function is plotted in Fig. \[fig\_norstrfun\]. The local scaling dimension is derived from this normalized strength function.

Behavior of the local scaling dimension
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![ Partition function (a) and local scaling dimension (b) for the IS GQR in $^{208}$Pb, and those in $^{40}$Ca are also shown at (c) and (d). Curves in each figure correspond to $m=2$ - 5 from upper to lower. Dotted curves in (b) and (d) represent $D_2(\epsilon)$ for the GOE. []{data-label="fig_pat_ldim"}](fig7.eps){width="9.2cm"}

Figure \[fig\_pat\_ldim\] (a) and (b) represent the partition function and the local scaling dimension, respectively, of IS GQR in $^{208}$Pb. The horizontal axes in both figures represent the bin width $\epsilon$ of energy in unit of $d$, where $d$ represents the average level spacing over the energy range 9.9 - 13.1 MeV ($d=0.96$ keV). The partition function clearly deviates from the linear relation in the log-log plot. This means that for the GQR strength function the self-similar property does not hold. We can also see a more detailed structure in the figure of the local scaling dimension. At the smallest energy scale $\epsilon\simeq d$, the value of the local scaling dimension is small, $D_2\simeq 0.35$, which means that the fluctuation is very large at small 