Multiple solutions for a class of nonlinear elliptic equations on the Sierpinski gasket

This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of △u+ c(x)u = f(x, u), with zero Dirichlet boundary conditions on the Sierpihski gasket. Our existence results do not require any growth conditions of f(x,t) in t, in contrast to the classical theory of elliptic equations on smooth domains.     

一个广义Siperpinski地毯的Hausdorff测度

§ 1 .　Introduction　　Infractalgeometry ,thestudyingonthetheoryandcalculationofHausdorffdimensionhasmadegreatprogress.ButasfarasHausdorffmeasureisconcerned ,onlyafewresultsareknown .ZHOUZuo linghasobtainedtheexactvalueofHausdorffmeasureforaSierpinskicar pet(see [3]) .MADong kuihasdevelopedthetechniquesin [3](see [4 ]) .ButineverystepofconstructingtheSierpinskicarpetsin [4 ],theremaininglittlesquaressatisfytheconditionthatfourofthemareatthefourvertexanglesoftheirimmediatepredecessorrespe…     

Some results on the upper convex densities for the self-similar sets

In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439-3444].     

基于Sierpinski地毯结构的类分形光子晶体特性研究

将谢尔宾斯基(Sierpinski)地毯结构引入二维光子晶体,设计了一种具有规则分形结构特征的光子晶体-Sierpinski类分形光子晶体.采用时域有限差分法仿真分析了空气背景介质柱和介质背景空气孔结构Sierpinski类分形光子晶体的透射谱.结果表明,只有当TM波入射空气背景介质柱结构时,Sierpinski类分形...     

谢宾斯基分形结构声散射特性与视觉偏好评价*

声扩散界面设计应兼顾声学和美学需求。结合参数化设计方法,谢宾斯基分形的模块化结构具有灵活生成个性化界面的潜力。然而,针对分形结构声学特性与视觉效益尚缺乏综合的量化研究。本文根据谢宾斯基分形的构造规则提出了一种宽频的扩散结构设计方法,分别从声学性能与视觉偏好的角度,研究模块形状、模块高度和随机排布对性能的影响。采用边界元素法的数值仿真与实测验证、结合虚拟仿真技术的主观调查、以及多目标综合分析,探究分形设计要素的作用机制和最佳阈值范围。结果表明,与相同体量的传统二次残差扩散体相比,具有不同平面尺寸和高度模块的分形结构改善了扩散性能,特别是在1 kHz以下的中低频范围。此外,发现在室内使用中高复杂度（迭代三次）、低随机排布、谢宾斯基三角截锥分形结构,有助于提高居住者对环境的偏好度。     

Surface structures of equilibrium restricted curvature model on two fractal substrates

With the aim to probe the effects of the microscopic details of fractal substrates on the scaling of discrete growth models, the surface structures of the equilibrium restricted curvature(ERC) model on Sierpinski arrowhead and crab substrates are analyzed by means of Monte Carlo simulations. These two fractal substrates have the same fractal dimension df, but possess different dynamic exponents of random walk zrw. The results show that the surface structure of the ERC model on fractal substrates are related to not only the fractal dimension df, but also to the microscopic structures of the substrates expressed by the dynamic exponent of random walk zrw. The ERC model growing on the two substrates follows the well-known Family–Vicsek scaling law and satisfies the scaling relations 2α + df≈ z ≈ 2zrw. In addition, the values of the scaling exponents are in good agreement with the analytical prediction of the fractional Mullins–Herring equation.     

The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

Let be a one-dimensional space which contains a copy of a circle and let it not be semi-locally simply connected at any point on Then the fundamental group of cannot be embeddable into a free -product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

     

Markov functions of a time-changed recurrent diffusion

Let (X _t ,P _x ) be a recurrent diffusion on the state spaceE. A necessary and sufficient condition on the continuous functionu:ER is given so thatu is a Markov function for a time-changed diffusionX . It is shown that no nonconstant continuous real-valued function is Markov for a Brownian motion on the Sierpinski gasket.     

经典分形集测度上估的计算机搜索I——对典型例子Sierpinski垫片编码技术的剖析

分形集的Hausdorff测度是一个非线性科学的理论课题,至今结果甚少,即使对于一些生成很有规则的经典分形集亦是如此[3].Sierpinski垫片就是这样一个经典分形集,但因其预分形的图形尚未被研究者解析地认识,所以对其Hausdorff测度的研究进展很慢.     

S紧locale与S紧空间

本文给出Ｓ紧ｌｏｃａｌｅ的内在刻画,作为推论,得到了一般Ｓ紧空间的内在刻画