In this Letter we discuss the classical three-dimensional XY model whose nearest-neighbor interaction is a mixture of $cos({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (ferromagnetic) and $cos2({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (nematic). This dual to theory with integer half-integer vortices. While both types vortices interact noncompact $...

let e be the ellipse with major and minor radii a and b respectively, and pbe its perimeter, then p = lim 4 tan(p/n)(a + b + 2) σ a2 cos2 (2k-2)pi/n+ sin2 (2k-2)pi/n; where n = 2m. so without considering the limit, it gives a reasonable approxi-mation for p, it means that we can choose n large enough such that the amountof error be less than any given small number. on the other hand, the formul...