Non-vanishing Theorems for Quadratic Twists of Elliptic Curves

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.Comment: Published versio

Knowledge diffusion vs. technological progress: the optimal strength of IPRs protection

By adjusting the strength of IPRs protection, the government can change the extent of knowledge spillovers in R&D. A large spillover rate helps to improve the productivity of the less efficient firms and save on the overall production costs. But, at the same time, it reduces the innovator's incentives to conduct R&D and results in a lower equilibrium innovation level. So, there is an inherent tension between knowledge diffusion and technological progress. In this paper, we formalized this relationship in a two stage asymmetric duopoly model and discussed the optimal IPRs protection policy. The main conclusion is that, to maximize social welfare the strength of IPRs protection should rise as the increase of the innovating firm's R&D efficiency.spillovers, knowledge diffusion, technological progress, IPRs