occur. It implies two cases: (a) The inner electrons have their $l_{i}$ remaining to be compactly aligned but with $l_{1}=k$ , and therefore have a (N$_{in})_{h}$-N$_{out}$ hollow structure .  (b) The $%
l_{i}$  of the core are no longer aligned compactly, e.g., $l_{1}=0$ while $%
l_{2}=2$, etc. .

It was found that, when $L$ is not large (say, $L\leq 101$ ), the first-states have either an inert core or an excited compact core with $%
l_{1}=1$ , as shown in Table 2 . However, when $L$ is large, higher core excitation with $l_{1}>1$ will emerge as shown later.

Incidentally,  due to eq.(10), the compact alignment of the angular momenta also implies a compact alignment of radial positions. Thus, in the core-ring structures, the groups of inner and outer electrons may each compactly aligned. The associated  BF can be simply denoted as $%
\{l_{1}-l_{N_{in}},l_{N_{in}+1}-l_{N}\}$ (e.g., {1,2,3, 6,7,$\cdot \cdot
\cdot $,15} $\equiv ${1-3,6-15}.). This is called a two-bunched BF by Ruan$^{23}$ (The one-bunched BF {$l_{1}-l_{N}$} is a special case of two-bunched BF with $l_{N_{in}}=l_{N_{in}+1}-1$ ). It is straight forward to prove that, for a CFS of a $\nu \leq 1$ state,  among the BFs of the CFS, one and only one of 