AI 中文总结
本文通过生成函数和渐近分析，研究了哈密顿Sturm排列的计数，证明了其概率分布在大n时趋于高斯分布，并给出了均值和方差的渐近估计。
AI 中文摘要
我们的组合分析源于偏微分方程动力学 \begin{equation} \mathbf{u_t} = \mathbf{u_{xx}} + \mathbf{g}(\mathbf{u}),\qquad 0<\mathbf{x}<1, \end{equation} 在Neumann边界条件下解 $\mathbf{u}=\mathbf{u}(\mathbf{t},\mathbf{x}),\\ \mathbf{t}\geq 0$ 的行为。对于耗散非退化非线性项 $\mathbf{g}$，该PDE的全局吸引子 $\mathcal{A}=\mathcal{A}_\mathbf{g}$ 可通过其在边界 $\mathbf{x}=0,1$ 上的 $2n+1$ 个平衡点 $\mathbf{v}$ 的排序进行分类。我们将边界排序编码为哈密顿Sturm排列。名称“Sturm”指PDE解 $\mathbf{u}(\mathbf{t},\mathbf{x})$ 的节点性质。“哈密顿”指平衡点 $\mathbf{v}(\mathbf{x})$ 满足的二阶摆常微分方程：\begin{equation} 0 = \mathbf{v_{xx}} + \mathbf{g}(\mathbf{v})。 \end{equation} 我们确定了哈密顿Sturm排列计数 $a_n$ 的生成函数 $a(z)=\sum_n a_nz^n$。当 $n\rightarrow\infty$ 时，这提供了 $a_n$ 的显式渐近行为。我们将这些计数细化为 $a_n=\sum b_{rq}$，其中 $b_{rq}$ 计数具有 $2r+1$ 个空间齐次平衡点和 $2q$ 个空间非齐次平衡点（满足 $r+q=n$）的哈密顿Sturm排列。我们还确定了显式生成函数 $b(x,y)=\sum_{r,q} b_{rq}x^ry^q$。这意味着当 $n$ 很大时，概率 $p_{nr}=b_{rq}/a_n$（其中 $r+q=n$）渐近服从高斯分布。我们推导了均值和方差的渐近表达式，并给出了 $1/n$ 阶的误差估计。所有渐近结果均基于Flajolet和Sedgewick的工作。最后，我们给出数值示例，并讨论在周期边界条件 $\mathbf{x}\in\mathbb{S}^1=\mathbb{R}/2\mathbb{Z}$ 下非线性项 $\mathbf{g}(\mathbf{u},\mathbf{u_x})$ 的备注，其中会出现旋转波。
英文摘要
Our combinatorial analysis is motivated by the PDE dynamics \begin{equation} \mathbf{u_t} = \mathbf{u_{xx}} + \mathbf{g}(\mathbf{u}),\qquad 0<\mathbf{x}<1, \end{equation} of solutions $\mathbf{u}=\mathbf{u}(\mathbf{t},\mathbf{x}),\ \mathbf{t}\geq 0$, under Neumann boundary conditions. For dissipative nondegenerate nonlinearities $\mathbf{g}$, the global attractors $\mathcal{A}=\mathcal{A}_\mathbf{g}$ of the PDE can then be classified by the orderings of their $2n+1$ equilibria $\mathbf{v}$ at the boundaries $\mathbf{x}=0,1$. We encode the boundary orders as Hamiltonian Sturm permutations. The name ''Sturm'' refers to nodal properties of PDE solutions $\mathbf{u}(\mathbf{t},\mathbf{x})$. ''Hamiltonian'' refers to the second order pendulum ODE for equilibria $\mathbf{v}(\mathbf{x})$: \begin{equation} 0 = \mathbf{v_{xx}} + \mathbf{g}(\mathbf{v}). \end{equation} We determine the generating function $a(z)=\sum_n a_nz^n$ for the counts $a_n$ of Hamiltonian Sturm permutations. For $n\rightarrow\infty$, this provides explicit asymptotics of $a_n$. We refine these counts as $a_n=\sum b_{rq}$. Here $b_{rq}$ counts Hamiltonian Sturm permutations with $2r+1$ spatially homogeneous equilibria and $2q$ spatially non-homogeneous equilibria, such that $r+q=n$. We also determine the explicit generating function $b(x,y)=\sum_{r,q} b_{rq}x^ry^q$. This implies asymptotically Gaussian distributions of the probabilities $p_{nr}=b_{rq}/a_n$ with $r+q=n$, asymptotically for large $n$. We derive asymptotics for means and variances, with error estimates of order $1/n$. All asymptotics are based on work by Flajolet and Sedgewick. We conclude with numerical illustrations and remarks on nonlinearities $\mathbf{g}(\mathbf{u},\mathbf{u_x})$ under periodic boundary conditions $\mathbf{x}\in\mathbb{S}^1=\mathbb{R}/2\mathbb{Z}$, where rotating waves arise.
Comments39+ii pages, 5 figures, 7 tables
