Compactness and fractal dimensions of inhomogeneous continuum random trees

Topological conditions on inhomogeneous fractals in Martin boundary theory and their algorithmic testing

We want to consider Martin boundary theory applied to inhomogeneous fractals. This is under some conditions possible, but up to now it is not clear, how one can easily check, if a certain fractal fulfills those conditions. We want to simplify one condition and further develop a computer algorithm, which can check, if an attractor fulfills the condition.

Fractal percolation, porosity, and dimension

We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less than $\tfrac12-\varepsilon$, or the lower porosity is larger than $\varepsilon$. Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton-Watson process.

自仿海绵上分形渗透的Hausdorff维数与packing维数的变分原理

Variational principles for Hausdorff and packing dimensions of fractal percolation on self-affine sponges

本文建立了一类统计自仿海绵（包括Barański和Gatzouras-Lalley地毯与海绵的分形渗透集）的Hausdorff维数与packing维数的变分原理，通过计算非齐次Mandelbrot测度的维数并利用局部化数字频率构建有效覆盖，获得了维数的精确上界。

Quantifying inhomogeneity in fractal sets

An inhomogeneous fractal set is one which exhibits different scaling behaviour at different points. The Assouad dimension of a set is a quantity which finds the `most difficult location and scale' at which to cover the set and its difference from box dimension can be thought of as a first-level overall measure of how inhomogeneous the set is. For the next level of analysis, we develop a quantitative theory of inhomogeneity by considering the measure of the set of points around which the set exhibits a given level of inhomogeneity at a certain scale. For a set of examples, a family of $(\times …

Limits of multiplicative inhomogeneous random graphs and Lévy trees: The continuum graphs

Motivated by limits of critical inhomogeneous random graphs, we construct a family of sequences of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the Erdős--Rényi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model). At the discrete level, the construction relies on a new point of view on (discrete) inhomogeneous random graphs that involves an embedding into a Galton--Watson forest. The new representatio…

三维环面上分数阶随机Navier-Stokes方程的随机吸引子的正则性与分形维数

Regularity of random attractor and fractal dimension of fractional stochastic Navier-Stokes equations on three-dimensional torus

本文研究三维环面上分数阶随机Navier-Stokes方程，在特定噪声条件下证明了随机吸引子的存在性及其分形维数有限，通过构造有界吸收集和Lipschitz连续性实现。