20 research outputs found
Amicable pairs and aliquot cycles for elliptic curves
An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good
reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study
elliptic amicable pairs and analogously defined longer elliptic aliquot cycles.
We show that there exist elliptic curves with arbitrarily long aliqout cycles,
but that CM elliptic curves (with j not 0) have no aliqout cycles of length
greater than two. We give conjectural formulas for the frequency of amicable
pairs. For CM curves, the derivation of precise conjectural formulas involves a
detailed analysis of the values of the Grossencharacter evaluated at a prime
ideal P in End(E) having the property that #E(F_P) is prime. This is especially
intricate for the family of curves with j = 0.Comment: 53 page
Origin of springtime ozone enhancements in the lower troposphereover Beijing: in situ measurements and model analysis
Ozone (O3) concentrations in the lower troposphere (LT) over Beijing have significantly increased over the past two decades as a result of rapid industrialization in China, with important implications for regional air quality and photochemistry of the background troposphere. We characterize the vertical distribution of lower-tropospheric (0–6 km) O3 over Beijing using observations from 16 ozonesonde soundings made during a field campaign in April–May 2005 and MOZAIC (Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft) aircraft measurements over 13 days in the same period. We focus on the origin of LT O3 enhancements observed over Beijing, particularly in May. We use a global 3-D chemistry and transport model (GEOS-Chem CTM) driven by assimilated meteorological fields to examine the transport pathways for O3 pollution, and quantify the sources contributing to O3 and its enhancements in the springtime LT over Beijing. Output from the Global Modeling Initiative (GMI) CTM is also used. High O3 concentrations (up to 94.7 ppbv) were frequently observed at the altitude of ~1.5–2 km. The CTMs captured the timing of the occurrences but significantly underestimated their magnitude. GEOS-Chem simulations and a case study showed that O3 produced in the Asian troposphere (especially from Asian anthropogenic pollution) made major contributions to the observed O3 enhancements. Contributions from anthropogenic pollution in the European and North American troposphere were reduced during these events, in contrast with days without O3 enhancements, when contributions from Europe and North America were substantial. The O3 enhancements typically occurred under southerly wind and warmer conditions. It is suggested that an earlier onset of the Asian summer monsoon would cause more O3 enhancement events in the lower troposphere over the North China Plain in late spring and early summer.</p