We show that the tangent measure distribution of a self-conformal measure exists at almost all points of the support of the measure. Moreover, we prove, that it is the same for almost all points.     

The Tangent Measure Distribution of Self-Conformal Fractals

We show that the tangent measure distribution of a self-conformal measure exists at almost all points of the support of the measure. Moreover, we prove, that it is the same for almost all points.     

Fractals and Bernstein polynomials

Self-similarity of Bernstein polynomials, embodied in their subdivision property is used for construction of an Iterative (hyperbolic) Function System (IFS) whose attractor is the graph of a given algebraic polynomial of arbitrary degree. It is shown that such IFS is of just-touching type, and that it is peculiar to algebraic polynomials. Such IFS is then applied to faster evaluation of Bézier curves and to introduce interactive free-form modeling component into fractal sets.     

MoranFractals的推广及其多重分形分解

本文给出Ｍｏｒａｎｆｒａｃｔａｌｓ的一个推广并且研究了其多重分形分解问题。     

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．     

Science Conference Presenters’ Images of Inquiry

Inquiry‐focused professional development and conceptions of inquiry held by eight professional development leaders were investigated within the context of a state science teacher conference. The prominent session format involved session leaders modeling classroom experiences. In all sessions, classroom inquiry was portrayed as a teacher‐guided activity with the primary goal being to increase motivation for engaging students in classroom inquiry. The leaders’ conceptualized inquiry primarily as a teaching approach with various goals, characteristics, and potential barriers. The findings of this study provide evidence of how inquiry, a prominent feature of science education reform, was portrayed in sessions at a conference sponsored by a state affiliate of the National Science Teachers Association and thought about by persons who led these sessions. The findings have implications for teacher learning from conference‐based professional development and its potential influence on science teacher thinking and practice.     

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.     

由表示系统生成的分形的维数

在这篇文章里,由Ｒｎ中点的表示系统所生成的自仿射集中,给出了自仿射集满足Ｍｏｒａｎ开集条件的一个新的判别方法和不满足开集条件的自相似集的Ｈａｕｓｄｏｒｆｆ维数的上界和下界,并根据两个实例估计出其上下界是相等的．     

Codes as Fractals and Noncommutative Spaces

We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes.     

Symmetric Intersections of Rauzy Fractals

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

Fractals via iterated functions and multifunctions

Fractals have wide applications in biology, computer graphics, quantum physics and several other areas of applied sciences (see, for instance [Daya Sagar BS, Rangarajan Govindan, Veneziano Daniele. Preface – fractals in geophysics. Chaos, Solitons & Fractals 2004;19:237–39; El Naschie MS. Young double-split experiment Heisenberg uncertainty principles and cantorian space-time. Chaos, Solitons & Fractals 1994;4(3):403–09; El Naschie MS. Quantum measurement, information, diffusion and cantorian geodesics. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995. p. 191–205; El Naschie MS. Iterated function systems, information and the two-slit experiment of quantum mechanics. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995. p. 185–9; El Naschie MS, Rossler OE, Prigogine I. Forward. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995; El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36; El Naschie MS. Fractal black holes and information. Chaos, Solitons & Fractals 2006;29:23–35; El Naschie MS. Superstring theory: what it cannot do but E-infinity could. Chaos, Solitons & Fractals 2006;29:65–8). Especially, the study of iterated functions has been found very useful in the theory of black holes, two-slit experiment in quantum mechanics (cf. El Naschie, as mentioned above). The intent of this paper is to give a brief account of recent developments of fractals arising from IFS. We also discuss iterated multifunctions.     

分形集的拟一致不连通性

一致不连通性在分形几何的研究中起重要作用.本文讨论了拟一致不连通性:对于一类由数列定义的集合,得到其(拟)一致不连通性成立的充要条件;还引入了一类缓变莫朗集,得到其拟一致不连通性成立的一个充分条件.     

Probability Space of Stochastic Fractals

We develop a general method for the construction of a probability structure on the space F of random sets in ℝ. For this purpose, by using the introduced notion of c-system, we prove a theorem on the unique extension of a finite measure from a c-system to the minimal c-algebra. The obtained structure of measurability enables one to determine probability distributions of the c-algebra of random events sufficient, e.g., for the so-called fractal dimensionality of random realizations to be considered as a measurable functional on F.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1467–1483, November, 2004.     

一类推广的随机分形的 Hausdorff维数

该文研究一类推广的${bf R}^{d}$中具有有限记忆的随机递归模型,引入了一个与该结构有关的函数$Psi(beta),betageq 0$,构造了一个随机测度$mu_omega$,证明了由该结构产生的随机集 $K(omega)$的Hausdorff维数是$alpha:=inf{beta:Psi(beta)leq1}$.