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I'm trying to understand this article: http://imgur.com/a/HfJoY but I'm unsure what $\alpha(N)$ means in this context? Is it the algebraic multiplicity, that's pretty much the only $\alpha$ I have ever seen, but how does this make sense here? Also what is $o(N)$

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I believe it means that there exists a function of $N$ called $\alpha$ such that "$\alpha(N)$ is little-o of $g(N)$", where $g(N)=N$ and the little-o notation is explained here. –  Stefan Hansen Jan 15 '13 at 15:07

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As in the definition of clustering, we want $\rho$ to converge to $0$ when $N\to \infty$, it's better to write it as $\rho_N$. By definition, $\rho_N\in[0,1)$. I didn't red the rest of the paper, but it probably helps to measure how $\rho_N$ behaves with respect to $1/N$, and that's the role played by the $\alpha_N$. Maybe it will be better writing $\rho_N\color{red}:=\frac{\alpha(N)}N$ in order to see it's defined like that.

As in clustering $\rho_N\to 0$, we can write $\alpha(N)$ as a product of $N$ with a function of $N$ converging to $0$.

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