Mathieu equation and Elliptic curve

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both $q>1$, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We provide a fifth order proof.Comment: 12 pages; minor improvement of the Conclusion section, references adde

Multiple Timescale Dispatch and Scheduling for Stochastic Reliability in Smart Grids with Wind Generation Integration

Integrating volatile renewable energy resources into the bulk power grid is challenging, due to the reliability requirement that at each instant the load and generation in the system remain balanced. In this study, we tackle this challenge for smart grid with integrated wind generation, by leveraging multi-timescale dispatch and scheduling. Specifically, we consider smart grids with two classes of energy users - traditional energy users and opportunistic energy users (e.g., smart meters or smart appliances), and investigate pricing and dispatch at two timescales, via day-ahead scheduling and realtime scheduling. In day-ahead scheduling, with the statistical information on wind generation and energy demands, we characterize the optimal procurement of the energy supply and the day-ahead retail price for the traditional energy users; in realtime scheduling, with the realization of wind generation and the load of traditional energy users, we optimize real-time prices to manage the opportunistic energy users so as to achieve systemwide reliability. More specifically, when the opportunistic users are non-persistent, i.e., a subset of them leave the power market when the real-time price is not acceptable, we obtain closedform solutions to the two-level scheduling problem. For the persistent case, we treat the scheduling problem as a multitimescale Markov decision process. We show that it can be recast, explicitly, as a classic Markov decision process with continuous state and action spaces, the solution to which can be found via standard techniques. We conclude that the proposed multi-scale dispatch and scheduling with real-time pricing can effectively address the volatility and uncertainty of wind generation and energy demand, and has the potential to improve the penetration of renewable energy into smart grids.Comment: Submitted to IEEE Infocom 2011. Contains 10 pages and 4 figures. Replaces the previous arXiv submission (dated Aug-23-2010) with the same titl

Trade Liberalization and Trade Performance of Environmental Goods: Evidence from Asia-Pacific Economic Cooperation Members

In this article, we study the impact of trade liberalization, including reductions in both tariff and nontariff trade barriers, on environmental goods (EGs) exports. Using bilateral trade data from 20 Asia-Pacific Economic Cooperation members, we find that tariff reduction in an exporting country has a larger positive impact on its exports of EGs than tariff reduction in an importing country. Our results also show that a lower nontariff barrier in an importing country increases its imports of EGs. A considerable amount of heterogeneity also exists in subsample results based on countries’ income levels