An existential 0-definition of F_q[[t]] in F_q((t))

We show that the valuation ring F_q[[t]] in the local field F_q((t)) is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an existential-F_q-definable bounded neighbouhood of 0. Then we `tweak' this set by subtracting, taking roots, and applying Hensel's Lemma in order to find an existential-F_q-definable subset of F_q[[t]] which contains tF_q[[t]]. Finally, we use the fact that F_q is defined by the formula x^q-x=0 to extend the definition to the whole of F_q[[t]] and to rid the definition of parameters. Several extensions of the theorem are obtained, notably an existential 0-definition of the valuation ring of a non-trivial valuation with divisible value group.Comment: 9 page

The existential theory of equicharacteristic henselian valued fields

We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of Fq((t))

N-body simulations with two-orders-of-magnitude higher performance using wavelets

Noise is a problem of major concern for N-body simulations of structure formation in the early Universe, of galaxies and plasmas. Here for the first time we use wavelets to remove noise from N-body simulations of disc galaxies, and show that they become equivalent to simulations with two orders of magnitude more particles. We expect a comparable improvement in performance for cosmological and plasma simulations. Our wavelet code will be described in a following paper, and will then be available on request.Comment: Mon. Not. R. Astron. Soc., in press. The interested reader is strongly recommended to ignore the low-resolution Fig. 3 (and Fig. 4), and to download the full-resolution paper (700 kb) from http://www.oso.chalmers.se/~romeo/Paper_VI.ps.g