论文标题
部分可观测时空混沌系统的无模型预测
Asymptotic Betti numbers for hard squares in the homological liquid regime
论文作者
论文摘要
我们研究$ n $ n $ punge的配置空间$ c（n; p，q）$ $ p $ by $ q $矩形。我们的目标是估算大型$ n $，$ j $，$ p $和$ q $的贝蒂数字。我们考虑区域范围坐标的序列，其中$（\ frac {n} {pq}，\ frac {j} {pq} {pq}）$收敛为$ n $，$ j $，$ j $，$ p $和$ q $ off infinity。对于在$（x，y）$ - 飞机中收敛到“可行区域”中一个点的每个序列，我们表明，贝蒂数的阶乘增长率与循环增长率相同。这意味着（1）Betti数字大于飞机中$ n $订购点的配置空间，其阶乘增长率为$ j！$，（2）最终在同源液体状态下。
We study configuration spaces $C(n; p, q)$ of $n$ ordered unit squares in a $p$ by $q$ rectangle. Our goal is to estimate the Betti numbers for large $n$, $j$, $p$, and $q$. We consider sequences of area-normalized coordinates, where $(\frac{n}{pq}, \frac{j}{pq})$ converges as $n$, $j$, $p$, and $q$ approach infinity. For every sequence that converges to a point in the "feasible region" in the $(x,y)$-plane, we show that the factorial growth rate of the Betti numbers is the same as the factorial growth rate of $n!$. This implies that (1) the Betti numbers are vastly larger than for the configuration space of $n$ ordered points in the plane, which have the factorial growth rate of $j!$, and (2) every point in the feasible region is eventually in the homological liquid regime.
