Averages and moments associated to class numbers of imaginary quadratic fields

For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_\ell(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_\ell(-d)$ for all primes $\ell \geq 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geq 3$.Comment: 26 pages; minor edits to exposition and notation, to agree with published versio

Simultaneous Integer Values of Pairs of Quadratic Forms

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.Comment: 63 page

Seasonal and interannual variations in total ozone revealed by the Nimbus-4 backscattered ultraviolet experiment

The first two years of Backscattered Ultraviolet (BUV) ozone data from the Nimbus-4 spacecraft were reprocessed. The seasonal variations of total ozone for the period April 1970 to April 1972 are described using daily zonal means to 10 deg latitude zones and a time-latitude cross section. In addition, the BUV data are compared with analyzed Dobson data and with IRIS data also obtained from the Nimbus-4 spacecraft. A harmonic analysis was performed on the daily zonal means. Amplitudes, days of peaks, and percentage of variance were computed for annual and semi-annual waves and for higher harmonics of an annual period for the two years. Asymmetries are found in the annual waves in the two hemispheres, with a subtle interannual difference which may be due to changes in the general circulation. A significant semi-annual component is detected in the tropics for the first year, which appears to result from influences of the annual waves in the two hemispheres

Ten Years of Solar Change as Monitored by SBUV and SBUV/2

Observations of the Sun by the Solar Backscatter Ultraviolet (SBUV) instrument aboard Nimbus 7 and the SBUV/2 instrument aboard NOAA-9 reveal variations in the solar irradiance from 1978, to 1988. The maximum to minimum solar change estimated from the Heath and Schlesinger Mg index and wavelength scaling factors is about 4 percent from 210 to 260 nm and 8 percent for 180 to 210 nm; direct measurements of the solar change give values of 1 to 3 percent and 5 to 7 percent, respectively, for the same wavelength range. Solar irradiances were high from the start of observations, late in 1978, until 1983, declined until early 1985, remained approximately constant until mid-1987, and then began to rise. Peak-to-peak 27-day rotational modulation amplitudes were as large as 6 percent at solar maximum and 1 to 2 percent at solar minimum. During occasional intervals of the 1979 to 1983 maximum and again during 1988, the dominant rotational modulation period was 13.5 days. Measurements near 200 to 205 nm show the same rotational modulation behavior but cannot be used to track long-term changes in the Sun because of uncertainties in the characterization of long-term instrument sensitivity changes

The flashing ratchet and unidirectional transport of matter

We study the flashing ratchet model of a Brownian motor, which consists in cyclical switching between the Fokker-Planck equation with an asymmetric ratchet-like potential and the pure diffusion equation. We show that the motor really performs unidirectional transport of mass, for proper parameters of the model, by analyzing the attractor of the problem and the stationary vector of a related Markov chain.Comment: 11 page