Untitled

Abstract

Among all the restrictions of weight orders to the subsets of monomials with a fixed degree, we consider those that yield a total order. Furthermore, we assume that each weight vector consists of an increasing tuple of weights. Every restriction, which is shown to be achieved by some monomial order, is interpreted as a suitable linearization of the poset arising by the intersection of all the weight orders. In the case of three variables, an enumeration is provided. For a higher number of variables, we show a necessary condition for obtaining such restrictions, using deducibility rules applied to homogeneous inequalities. The logarithmic version of this approach is deeply related to classical results of Farkas type, on systems of linear inequalities. Finally, we analyze the linearizations determined by sequences of prime numbers and provide some connections with topics in arithmetic.

Comment on the first-order Fermi acceleration at ultra-relativistic shocks

“…Several authors (Bednarz & Ostrowski 1998) discussed this process leading to formation of the spectrum with the energy spectral index σ ≈ 2.2 at γ ≫ 1 . Essentially the same results were obtained within different approaches presented by the above authors and by Kirk et al (2000) and Vietri (2002). We do not consider here the papers postulating the acceleration process, but not discussing details of the proposed mechanism.…”

“…A process of the first-order Fermi acceleration in such shocks was discussed in a series of papers by Bednarz & Ostrowski (1997, 1998; see also Bednarz 2000a,b), see also Achterberg et al 2001), Kirk et al (2000) and Vietri (2002). Below, in Sect.…”

“…Ultra-relativistic shock waves suggested to be sources of gamma-ray bursts are also expected by some authors to produce ultra-high-energy cosmic ray particles. A process of the first-order Fermi acceleration in such shocks was discussed in a series of papers by Bednarz & Ostrowski (1997, 1998; see also Bednarz 2000a,b), see also Achterberg et al 2001), Kirk et al (2000) and Vietri (2002). Below, in section 2, we briefly compare and discuss different approaches to the considered acceleration process, leading to the asymptotic spectral index σ γ≫1 ≈ 2.2.…”

“…Two quasi-analytic approaches to the considered acceleration process were presented by Kirk et al (2000) and Vietri (2002). Both provide methods to solve the Fokker-Planck equation describing particle advection with the general plasma flow and the small amplitude scattering of particle pitch angle as measured with respect to the shock normal.…”