We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming "Swiss cheeses" in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in "most" valued fields F, if f(x),g(x) ∈ F[ x] and v is (...) the valuation map, then the set {x : v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of "valued trees", which we define formally. (shrink)

Van den Dries, L. and J. Holly, Quantifier elimination for modules with scalar variables, Annals of Pure and Applied Logic 57 161–179. We consider modules as two-sorted structures with scalar variables ranging over the ring. We show that each formula in which all scalar variables are free is equivalent to a formula of a very simple form, uniformly and effectively for all torsion-free modules over gcd domains . For the case of Presburger arithmetic with scalar variables the result takes a (...) still simpler form, and we derive in this way the polynomial-time decidability of the sets defined by such formulas. (shrink)

It is important to study movement data from individual subjects rather than by averaging data across subjects or trials, because averaged data may follow different laws than those followed by the individual data. This fact can be shown mathematically. In addition, clear assumptions and a thorough understanding of their consequences are a necessary component of any realistic model.

Elimination of imaginaries for 1-variable definable equivalence relations is proved for a theory of algebraically closed valued fields with new sorts for the disc spaces. The proof is constructive, and is based upon a new framework for proving elimination of imaginaries, in terms of prototypes which form a canonical family of formulas for defining each set that is definable with parameters. The proof also depends upon the formal development of the tree-like structure of valued fields, in terms of valued trees, (...) and a decomposition of valued trees which is used in the coding of certain sets of discs. (shrink)