Atomic Models

Atomic models higher up

There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ω CK 1 and where the LωCK 1 ,ω-theory of M is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ω-theory. Most well-known models with unbounded atoms below ωCK 1 also realize a non-principal LωCK 1 ,ω-type; such a model that preserves the Σ1-admissibility of ωCK 1 will hav...

Möbius: Framework and Atomic Models

This paper gives an overview of the Möbius framework, and gives a formal specification for defining atomic models within the framework. The framework is designed to be capable of incorporating multiple modeling formalisms, including atomic models (e.g, SPNs), composition formalisms (e.g., Replicate/Join), measure specification formalisms, connection formalisms, and solvers. We focus on atomic m...

Atomic Energy Models for Machine Learning: Atomic Restricted Boltzmann Machines

On the Existence of Atomic Models

We give an example of a countable theory T such that for every cardinal λ ≥ א2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T (A) is a theory of size λ where the principal types are dense, yet T (A) has no atomic model. If a complete theory T has an atomic model then the principal types a...

Bonding in Organic Molecules. Atomic-orbital Models

I n previous chapters, we have shown how you can use ball-and-stick models to predict the general arrangements in space of organic molecules. The sticks correspond to chemical bonds, which we represent in structural formulas as lines, or in Lewis struc'tures as pairs of dots denoting shared pairs of electrons. Remembering that electrons and nuclei are charged particles, and that it is electrica...