THERMODYNAMIQUE DES ENSEMBLES DE CANTOR AUTOSIMILAIRES（THERMODYNAMICS OF SELF-SIMILAR CANTOR SETS）

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Asymptotic dimension of finitely presented groups

We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than . It follows that a finitely presented group of asymptotic dimension is virtually free.

     

一维碳纳米管/二维二硫化钼混合维度异质结的原位制备及其电荷转移性能

一维(1D)材料与二维(2D)材料的结合可形成独特的混合维度异质结,其在继承2D/2D范德瓦尔斯异质结的独特物性之外,还具有丰富的堆叠构型,为进一步调控异质结的结构及性能提供了新的可操控自由度。p型1D单壁碳纳米管(SWCNT)与n型2D二硫化钼(MoS₂)的结合,为调控异质结的能带结构及器件性能提供了丰富的选择。本文直接在高密度、手性窄分布的SWCNT定向阵列及无序薄膜表面原位生长MoS₂,制备出高质量1D SWCNT/2D MoS₂混合维度异质结。深入分析形核点的表面形貌与结构,提出了“吸附-扩散-吸附”生长机制,用于解释混合维度异质结的生长。利用拉曼光谱分析,证实SWCNT与MoS₂间存在显著的电荷转移作用,载流子可在界面处快速传输,为后续基于此类1D/2D异质结的新型电子及光电器件的设计与制备提供了新思路。     

Bayesian Dimensionality Reduction With PCA Using Penalized Semi-Integrated Likelihood

We discuss the problem of estimating the number of principal components in principal components analysis (PCA). Despite the importance of the problem and the multitude of solutions proposed in literature, it comes as a surprise that there does not exist a coherent asymptotic framework, which would justify different approaches depending on the actual size of the dataset. In this article, we address this issue by presenting an approximate Bayesian approach based on Laplace approximation and introducing a general method of developing criteria for model selection, called PEnalized SEmi-integrated Likelihood (PESEL). Our general framework encompasses a variety of existing approaches based on probabilistic models, like the Bayesian Information Criterion for Probabilistic PCA (PPCA), and enables the construction of new criteria, depending on the size of the dataset at hand and additional prior information. Specifically, we apply PESEL to derive two new criteria for datasets where the number of variables substantially exceeds the number of observations, which is out of the scope of currently existing approaches. We also report results of extensive simulation studies and real data analysis, which illustrate the desirable properties of our proposed criteria as compared to state-of-the-art methods and very recent proposals. Specifically, these simulations show that PESEL-based criteria can be quite robust against deviations from the assumptions of a probabilistic model. Selected PESEL-based criteria for the estimation of the number of principal components are implemented in the R package pesel, which is available on github (https://github.com/psobczyk/pesel). Supplementary material for this article, with additional simulation results, is available online. The code to reproduce all simulations is available at https://github.com/psobczyk/pesel_simulations.     

Dimension of Slices Through Fractals with Initial Cubic Pattern

In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R~n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by "multirules" take the value in Marstrand's theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure L~m. When μV《 L~m, the connection of the local dimension ofμVand the box dimension of slices is given.     

On Dimension and Metric Properties of Trajectory Attractors

We study metric properties of trajectory attractors for infinite-dimensional dissipative systems. Under natural conditions we show that in the appropriate topology the functional dimension of this attractor is not greater than 1 and the metric order is 0. We also prove that every finite (in time) “piece” of the trajectory attractor has finite fractal dimension. As examples we consider a reaction-diffusion system, the 2D Navier-Stokes equation and also 3D Navier-Stokes equation under an additional regularity assumption concerning the corresponding trajectory attractor which is valid in the case of thin domains 2000 Mathematics Subject Classification: 37C45; 37L30.     

Transitive closure and betweenness relations

Indistinguishability operators fuzzify the concept of equivalence relation and have been proved a useful tool in theoretical studies as well as in different applications such as fuzzy control or approximate reasoning. One interesting problem is their construction. There are different ways depending on how the data are given and on their future use. In this paper, the length of an indistinguishability operator is defined and it is used to relate its generation via max-T product and via the representation theorem when T is an Archimedean t-norm. The link is obtained taking into account that indistinguishability operators generate betweenness relations. The study is also extended to decomposable operators.     

交错跳跃函数之例外集的Hausdorff维数

在作者本人提出的交错跳跃函数及其Cantor型导数的基础上,得到不存在Cantor型导数的点的全体（即例外集）的Hausdorff维数。     

关于修改的盒维数的一个结果

首先证明了修改的下盒维数的乘积公式,继而给出了R上一类随机分形集的修改的盒维数的维数性质。     

Sobolev spaces, dimension, and random series

We investigate dimension-increasing properties of maps in Sobolev spaces; we obtain sharp results with a random process somewhat like Brownian motion.