论文标题
部分可观测时空混沌系统的无模型预测
On sums of $k$-th powers with almost equal primes
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。
For "almost all" sufficiently large $N,$ satisfying necessary congruence conditions and $k\geq 2$, we show that there is an {\bf asymptotic formula} for the number of solutions of the equation \begin{align*} \begin{split} &N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k}, \\ &\left|p_{i}-( N/s)^{1/k}\right|\leq (N/s)^{θ/k},\ (1\leq i\leq s) \end{split} \end{align*} with \begin{align*} s\geq \frac{k(k+1)}{2}+1\ \ \textup{and}\ \ θ\geq {\bf 2/3}+\varepsilon. \end{align*} This enlarges the effective range of $s$ for which can be obtained by the method of Mätomaki and Xuancheng Shao \cite{MS}. [Discorrelation between primes in short intervals and polynomial phase, Int. Math. Res. Not. IMRN 2021, no. 16, 12330-12355.] The idea is to avoid using the exponential sums (1.2) and Vinogradov mean value theorems in Lemma 2.4 simultaneously. And the main new ingredient is from Kumchev and Liu \cite{KL} (see Lemma 2.2).
