Let M1 and M2 be closed, irreducible 3-manifolds. Let Hi and Gi denote non-isotopic Heegaard surfaces in Mi, where Hi is unstabilized. Let H be the amalgamation of H1 and H2 and G be the amalgamation of G1 and G2 in M1#M2. Then H is not a stabilization of G. It follows that H and G are not isotopic, and that H is unstabilized. This answers a question of C. Gordon (Problem 3.91 from [Kir97]) in ...

The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assu...

Grossberg [Gro02, Conjecture 9.3] has raised the question of the existence of Hanf numbers for joint embedding and amalgamation in Abstract Elementary Classes (AEC). Various authors have given lower bounds (discussed in Section 5), usually for the disjoint version of these properties. Here we show a strongly compact cardinal κ is an upper bound for various such Hanf numbers. We define 4 kinds a...

We present several examples of hereditary classes finite structures satisfying the joint embedding property and weak amalgamation property, but failing cofinal property. These include a continuum-sized family undirected graphs, as well an example due to Pouzet with countably categorical generic limit.

This paper proposes a technique for constructing programming languages under object-oriented operating systems. Such operating systems consist of independent modules, so one can build exible systems and applications. However, traditional object-oriented operating systems provide users with system's object model (system object), so that programming language designers have to map their object mod...

We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra D (under an hypothesis of unitary invariance). Then we consider elements of a finite von Neumann algebra containing D, which have kernel and range projection in D. We associate them a free entropy wi...