On planar self-similar sets with a dense set of rotations

$L^q$-谱与箱状图指导自仿射测度：闭形式，含旋转

$L^q$-spectra of box-like graph-directed self-affine measures: closed forms, with rotation

本文研究平面图指导自仿射测度的$L^q$-谱，推导出闭形式表达式，并给出关联的箱状图指导自仿射集的盒维数。同时回答了Fraser在2016年关于对角矩阵生成的平面自仿射测度的$L^q$-谱问题。

A density problem for Sobolev spaces on planar domains

We prove that for a bounded simply connected domain $Ω\subset \mathbb R^2$, the Sobolev space $W^{1,\,\infty}(Ω)$ is dense in $W^{1,\,p}(Ω)$ for any $1\le p<\infty$. Moreover, we show that if $Ω$ is Jordan, then $C^{\infty}(\mathbb R^2)$ is dense in $W^{1,\,p}(Ω)$ for $1\le p<\infty$.

Symmetric Regularization, Reduction and Blow-Up of the Planar Three-Body Problem

We carry out a sequence of coordinate changes for the planar three-body problem which successively eliminate the translation and rotation symmetries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regularization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while emphasizing the role of the shape sphere. By using homogeneous coordinates to desc…

The Assouad dimensions of projections of planar sets

We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension $s \in [0,2]$, then the projections have Assouad dimension at least $\min\{1,s\}$ almost surely. Compared to the famous analogue for Hausdorff dimension -- namely \emph{Marstrand's Projection Theorem} -- a striking difference is that the words `at least' cannot be dispensed with: in fact, for many planar self-similar sets of dimension $s < 1$, we prove …