论文标题
一致性子组的最佳独立生成系统$γ_0（p）$和$γ_0（p^2）$
Optimal independent generating system for the congruence subgroups $Γ_0(p)$ and $Γ_0(p^2)$
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Let $n$ be a prime or its square. We prove that the congruence subgroup $Γ_0(n)$ admits a free product decomposition into cyclic factors in such a way that the $(2,1)$-component of each cyclic generator is either $n$ or $0$, answering a conjecture of Kulkarni. We can also require that the Frobenius norm of each generator is less than $2n-1$. A crucial observation is that if $P$ denotes the convex hull of the extended Farey sequence of order $\lfloor \sqrt{n} \rfloor$ in the hyperbolic plane $\mathbb{H}^2$, then the projection $π: \mathbb{H}^2\to \mathbb{H}^2/Γ_0(n)$ is injective on the interior of $P$ and each connected component of $π(\mathbb{H}^2)\setminusπ(P)$ is either an order-three cone of area $π/3$ or an ideal triangle. Denoting by $m(Γ_0(n))$ the minimum of the largest denominator in the cusp set of $Q$ where $Q$ ranges over all possible special (fundamental) polygons for $Γ_0(n)$, we establish the inequality $ \lfloor \sqrt{n} \rfloor \le m(Γ_0(n))\le \lfloor \sqrt{4n/3} \rfloor$, and completely characterize the cases in which the bounds are achieved. We also prove analogous results when $n$ is the multiplication of two sufficiently close odd primes.
