We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $ -rank.

In a stable theory, a stationary type $q \in S\left( A \right)$ internal to a family of partial types ${\cal P}$ over A gives rise to a type-definable group, called its binding group. This group is isomorphic to the group $Aut\left( {q/{\cal P},A} \right)$ of permutations of the set of realizations of q, induced by automorphisms of the monster model, fixing ${\cal P}\,\mathop \cup \nolimits \,A$ pointwise. In this article, we investigate families of internal types varying uniformly, what we will call relative internality. We prove that the binding groups also vary uniformly, and are the isotropy groups of a natural type-definable groupoid (and even more). We then investigate how properties of this groupoid are related to properties of the type. In particular, we obtain internality criteria for certain 2-analysable types, and a sufficient condition for a type to preserve internality.

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

We compare dewetting characteristics of a thin nonwetting solid film in theabsence of stress, fortwo models of a wetting potential: the exponential and the algebraic.The exponential model is a one-parameter (r) model, and the algebraic model isa two-parameter (r, m)model, where r is the ratio of the characteristic wetting length to the heightof the unperturbed film,and m is the exponent of h (film height) in a smooth function thatinterpolates the system's surfaceenergy above and below the film-substrate interface at z = 0. The exponentialmodelgives monotonically decreasing (with h) wetting chemical potential, whilethis dependence is monotonic only for the m = 1 case of the algebraic model.Linear stability analysis of the planar equilibrium surface is performed.Simulations of the surface dynamics in the strongly nonlinear regime(large deviations from the planar equilibrium) and for large surface energyanisotropies demonstrate that for any m the film is less prone to dewettingwhen it is governed by the algebraic model. Quasiequilibrium states similar tothe one found in the exponential model [M. Khenner, Phys. Rev. B, 77 (2008), 245445.] exist in the algebraic model as well, and the film morphologies are similar.