具有大拉普拉斯特征值的平面环面,维数可达8

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED
C. Kao, B. Osting, J. C. Turner
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引用次数: 0

摘要

我们考虑在固定体积的平坦$d$维环面上最大化$k$ -拉普拉斯特征值$\lambda_{k}$的优化问题。对于$k=1$,这个问题等价于最密集晶格球填充问题。对于较大的$k$,这相当于寻找具有最长$k$ -最短晶格向量的$d$维(对偶)晶格的np困难问题。作为大量计算的结果,对于$d \leq 8$,我们得到一个平面环面序列$T_{k,d}$,每个卷一,使得$T_{k,d}$的$k$ -第拉普拉斯特征值非常大;对于每个(有限)$k$, $k$ -th特征值超过($k\to \infty$渐近)Weyl定律中的值,根据维度在1.54和2.01之间。推导了$T_{k,d}$的平稳性条件并进行了数值验证,我们将环面退化描述为$k \to \infty$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Flat tori with large Laplacian eigenvalues in dimensions up to eight
We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $\lambda_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d \leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k \to \infty$.
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来源期刊
CiteScore
2.20
自引率
0.00%
发文量
19
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