a uniform rotation, i.e., $\omega _{out}\approx \omega _{in}$ .

From the point of view of quantum mechanics, the low-lying states are mainly dominated by the BFs belonging to the LLL. In these BFs, all the single-particle state $\phi _{mk}$ have  $k=0$ and angular momentum $%
l=m-k=m $. For each $\phi _{mk}$ , the angular velocity can be defined as $%
\omega =<l>/(m^{\ast }<r^{2}>)$, which is proportional to $\frac{l}{l+1}$ if $k=0$.  Evidently, $\omega $ is close to a constant unless $l$ is very small. Thus , for the BFs of the LLL, all the electrons rotate with similar angular velocities, and we have the uniform rotation $\omega _{out}\approx
\omega _{in}$ .

\(iii) [**Symmetry constraints and the  favorable structures**]{}.

It has been found that[* inherent nodal surfaces are imposed in wave functions by symmetry, thereby the structures of quantum states are seriously affected*]{}.$^{12,19-21}$ In the case of 2-dimensional polarized quantum dots , it was found that a wave function would be zero when the electrons locate at the vertexes of a regular N-side polygon if $L\neq $N$%
(j+\frac{1+(-1)^{N}}{4})$ , where $j$ is an integer$^{6,11,22}$. This constraint can be generalized to the core-ring structures. Let the ring has an angular momentum $l_{out}$ , 