Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations
Abstract In this paper we show that the index of a 1-reducible subgroup of the differential Galois group of an ordinary homogeneous linear differential equation L ( y ) = 0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L ( y ) = 0. For an irreducible third order equation we show that this degree belongs to {3,6,9,21,36}. When the Galois group is a finite primitive group, we reformulate and generalize work of L. Fuchs to show how to compute the minimal polynomial of a solution instead of the minimal polynomial of the logarithmic derivative of a solution. These results lead to an effective algorithm to compute Liouvillian solutions of second and third order linear differential equations.
Self organizing anisotropic nematic liquid crystals (LCs) induce self-assembly on quantum dots (QDs) to form one-dimensional chains along the nematic director. Spatial ordering of QDs, achieved in the nematic LC matrix, can be controlled in a preferred direction on application of electric fields, gaining better self-assembly. Once the field goes off, the LC+QD system relaxes back to the original state, revealing the intrinsic dynamics. Due to the dielectric anisotropy of the system, this dynamic response can be captured by studying dielectric relaxation. The studied dynamic response reveals insights on the field-induced self-assembly mechanism and the stability of the LC+QD system.