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The derivatives of the nearest neighbour scaling function

Nearest neighbour distances are assumed to scale as (see Section 3.2):
\begin{displaymath}
\ln {<r_{k}^{\gamma}>}^{1/\gamma} =
C + \frac{\ln(k) - \ln(N)} {D(\gamma)} + \ln G(k,\gamma,D(\gamma))
\end{displaymath} (A.1)

where
\begin{displaymath}
G(k,\gamma,D(\gamma)) = {\left[
\frac{ \Gamma(k+\gamma/D(\gamma)) } { \Gamma(k) k^{\gamma/D(\gamma)} }
\right]}^{1/\gamma}
\end{displaymath} (A.2)

For the estimation of the dimension function $D(\gamma)$ and the constant $C$, the derivatives of the nearest neighbour scaling function are required with respect to these parameters. We used, for the different fit methods (section 3.3)
N-method
:
For this method, $k$ is fixed and the scaling function is
\begin{displaymath}
F(\ln(N); D(\gamma), C) = C + \frac{ \ln(k) - \ln(N)}{D(\gamma)}
\end{displaymath} (A.3)

so
\begin{displaymath}
\frac{\partial F(\ln(N); D(\gamma),C)}{\partial D(\gamma)} =
\frac{\ln(N) - \ln(k)}{ {D^2(\gamma)}}
\end{displaymath} (A.4)

and
\begin{displaymath}
\frac{\partial F(\ln(N); D(\gamma),C)}{\partial C} = 1
\end{displaymath} (A.5)

k-method
:

\begin{displaymath}
F(\ln(k); D(\gamma), C) = C + \frac{ \ln(k) - \ln(N)}{D(\gamma)}
+ \ln G(k,\gamma,D(\gamma))
\end{displaymath} (A.6)

so
\begin{displaymath}
\frac{\partial F(\ln(k); D(\gamma),C)}{\partial D(\gamma)} =...
...{\ln(N) - \psi(k + \frac{\gamma}{D(\gamma)}) }{ {D(\gamma)}^2}
\end{displaymath} (A.7)

and
\begin{displaymath}
\frac{\partial F(\ln(k); D(\gamma),C)}{\partial C} = 1
\end{displaymath} (A.8)

Here
\begin{displaymath}
\psi(x) = \frac{ d \ln \Gamma(x) } { d x }
\end{displaymath} (A.9)

k(N)-method
:

\begin{displaymath}
F(\ln(\frac{k}{N}); D(\gamma), C) = C + \frac{ \ln(\frac{k}{N})}{D(\gamma)}
+ \ln G(k,\gamma,D(\gamma))
\end{displaymath} (A.10)

so
\begin{displaymath}
\frac{\partial F(\ln(\frac{k}{N}); D(\gamma),C)}{\partial D(...
...frac{\gamma}{D(\gamma)}) -
\ln(\frac{k}{N})) }{ {D^2(\gamma)}}
\end{displaymath} (A.11)

and
\begin{displaymath}
\frac{\partial F(\ln(\frac{k}{N}); D(\gamma),C)}{\partial C} = 1
\end{displaymath} (A.12)

The correction function is considered to be inappropriate when $k+\gamma/D(\gamma) \leq 0$; its implementation then returns the value 1. The derivatives are returned as zero under that condition. For the exact evaluation of the correction function, see [Olofsen, 1987,van de Water, 1987].


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Next: Information from outside the Up: The identification of strange Previous: CONCLUSIONS AND SUGGESTIONS   Contents
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