Muckenhoupt weights with singularities on closed lower dimensional sets in spaces of homogeneous type

We give sufficient conditions on a real number β and on a closed set F   in a general space of homogeneous type (X,d,μ)$(X,d,\mu )$ in such a way that μ(B(x,d(x,F)))^β$\mu {(B(x,d(x,F)))}^{\beta }$ becomes a Muckenhoupt weight. In order to prove our result, we modify the underlying space so that it becomes 1-Ahlfors regular.     

How Do Cultural Classes Emerge from Assimilation and Distinction? An Extension of the Cucker-Smale Flocking Model

When cultural tastes are not neutral but hierarchically matched to social status, people assimilate themselves to higher status by consuming cultural goods while distinguishing themselves from lower status by developing new tastes. Extending the Cucker-Smale model for mutual influence among agents, we examine when and how many cultural classes emerge from continuous distributions of tastes and what conditions those classes satisfy, through the assimilation-distinction mechanism. We simulate the models with different initial distributions of tastes (uniform, normal, and chi-square), given various ranges of 2 parameters: (a) the strength and (b) the range of distinction relative to assimilation. Tastes are flocking and cultural classes emerge when the range of assimilation is much larger than that of distinction. The number of classes increases with the strength of distinction, whereas the distance between classes equals the range of distinction. Some properties of emergent classes are mathematically proved. First, in a two-class system, the stronger distinction, the larger the upper class. Second, in a three-class system, the middle class is necessarily larger than the lower class and likely larger than the upper class. Third, a 3-class system cannot emerge if distinction is weaker than assimilation. These properties are universal and do not depend on the initial distribution of cultural tastes. This independence predicts homogeneous cultural classes emerging across different social conditions. Also, the cultural middle class as the largest group may explain why subjective class consciousness is often higher than objective position. Unless assimilating efforts can reach an infinite range, there emerges a cultural outcast at the lowest end of the cultural hierarchy.     

不确定信息处理及在归纳学习中的应用

信息的不确定性是由于模糊性、随机性、不完全性、不精确性等因素造成的,不确定性信息在现实世界中广泛存在.关于这方面的研究主要集中在模糊性这种不确定性上.我们课题组在此领域的研究已经持续20余年,概括起来讲,早期的研究兴趣在于模糊(非可加)测度与积分的理论建立,主要研究了传统的可加测度与积分理论到非可加情况的推广;尔后的兴趣转向为如何用数值方法确定非可加测度;近期的研究兴趣在于从已有数据中获取模糊控制规则以及具有模糊表示的归纳学习问题.近10年的研究曾得到了多项基金项目资助,出版专著3部,发表学术论文100余篇,30余篇被SCI收录,20余篇被EI收录,省级以上奖励3项.以下是我们近期研究工作的一个总结,主要包括模糊决策树归纳;基于模糊信息的决策表简化;产生模糊规则的扩张矩阵算法;模糊归纳学习在其他领域中的应用;基于示例学习的模糊控制等.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

含改性松香的丁二醇二缩水甘油醚固化反应动力学研究

设计了含改性松香无苯环的环氧固化体系和无松香的对照体系,分别是无松香体系Ⅰ：丁二醇二缩水甘油醚（BDGE）和甲基六氢苯酐（MeHHPA）;无规体系Ⅱ：丙烯酸松香（AR）、BDGE和MeHHPA;有规体系Ⅲ：丙烯酸松香基环氧树脂（ARE,由AR和BDGE预聚而来）和MeHHPA。对三体系的固化反应应用动态示差扫描量热仪（DSC）进行了研究,利用KAS法求得不同转化率下的表观活化能,通过整个反应过程反应活化能的变化,得到了三体系反应的内在机理,结果发现松香基的引入及引入顺序都对固化反应产生影响。     

分数阶模糊时滞神经网络模型解的存在唯一性和有限时间稳定性

本文介绍了一类分数阶模糊时滞神经网络模型.利用压缩映射原理,讨论了带时滞的分数阶神经网络模型解的存在性和唯一性,并根据Gronwall不等式结合分数阶微分方程的性质,证明了分数阶神经网络模型平衡点的有限时间稳定性,给出了有限时间稳定性的判断准则.最后,给出数值仿真说明了理论结果的正确性.     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .