Beyond abstract elementary classes: On the model theory of geometric lattices

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $({\mathbf{K}}^3,?)$, is ω-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $({\mathbf{K}}^3,?)$ the Smoothness Axiom fails, and so $({\mathbf{K}}^3,?)$ is not an AEC.     

Forcing a Boolean algebra with predesigned automorphism group

For suitable groups we will show that one can add a Boolean algebra by forcing in such a way that is almost isomorphic to . In particular, we will give a positive answer to the following question due to J. Roitman: Is a possible number of automorphisms of a rich Boolean algebra?

     

Categoricity in homogeneous complete metric spaces

We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of the set instead of the cardinality, is ${\aleph_0}$ . In these settings we prove an analogue of Morley’s categoricity transfer theorem. We also give concrete examples of homogeneous MAECs.     

On eigenvectors,approximations and the Feynman propagator

Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product $〈x_0|K^t|x_1〉$ of the eigenvector $|x_0〉$ of the position operator with eigenvalue $x_0$ and $K^t(|x_1〉)$, where $K^t$ is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new.     

On ‐complete equivalence relations on the generalized Baire space

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if , then many of them are ‐complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is ‐complete (it is, if , but can be forced not to be).     

On stability in finite models

We search for a set-up in which results from the theory of infinite models hold for finite models. As an example we prove results from stability theory. Received: 23 February 1998     

A Comparative Study of Pyrolysis Liquids by Slow Pyrolysis of Industrial Hemp Leaves,Hurds and Roots

This study assessed the pyrolysis liquids obtained by slow pyrolysis of industrial hemp leaves, hurds, and roots. The liquids recovered between a pyrolysis temperature of 275–350 °C, at two condensation temperatures 130 °C and 70 °C, were analyzed. Aqueous and bio-oil pyrolysis liquids were produced and analyzed by proton nuclear magnetic resonance (NMR), gas chromatography–mass spectrometry (GC-MS), and atmospheric pressure photoionization Fourier transform ion cyclotron resonance mass spectrometry (APPI FT-ICR MS). NMR revealed quantitative concentrations of the most abundant compounds in the aqueous fractions and compound groups in the oily fractions. In the aqueous fractions, the concentration range of acetic acid was 50–241 gL⁻¹, methanol 2–30 gL⁻¹, propanoic acid 5–20 gL⁻¹, and 1-hydroxybutan-2-one 2 gL⁻¹. GC-MS was used to compare the compositions of the volatile compounds and APPI FT-ICR MS was utilized to determine the most abundant higher molecular weight compounds. The different obtained pyrolysis liquids (aqueous and oily) had various volatile and nonvolatile compounds such as acetic acid, 2,6-dimethoxyphenol, 2-methoxyphenol, and cannabidiol. This study provides a detailed understanding of the chemical composition of pyrolysis liquids from different parts of the industrial hemp plant and assesses their possible economic potential.     

On potential isomorphism and non-structure

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game. The research of the first and second author was partially supported by Academy of Finland grant 40734 Mathematics Subject Classification (2000): 03C55, 03C45     

On Borel equivalence relations in generalized Baire space

We construct two Borel equivalence relations on the generalized Baire space κ ^κ , κ ^<κ ?=?κ >?ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails.