Lagrange's four squares theorem with one prime and three almost--prime variables

It is conjectured that every sufficiently large integer $N\equiv 4\pmod{24}$ should be a sum of the squares of 4 primes. The best approximation to this in the literature is the result of Brüdern and Fouvry [J. Reine Angew. Math., 454 (1994), 59--96] who showed that every sufficiently large integer $N\equiv 4\pmod{24}$ is a sum of the squares of 4 almost-primes, each of which has at most 34 prime factors. The present paper proves such a result with the square of one prime and 3 almost-primes, which in this case have at most 101 prime factors each. The work of Brüdern and Fouvry was based on Kloosterman's approach to representations by quaternary forms, but this does not lend itself to situations in which one of the variables is restricted to be a prime. Instead the present paper works with an `almost all' result for the representation of numbers $m$ as sums of 3 squares. To use this approach one has to take $m$ of the form $N-p^2$, and such numbers are too sparse for the standard theory. It is therefore necessary to use an `amplification' procedure, which emphasizes those integers $m$ for which $N-m$ is a square. All this machinery is coupled with Kloosterman's version of the circle method. There are considerable technical complications, in which bounds for the Kloosterman sum play a key rôle. At one point in the argument a saving has to be extracted from a non-trivial averaging over the denominators of the Farey arcs. This is an instance of `the second Kloosterman refinement'

Higher Descent Data as a Homotopy Limit

We define the 2-groupoid of descent data assigned to a cosimplicial 2-groupoid and present it as the homotopy limit of the cosimplicial space gotten after applying the 2-nerve in each cosimplicial degree. This can be applied also to the case of $n$-groupoids thus providing an analogous presentation of "descent data" in higher dimensions.Comment: Appeared in JHR

Children's Databases - Safety and Privacy

This report describes in detail the policy background, the systems that are being built, the problems with them, and the legal situation in the UK. An appendix looks at Europe, and examines in particular detail how France and Germany have dealt with these issues. Our report concludes with three suggested regulatory action strategies for the Commissioner: one minimal strategy in which he tackles only the clear breaches of the law, one moderate strategy in which he seeks to educate departments and agencies and guide them towards best practice, and finally a vigorous option in which he would seek to bring UK data protection practice in these areas more in line with normal practice in Europe, and indeed with our obligations under European law

Gauge Invariant Variational Approach with Fermions: the Schwinger Model

We extend the gauge invariant variational approach of Phys. Rev. D52 (1995) 3719, hep-th/9408081, to theories with fermions. As the simplest example we consider the massless Schwinger model in 1+1 dimensions. We show that in this solvable model the simple variational calculation gives exact results.Comment: 14 pages, 1 figur

Rescattering Effects on Intensity Interferometry

We derive a general formula for the correlation function of two identical particles with the inclusion of multiple elastic scatterings in the medium in which the two particles are produced. This formula involves the propagator of the particle in the medium. As illustration of the effect we apply the formula to the special case where the scatterers are static, localized 2-body potentials. In this illustration both $R^2_{\rm side}$ and $R^2_{\rm out}$ are increased by an amount proportional to the square of the spatial density of scatterers and to the differential cross section. Specific numbers are used to show the expected magnitude of the rescattering effect on kaon interferometry.Comment: 15 pages, 4 figure

Keck Pencil-Beam Survey for Faint Kuiper Belt Objects

We present the results of a pencil-beam survey of the Kuiper Belt using the Keck 10-m telescope. A single 0.01 square degree field is imaged 29 times for a total integration time of 4.8 hr. Combining exposures in software allows the detection of Kuiper Belt Objects (KBOs) having visual magnitude V < 27.9. Two new KBOs are discovered. One object having V = 25.5 lies at a probable heliocentric distance d = 33 AU. The second object at V = 27.2 is located at d = 44 AU. Both KBOs have diameters of about 50 km, assuming comet-like albedos of 4%. Data from all surveys are pooled to construct the luminosity function from red magnitude R = 20 to 27. The cumulative number of objects per square degree, N (< R), is fitted to a power law of the form log_(10) N = 0.52 (R - 23.5). Differences between power laws reported in the literature are due mainly to which survey data are incorporated, and not to the method of fitting. The luminosity function is consistent with a power-law size distribution for objects having diameters s = 50 to 500 km; dn ~ s^(-q) ds, where the differential size index q = 3.6 +/- 0.1. The distribution is such that the smallest objects possess most of the surface area, but the largest bodies contain the bulk of the mass. Though our inferred size index nearly matches that derived by Dohnanyi (1969), it is unknown whether catastrophic collisions are responsible for shaping the size distribution. Implications of the absence of detections of classical KBOs beyond 50 AU are discussed.Comment: Accepted to AJ. Final proof-edited version: references added, discussion of G98 revised in sections 4.3 and 5.