On Partitioning Colored Points

P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can be separated along the colors. In this paper, we show a more colorful theorem

Design of resonant microcavities: application to optical gyroscopes

We study theoretically and numerically the effect of rotation on resonant frequencies of microcavities in a rotating frame of reference. Cavity rotation causes the shifts of the resonant frequencies proportional to the rotation rate if it is larger than a certain value. Below the value, a region of rotation rate exists where there is no resulting the frequency shifts proportional to the rotation rate. We show that designing cavity symmetry as $C_{nv}$ ($n\ge 3$) can eliminate this region.Comment: 10 page