Beautiful pairs

We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called beautiful pairs, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with classical Fra\"{i}ss\'{e} properties. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically, $p$-adically and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces such as Berkovich and Huber's analytifications. In particular, we recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields.Comment: 40 page

On structures in hypergraphs of models of a theory

We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types of models of a theory, are given

SMT Solving for Functional Programming over Infinite Structures

We develop a simple functional programming language aimed at manipulating infinite, but first-order definable structures, such as the countably infinite clique graph or the set of all intervals with rational endpoints. Internally, such sets are represented by logical formulas that define them, and an external satisfiability modulo theories (SMT) solver is regularly run by the interpreter to check their basic properties. The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038

Wick's Theorem at Finite Temperature

We consider Wick's Theorem for finite temperature and finite volume systems. Working at an operator level with a path ordered approach, we show that contrary to claims in the literature, expectation values of normal ordered products can be chosen to be zero and that results obtained are independent of volume. Thus the path integral and operator approaches to finite temperature and finite volume quantum field theories are indeed seen to be identical. The conditions under which normal ordered products have simple symmetry properties are also considered.Comment: 15 pages, LaTeX (no figures), available through anonymous ftp as LaTeX from ftp://euclid.tp.ph.ic.ac.uk/papers/95-6_18.tex or as LaTeX or postscript at http://euclid.tp.ph.ic.ac.uk/Papers/index.htm