Packing dimension of the range of a Lévy process

Let denote a Lévy process in with exponent . Taylor (1986) proved that the packing dimension of the range is given by the index

We provide an alternative formulation of in terms of the Lévy exponent . Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.

     

Schur functors on QF-3 standardly stratified algebras

Let A be a QF-3 standardly stratified algebra and f be a Schur functor corresponding to some projective-injective faithful A-module, denoted by Ae. The main result of this paper is to prove that, if the dominant dimension of A is sufficiently large, then ] induces a full embedding from ￡（△） to eAe-mod which preserves Ext-groups up to certain degrees, where ￡（△） denotes the full subcategory of A-mod whose objects are filtered by standard A-modules. We check this criterion on some typical examples, quantized Schur algebras Sq（n,r） with n≥r and finite-dimensional algebras associated with the Bernstein-Gelfand-Gelfand category O of semisimple complex Lie algebras.     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.      

Designing Thiophene-Enriched Fully Conjugated 3D Covalent Organic Framework as Metal-Free Oxygen Reduction Catalyst for Hydrogen Fuel Cells

The application of three-dimensional (3D) covalent organic frameworks (COFs) in renewable energy fields is greatly limited due to their non-conjugated skeletons. Here, we design and successfully synthesize a thiophene-enriched fully conjugated 3D COF (BUCT-COF-11) through an all-thiophene-linked saddle-shaped building block (COThTh-CHO). The BUCT-COF-11 exhibits excellent semiconducting property with intrinsic metal-free oxygen reduction reaction (ORR) activity. Using the COF as cathode catalyst, the assembled anion-exchange membrane fuel cells (AEMFCs) exhibited a high peak power density up to 493 mW cm⁻². DFT calculations reveal that thiophene introduction in the COF not only improves the conductivity but also optimizes the electronic structure of the sample, which therefore boosts the ORR performance. This is the first report on the application of COFs as metal-free catalysts in fuel cells, demonstrating the great potential of fully conjugated 3D COFs as promising semiconductors in energy fields.     

基于分形理论的尖-板电极短空气隙放电现象研究

气体放电的发展过程非常复杂, 实验观测结果表明非均匀场短空气隙放电过程中经常产生随机并伴有分岔现象的放电通道. 由于这些放电现象的整体结构表现出一定的自相似性, 因此, 结合经典流注放电与分形理论而建立的介质击穿模型成为描述气体中放电通道分岔现象的一种有效的分析方法, 而放电通道的复杂程度可以用分形维数表征. 为了明确此模型参数中发展概率指数的取值, 本文对直流高压作用下的10 mm针板电极空气隙放电通道图像进行了分析、处理和分形计算, 并与理论模型的仿真结果对比. 由于空气隙主放电通道的亮度明显高于其他弱分支, 导致各放电通道的宽度各异, 需要对所拍摄图像进行灰度变换和边界识别处理, 最后运用盒维数法计算出分形维数. 研究结果表明在其他参数与实验条件相同条件下, 调整理论模型中的发展概率指数, 使得仿真结果与实验结果相吻合, 依据本文实验条件下和理论模型分析, 发展概率指数η 在0.04-0.05范围. 本文的研究印证了尖-板电极短空气隙放电复杂现象的可测性并提供了一种分析方法.     

具有特殊伸缩矩阵的Parseval框架小波集的结构

揭示具有特殊伸缩矩阵的Parseval框架小波集的丰富结构.借助于平移不变空间和维数函数,研究了具有特殊伸缩矩阵M的Parseval框架小波(M-PFW)、半正交M-PFW和MRA M-PFW的各种性质,探讨了M-PFW集合的各种子类,给出了这些子类的构造性算例.     

关于Gorenstein投射维数的一个注记

本文研究了Gorenstein投射维数的相关问题.利用经典同调维数的研究方法,给出了Gorenstein投射维数有限模的Gorenstein投射维数的一个刻画,并利用这一结果证明了Gorenstein完全环和Artin环的Gorenstein整体维数分别由各自的循环模和单模的Gorenstein投射维数来确定.这些结论丰富了Gorenstein同调代数理论.     

Minimal dimensions of faithful representations for abelian Lie superalgebras

This paper determines the minimal dimensions of faithful representations for abelian Lie superalgebras of finite dimensions over an algebraically closed field of characteristic zero. In particular, we also obtain the maximal dimensions of abelian subalgebras for the general linear Lie superalgebras.     

Adic finiteness: Bounding homology and applications

We prove the versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring R of prime characteristic is regular if and only if for some proper ideal 𝔟 the derived local cohomology complex RΓ_𝔟(R) has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.     

On Gorenstein Resolution Dimensions of Complexes

Let W be a self-orthogonal class of R-modules. We prove that W-Gorenstein res-olution dimension of a complex X is equivalent to the supremum of W-Gorenstein resolution dimension of modules Xi for all i∈Z.