The following note is a supplement to [1]. We will maintain the notation of [1] throughout, and all numbered items to which we refer are from that paper. Theorem 6.11 states that if M is a finitely generated R-module then Th(M,F) is stable. As pointed out in Remark 6.12, the proof Theorem 6.11 yields a stronger conclusion: Th(M,F) is superstable. These notes consist of detailed arguments for th...

We give bounds on finite volume expectations for a set of boundary conditions containing the support of any tempered Gibbs state and prove a theorem connecting the behavior of Gibbs states to the differentiability of the pressure for continuum statistical mechanical systems with long range superstable potentials. Convergence of grandcanonical Gibbs states is also studied. This is an extended ve...

In [Vau61] Vaught conjectured that a countable first order theory has countably many or 2א0 many countable models. Here, the following special case is proved. Theorem. If T is a superstable theory of finite rank with < 2א0 many countable models, then T has countably many countable models. The basic idea is to associate with a theory a ∧ − definable group G (called the structure group) which con...

Let gα be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call ε the perturbing param...

In the early days of the development of model theory it was considered natural and was certainly beneficial to assume that the theories under investigation were in a countable language. The primary advantage of this assumption was the presence of the Omitting Types Theorem of Grzegorczyk, Mostowski, and Ryll-Nardzewski [1], which generalized arguments of Henkin [3] and Orey [8]. Following this,...

For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z n + λ/z gives a 1-parameter, n+ d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z. This work presents several results related to these g...

ELEMENTARY CLASSES STABLE IN א0 SAHARON SHELAH AND SEBASTIEN VASEY Abstract. We study abstract elementary classes (AECs) that, in א0, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). We prove that such classes exhibit superstable-like behavior at א0. More precisely, there is a superlimit model of cardinality א0 and the class generat...

The F-box protein superfamily represents one of the largest families in the plant kingdom. F-box proteins phylogenetically organize into numerous subfamilies characterized by their carboxyl (C)-terminal protein-protein interaction domain. Among the largest F-box protein subfamilies in plant genomes are those with C-terminal kelch repeats. In this study, we analyzed the phylogeny and evolution o...