2,115 research outputs found
On finite imaginaries
We study finite imaginaries in certain valued fields, and prove a conjecture
of Cluckers and Denef.Comment: 15p
Amalgamation of types in pseudo-algebraically closed fields and applications
This paper studies unbounded PAC fields and shows an amalgamation result for
types over algebraically closed sets. It discusses various applications, for
instance that omega-free PAC fields have the property NSOP3. It also contains a
description of imaginaries in PAC fields.Comment: Minor changes in v3. Accepted versio
Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics
Before we dive into the accessibility stream of nowadays indicatory
applications of octonions to computer and other sciences and to quantum physics
let us focus for a while on the crucially relevant events for todays revival on
interest to nonassociativity. Our reflections keep wandering back to the
two square identity and then via the four
square identity up to the eight square identity.
These glimpses of history incline and invite us to retell the story on how
about one month after quaternions have been carved on the bridge
octonions were discovered by , jurist and
mathematician, a friend of . As for today we just
mention en passant quaternionic and octonionic quantum mechanics,
generalization of equations for octonions and triality
principle and group in spinor language in a descriptive way in order not
to daunt non specialists. Relation to finite geometries is recalled and the
links to the 7stones of seven sphere, seven imaginary octonions units in out of
the cave reality applications are appointed . This way we are welcomed
back to primary ideas of , and other distinguished
fathers of quantum mechanics and quantum gravity foundations.Comment: 26 pages, 7 figure
Groupoids, imaginaries and internal covers
Let be a first-order theory. A correspondence is established between
internal covers of models of and definable groupoids within . We also
consider amalgamations of independent diagrams of algebraically closed
substructures, and find strong relation between: covers, uniqueness for
3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and
definable groupoids. As a corollary, we describe the imaginary elements of
families of finite-dimensional vector spaces over pseudo-finite fields.Comment: Local improvements; thanks to referee of Turkish Mathematical
Journal. First appeared in the proceedings of the Paris VII seminar:
structures alg\'ebriques ordonn\'ee (2004/5
Galois-stability for Tame Abstract Elementary Classes
We introduce tame abstract elementary classes as a generalization of all
cases of abstract elementary classes that are known to permit development of
stability-like theory. In this paper we explore stability results in this
context. We assume that \K is a tame abstract elementary class satisfying the
amalgamation property with no maximal model. The main results include:
(1) Galois-stability above the Hanf number implies that \kappa(K) is less
than the Hanf number. Where \kappa(K) is the parallel of \kapppa(T) for f.o. T.
(2) We use (1) to construct Morley sequences (for non-splitting) improving
previous results of Shelah (from Sh394) and Grossberg & Lessmann.
(3) We obtain a partial stability-spectrum theorem for classes categorical
above the Hanf number.Comment: 23 page
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