Warings problem for fields

Denote by P(K, k) the members of the field K which are sums of kth powers of field elements, by P+(K, k) the set of members of K which are sums of kth powers of totally positive elements of K. We are interested in deciding whether or not there exist integers w(K, k) and g(K, k) such that: a in P(K, k) implies that a is the sum of at most w(K, k) kth powers; a in P+(K, k) implies that a is the sum of at most g(K, k) totally positive kth powers. We will show that if w(K, 2) is finite and provided that the kth powers are dense (in a sense described explicitly in theorem 3) in K, then w(K, k) is also finite for k > 2. The proofs are constructive, but the implied upper bounds for w(K, k) are large. This is to be expected since the method of proof does not use any deep arithmetical or algebraic properties of the field K.Comment: 14 page

Outreach Education: Making Neuroscience Readily Available for Rural Students and Communities of Alaska

WHAT is neuroscience? Neuroscience is the scientific study of the structure or function of the nervous system and brain. To get a grasp on the vast study of neuroscience, first think of the human body and how complex it is. Think of the skeletal system and the muscular, nervous, digestive and respiratory systems that make up the human body and keep it running in tip top shape. Now think of the brain, a singular organ of soft nervous tissue that one-handedly controls all of those other systems, including both mental and physical actions. Sounds a bit daunting doesn’t it

W and Z Properties at the Tevatron

We present recent results from CDF and D0 on W and Z production cross sections, the width of the W boson, tau-e universality in W decays, trilinear gauge boson couplings, and on the observation of Z -> b bbar.Comment: Paper presented at the International Europhysics Conference on High Energy Physics, Tampere, Finland, 15-21 July, 199

Nonlinear Particle Acceleration in Relativistic Shocks

Monte Carlo techniques are used to model nonlinear particle acceleration in parallel collisionless shocks of various speeds, including mildly relativistic ones. When the acceleration is efficient, the backreaction of accelerated particles modifies the shock structure and causes the compression ratio, r, to increase above test-particle values. Modified shocks with Lorentz factors less than about 3 can have compression ratios considerably greater than 3 and the momentum distribution of energetic particles no longer follows a power law relation. These results may be important for the interpretation of gamma-ray bursts if mildly relativistic internal and/or afterglow shocks play an important role accelerating particles that produce the observed radiation. For shock Lorentz factors greater than about 10, r approaches 3 and the so-called `universal' test-particle result of N(E) proportional to E^{-2.3} is obtained for sufficiently energetic particles. In all cases, the absolute normalization of the particle distribution follows directly from our model assumptions and is explicitly determined.Comment: Updated version, Astroparticle Physics, in press, 29 pages, 13 figure

The Cosmic Ray - X-ray Connection: Effects of Nonlinear Shock Acceleration on Photon Production in SNRs

Cosmic-ray production in young supernova remnant (SNR) shocks is expected to be efficient and strongly nonlinear. In nonlinear, diffusive shock acceleration, compression ratios will be higher and the shocked temperature lower than test-particle, Rankine-Hugoniot relations predict. Furthermore, the heating of the gas to X-ray emitting temperatures is strongly coupled to the acceleration of cosmic-ray electrons and ions, thus nonlinear processes which modify the shock, influence the emission over the entire band from radio to gamma-rays and may have a strong impact on X-ray line models. Here we apply an algebraic model of nonlinear acceleration, combined with SNR evolution, to model the radio and X-ray continuum of Kepler's SNR.Comment: 7 pages including 4 figures; to appear in ``The Acceleration and Transport of Energetic Particles Observed in the Heliosphere,'' Proceedings of the ACE-2000 Symposium held on January 5 - 8, 2000, Indian Springs, C

Stabilisation bias in monetary policy under endogenous price stickiness

This paper investigates the consequences of introducing endogenous price stickiness into a standard monetary policy model. We find that the modification reduces the optimal degree of inflation stabilization to which the central bank should commit. The reason is that less inflation stabilization encourages firms to review their prices more frequently. The economy becomes more flexible and the Phillips-curve tradeoff is improved, making it easier for the central bank to control inflation. This reduces, and may even reverse, the stabilization bias that is present in models with exogenous price stickiness and that recommends that the central bank generally commit to tighter stabilization of inflation than it would in a discretionary policy regime.price stickiness; monetary policy; stabilization bias