76,697 research outputs found
On the elementary theory of pairs of real closed fields. II
Let ℒ be the first order language of field theory with an additional one place predicate symbol. In [B2] it was shown that the elementary theory T of the class of all pairs of real closed fields, i.e., ℒ-structures ‹K, L›, K a real closed field, L a real closed subfield of K, is undecidable. The aim of this paper is to show that the elementary theory Ts of a nontrivial subclass of containing many naturally occurring pairs of real closed fields is decidable (Theorem 3, §5). This result was announced in [B2]. An explicit axiom system for Ts will be given later. At this point let us just mention that any model of Ts , is elementarily equivalent to a pair of power series fields ‹R 0((TA )), R 1((TB ))› where R 0 is the field of real numbers, R 1 = R 0 or the field of real algebraic numbers, and B ⊆ A are ordered divisible abelian groups. Conversely, all these pairs of power series fields are models of T s. Theorem 3 together with the undecidability result in [B2] answers some of the questions asked in Macintyre [M]. The proof of Theorem 3 uses the model theoretic techniques for valued fields introduced by Ax and Kochen [A-K] and Ershov [E] (see also [C-K]). The two main ingredients are (i) the completeness of the elementary theory of real closed fields with a distinguished dense proper real closed subfield (due to Robinson [R]), (ii) the decidability of the elementary theory of pairs of ordered divisible abelian groups (proved in §§1-4). I would like to thank Angus Macintyre for fruitful discussions concerning the subject. The valuation theoretic method of classifying theories of pairs of real closed fields is taken from [M
Brane and string field structure of elementary particles
The two quantizations of QFT,as well as the attempt of unifying it with
general relativity,lead us to consider that the internal structure of an
elementary fermion must be twofold and composed of three embedded internal
(bi)structures which are vacuum and mass (physical) bosonic fields decomposing
into packets of pairs of strings behaving like harmonic oscillators
characterized by integers mu corresponding to normal modes at mu (algebraic)
quanta.Comment: 50 page
Dimension, matroids, and dense pairs of first-order structures
A structure M is pregeometric if the algebraic closure is a pregeometry in
all M' elementarily equivalent to M. We define a generalisation: structures
with an existential matroid. The main examples are superstable groups of U-rank
a power of omega and d-minimal expansion of fields. Ultraproducts of
pregeometric structures expanding a field, while not pregeometric in general,
do have an unique existential matroid.
Generalising previous results by van den Dries, we define dense elementary
pairs of structures expanding a field and with an existential matroid, and we
show that the corresponding theories have natural completions, whose models
also have a unique existential matroid. We extend the above result to dense
tuples of structures.Comment: Version 2.8. 61 page
Generic derivations on o-minimal structures
Let be a complete, model complete o-minimal theory extending the theory
RCF of real closed ordered fields in some appropriate language . We study
derivations on models . We introduce the notion
of a -derivation: a derivation which is compatible with the
-definable -functions on . We show
that the theory of -models with a -derivation has a model completion
. The derivation in models
behaves "generically," it is wildly discontinuous and its kernel is a dense
elementary -substructure of . If RCF, then
is the theory of closed ordered differential fields (CODF) as introduced by
Michael Singer. We are able to recover many of the known facts about CODF in
our setting. Among other things, we show that has as its open
core, that is distal, and that eliminates
imaginaries. We also show that the theory of -models with finitely many
commuting -derivations has a model completion.Comment: 29 page
- …