513 research outputs found
The model theory of Commutative Near Vector Spaces
In this paper we study near vector spaces over a commutative from a model
theoretic point of view. In this context we show regular near vector spaces are
in fact vector spaces. We find that near vector spaces are not first order
axiomatisable, but that finite block near vector spaces are. In the latter case
we establish quantifier elimination, and that the theory is controlled by which
elements of the pointwise additive closure of are automorphisms of the near
vector space
Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem
Kochen and Specker's theorem can be seen as a consequence of Gleason's
theorem and logical compactness. Similar compactness arguments lead to stronger
results about finite sets of rays in Hilbert space, which we also prove by a
direct construction. Finally, we demonstrate that Gleason's theorem itself has
a constructive proof, based on a generic, finite, effectively generated set of
rays, on which every quantum state can be approximated.Comment: 14 pages, 6 figures, read at the Robert Clifton memorial conferenc
Pseudofinite structures and simplicity
We explore a notion of pseudofinite dimension, introduced by Hrushovski and
Wagner, on an infinite ultraproduct of finite structures. Certain conditions on
pseudofinite dimension are identified that guarantee simplicity or
supersimplicity of the underlying theory, and that a drop in pseudofinite
dimension is equivalent to forking. Under a suitable assumption, a
measure-theoretic condition is shown to be equivalent to local stability. Many
examples are explored, including vector spaces over finite fields viewed as
2-sorted finite structures, and homocyclic groups. Connections are made to
products of sets in finite groups, in particular to word maps, and a
generalization of Tao's algebraic regularity lemma is noted
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