论文标题
HEUNB功能的应用
An application of the HeunB function
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。
How does the inclusion of the gravitational potential alter an otherwise exact quantum mechanical result? This question motivates this report, with the answer determined from an edited version of problem #12 on p.273 of Ref.1. To elaborate, we begin with the Hamiltonian associated with the system of two masses in the problem obeying Hooke's law and vibrating about their equilibrium positions in one dimension; the Schrodinger equation for the reduced mass is then solved to obtain the parabolic cylinder functions as eigenfunctions and the eigenvalues of the reduced Hamiltonian are calculated exactly. Parenthetically,the quantum mechanics of a bounded linear harmonic oscillator was perhaps first studied by Auluck and Kothari[2]. The introduction of the gravitational potential in the aforesaid Schrodinger equation alters the eigenfunctions to the biconfluent HeunB function[3]; and the eigenvalues are the determined from a recent series expansion[4] in terms of the Hermite functions for the solution of the differential equation whose exact solution is the aforesaid HeunB function.
