论文标题
laplacian基质的伪变种：其痕迹的渐近行为
The pseudoinverse of the Laplacian matrix: Asymptotic behavior of its trace
论文作者
论文摘要
在本文中，我们关注\ [ \ operatotorname {tr}（\ Mathcal {l}^+_ {\ rm sq}） = \ frac {1} {4} \ sum_ {j，k = 0 \ atop（j，k）\ neq（0,0）}^{n-1} \ frac {1} {1- \ frac {1} \ frac {2πk} {n} \ big）}，\] \] laplacian矩阵的伪内词的跟踪与方形晶格相关，如$ n \ to \ infty $。我们在以前的论文中开发的此类总和的方法取决于泰勒近似值的使用。结果表明，误差项取决于使用的泰勒多项式是否为二维程度或更高。在这里，我们对泰勒多项式的第四级正方形晶格执行此操作，从而获得了带有改进的误差项的结果，这也许是最精确的人所希望的。
In this paper we are concerned with the asymptotic behavior of \[ \operatorname{tr}(\mathcal{L}^+_{\rm sq}) = \frac{1}{4} \sum_{j,k=0 \atop (j,k) \neq (0,0)}^{n-1} \frac{1}{1-\frac{1}{2} \big( \cos \frac{2πj}{n} + \cos \frac{2πk}{n} \big)}, \] the trace of the pseudoinverse of the Laplacian matrix related with the square lattice, as $n \to \infty$. The method we developed for such sums in former papers depends on the use of Taylor approximations for the summands. It was shown that the error term depends on whether the Taylor polynomial used is of degree two or higher. Here we carry this out for the square lattice with a fourth degree Taylor polynomial and thereby obtain a result with an improved error term which is perhaps the most precise one can hope for.
