1,721,035 research outputs found
Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem
Get PDFKochen 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
Totally categorical structures
No full textA first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every
ℵ
0
{\aleph _0}
-stable,
ℵ
0
{\aleph _0}
-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.</p
Stable group theory and approximate subgroups
No full textWe note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group
G
G
, we show that a finite subset
X
X
with
|
X
X
−
1
X
|
/
|
X
|
|X X ^{-1}X |/ |X|
bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of
G
G
. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.</p
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