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

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

Improving Performance of Iterative Methods by Lossy Checkponting

Iterative methods are commonly used approaches to solve large, sparse linear systems, which are fundamental operations for many modern scientific simulations. When the large-scale iterative methods are running with a large number of ranks in parallel, they have to checkpoint the dynamic variables periodically in case of unavoidable fail-stop errors, requiring fast I/O systems and large storage space. To this end, significantly reducing the checkpointing overhead is critical to improving the overall performance of iterative methods. Our contribution is fourfold. (1) We propose a novel lossy checkpointing scheme that can significantly improve the checkpointing performance of iterative methods by leveraging lossy compressors. (2) We formulate a lossy checkpointing performance model and derive theoretically an upper bound for the extra number of iterations caused by the distortion of data in lossy checkpoints, in order to guarantee the performance improvement under the lossy checkpointing scheme. (3) We analyze the impact of lossy checkpointing (i.e., extra number of iterations caused by lossy checkpointing files) for multiple types of iterative methods. (4)We evaluate the lossy checkpointing scheme with optimal checkpointing intervals on a high-performance computing environment with 2,048 cores, using a well-known scientific computation package PETSc and a state-of-the-art checkpoint/restart toolkit. Experiments show that our optimized lossy checkpointing scheme can significantly reduce the fault tolerance overhead for iterative methods by 23%~70% compared with traditional checkpointing and 20%~58% compared with lossless-compressed checkpointing, in the presence of system failures.Comment: 14 pages, 10 figures, HPDC'1

The design and fabrication of the Brayton Rotating Unit operating on Roller Element Bearings /BRU-R/ Final report

Design and fabrication of Brayton rotating unit operating on oil lubricated rolling element bearing