76,686 research outputs found

    On the elementary theory of pairs of real closed fields. II

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    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

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    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

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    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

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    Let TT be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language LL. We study derivations δ\delta on models M⊨T\mathcal{M}\models T. We introduce the notion of a TT-derivation: a derivation which is compatible with the L(∅)L(\emptyset)-definable C1\mathcal{C}^1-functions on M\mathcal{M}. We show that the theory of TT-models with a TT-derivation has a model completion TGδT^\delta_G. The derivation in models (M,δ)⊨TGδ(\mathcal{M},\delta)\models T^\delta_G behaves "generically," it is wildly discontinuous and its kernel is a dense elementary LL-substructure of M\mathcal{M}. If T=T = RCF, then TGδT^\delta_G 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 TGδT^\delta_G has TT as its open core, that TGδT^\delta_G is distal, and that TGδT^\delta_G eliminates imaginaries. We also show that the theory of TT-models with finitely many commuting TT-derivations has a model completion.Comment: 29 page
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