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1. CJM 2011 (vol 64 pp. 24)
Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves We consider the problem of minimizing the energy of $N$ points
repelling each other on curves in $\mathbb{R}^d$ with the potential
$|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ is
the Euclidean norm. For a sufficiently smooth, simple, closed,
regular curve, we find the next order term in the asymptotics of the
minimal $s$-energy. On our way, we also prove that at
least for $s\geq 2$, the minimal pairwise distance in optimal configurations
asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length
of the curve.
Keywords:minimal discrete Riesz energy, lower order term, power law potential, separation radius Categories:31C20, 65D17 |
2. CJM 2004 (vol 56 pp. 529)
Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$ We consider the $s$-energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\|z_{i,n}-z_{j,n}\|;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite one-dimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{-s} ,& \mbox{if } s>0, \\
-\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$-energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.
Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Best-packing on curves Categories:52A40, 31C20 |