A Three-Dimensional Voting System in Hong Kong

The voting system in the Legislative Council of Hong Kong (Legco) is sometimes unicameral and sometimes bicameral, depending on whether the bill is proposed by the Hong Kong government. Therefore, although without any representative within Legco, the Hong Kong government has certain degree of legislative power --- as if there is a virtual representative of the Hong Kong government within the Legco. By introducing such a virtual representative of the Hong Kong government, we show that Legco is a three-dimensional voting system. We also calculate two power indices of the Hong Kong government through this virtual representative and consider the $C$-dimension and the $W$-dimension of Legco. Finally, some implications of this Legco model to the current constitutional reform in Hong Kong will be given

Meromorphic solutions of a third order nonlinear differential equation

We prove that all the meromorphic solutions of the nonlinear differential equation c0 u"' + 6 u^4 + c1 u" + c2 u u' + c3 u^3 + c4 u'+ c5 u^2 + c6 u +c7=0 are elliptic or degenerate elliptic, and we build them explicitly.Comment: 12 pages, to appear, Journal of Mathematical Physic

Smale's mean value conjecture for finite Blaschke products

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.Comment: To appear in an issue of Journal of Analysis denoted to the Proceedings of the Conference on Modern Aspects of Complex Geometry (MindaFest)

Hayman's classical conjecture on some nonlinear second order algebraic ODEs

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second order nonlinear algebraic ordinary differential equations. We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the {\it classical conjecture} proposed by Hayman in 1996.Comment: 15 pages, to appear, Complex variables and elliptic equation

A Uniqueness Theorem for Holomorphic Mappings in the Disk Sharing Totally Geodesic Hypersurfaces

In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces