完全图上的并行Chip-Fireing：Devil's Staircase和Poincare旋转数

从旋转和刘维尔数的无理测度看魔鬼楼梯

A devil's staircase from rotations and irrationality measures for Liouville numbers

从Sturmian和Christoffel词出发，我们导出了一个严格递增函数$Δ:[0,\infty)\to\mathbb{R}$ .该函数在任意无理点处连续，在有理点处左连续但右不连续。此外，它假定有理数为代数整数，无理数为超越数。我们还看到，$Δ$的微分区分了实数的一些无理测度。

随机图与图极限上芯片燃烧的魔鬼阶梯

The devil's staircase for chip-firing on random graphs and on graphons

该研究将并行芯片燃烧推广到graphon，证明了Erdős–Rényi随机图上其活跃度图随顶点数增加收敛为魔鬼阶梯，还明确了graphon上活跃度的存在性条件。

Univoque bases of real numbers: local dimension, Devil's staircase and isolated points

Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal U(x)$ such that when $x$ passes the first critical value the set $\mathcal U(x)$ c…

满足$2|E|-|V|<|σ|<2|E|$的所有并行芯片点火游戏都具有周期2

All parallel chip-firing games with $2|E|-|V|<|σ|<2|E|$ have period $2$

该研究使用GPT-5.6-Sol证明了2024年Ji等人关于并行芯片点火游戏活动为$\tfrac12$时存在通用界的猜想，推广了魔鬼阶梯的中间层并统一了多类图的相关结果。

非稳定化的并行点火博弈

Non-Stabilizing Parallel Chip-Firing Games

本文研究了并行点火博弈的非稳定化条件，证明了在2|E|-|V|<|σ|<2|E|范围内博弈周期不为3或4，并扩展共轭配置方法证明了完全二部图上的相关猜想。