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

An invariant for difference field extensions

In this paper we introduce a new invariant (the distant degree) for difference field extensions of finite transcendence degree, and we explore some of its properties. We also discuss a generalisation of this invariant and of the limit degree to groups with an automorphism.Comment: After posting the previous version, we discovered the work of Willis on totally disconnected locally compact groups. Over a large area of overlap, our "distant degree" invariant of an automorphism agrees with Willis' {\em scale}. In the new version we describe the relations between the two frameworks. 14pp. July 2011: Small change

On subgroups of semi-abelian varieties defined by difference equations

Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics onto (T,M) arises from a finite isogeny T \to T. A similar and more general statement is shown for Abelian and semi-abelian varieties. In model-theoretic terms, our result says: Working in an existentially closed difference field, we consider a definable subgroup B of a semi-abelian variety A; assume B does not have a subgroup isogenous to A'(F) for some twisted fixed field F, and some semi-Abelian variety A'. Then B with the induced structure is stable and stably embedded. This implies in particular that for any n>0, any definable subset of B^n is a Boolean combination of cosets of definable subgroups of B^n. This result was already known in characteristic 0 where indeed it holds for all commutative algebraic groups ([CH]). In positive characteristic, the restriction to semi-abelian varieties is necessary.Comment: Revised version, to appea

Differential-algebraic jet spaces preserve internality to the constants

This paper concerns the model theory of jet spaces (i.e., higher-order tangent spaces) in differentially closed fields. Suppose p is the generic type of the jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constant field called "preserving internality to the constants". This strengthening is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. A counterexample is constructed showing that only this generic analogue holds in differential-algebraic geometry.Comment: 13 page

Cosmic ray muons for spent nuclear fuel monitoring

There is a steady increase in the volume of spent nuclear fuel stored on-site (at reactor) as currently there is no permanent disposal option. No alternative disposal path is available and storage of spent nuclear fuel in dry storage containers is anticipated for the near future. In this dissertation, a capability to monitor spent nuclear fuel stored within dry casks using cosmic ray muons is developed. The motivation stems from the need to investigate whether the stored content agrees with facility declarations to allow proliferation detection and international treaty verification. Cosmic ray muons are charged particles generated naturally in the atmosphere from high energy cosmic rays. Using muons for proliferation detection and international treaty verification of spent nuclear fuel is a novel approach to nuclear security that presents significant advantages. Among others, muons have the ability to penetrate high density materials, are freely available, no radiological sources are required and consequently there is a total absence of any artificial radiological dose. A methodology is developed to demonstrate the applicability of muons for nuclear nonproliferation monitoring of spent nuclear fuel dry casks. Purpose is to use muons to differentiate between spent nuclear fuel dry casks with different amount of loading, not feasible with any other technique. Muon scattering and transmission are used to perform monitoring and imaging of the stored contents of dry casks loaded with spent nuclear fuel. It is shown that one missing fuel assembly can be distinguished from a fully loaded cask with a small overlapping between the scattering distributions with 300,000 muons or more. A Bayesian monitoring algorithm was derived to allow differentiation of a fully loaded dry cask from one with a fuel assembly missing in the order of minutes and negligible error rate. Muon scattering and transmission simulations are used to reconstruct the stored contents of sealed dry casks from muon measurements. A combination of muon scattering and muon transmission imaging can improve resolution and thus a missing fuel assembly can be identified for vertical and horizontal dry casks. The apparent separation of the images reveals that the muon scattering and transmission can be used for discrimination between casks, satisfying the diversion criteria set by IAEA

A note on the non-existence of prime models of theories of pseudo-finite fields

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we generalise this result to \kappa-prime models, for \kappa a regular uncountable cardinal or \aleph_\epsilon.Comment: Submitte

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe