Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counter example of the Br\'ezis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina.Comment: 10 page

Description and Realization for a Class of Irrational Transfer Functions

This paper proposes an exact description scheme which is an extension to the well-established frequency distributed model method for a class of irrational transfer functions. The method relaxes the constraints on the zero initial instant by introducing the generalized Laplace transform, which provides a wide range of applicability. With the discretization of continuous frequency band, the infinite dimensional equivalent model is approximated by a finite dimensional one. Finally, a fair comparison to the well-known Charef method is presented, demonstrating its added value with respect to the state of art.Comment: 9 pages, 9 figure