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Search: MSC category 31C20 ( Discrete potential theory and numerical methods )

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1. CJM 2011 (vol 64 pp. 24)

Borodachov, S. V.
 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 radiusCategories:31C20, 65D17

2. CJM 2004 (vol 56 pp. 529)

Martínez-Finkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.
 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 curvesCategories:52A40, 31C20