We show that an infinite number of non-unitary minimal models may describe two dimensional turbulent magnetohydrodynamics (MHD), both in the presence and absence of the Alf'ven effect. We argue that the existence of a critical dynamical index results in the Alf'ven effect or equivelently the equipartition of energy. We show that there are an infinite number of conserved quantities in 2D − M HD ...

Assume G is a group definable in a model M of a stable theory T . We prove that the semigroup SG(M) of complete G-types over M is an inverse limit of some semigroups type-definable in M eq. We prove that the maximal subgroups of SG(M) are inverse limits of some definable quotients of subgroups of G. We consider the powers of types in the semigroup SG(M) and prove that in a way every type in SG(...

In Bue93b] we proved Vaught's conjecture for all superstable theories of nite rank; that is, such a theory has countably many or continuum many countable models. While this proof does settle Vaught's conjecture for unidimensional theories a sharper result can be obtained for these theories and in places the proof can be simpliied. Let T be a properly unidimensional theory with < 2 @ 0 many coun...

In 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e. for every complete atomic decidable theory T , there is a prime model M computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for ∆2 sets. Some of these predicates, such as prime bounding, and others ...

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G00, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, Mext, of a model M . In the light of these results we continue the study of the “definable topological dynamics” of groups in NIP theories. In particular we prove the Ellis gro...

Here, T is the time of an event and Z a p-dimensional vector of covariates, φ a square integrable function, and F(t | z) = P (T ≤ t | Z = z) and H( dz) = P (Z ∈ dz) denote the conditional survival distribution and the marginal law of Z , respectively. For example, θ(I {t > t∗}) is the marginal survival probability at time t∗, θ(I {t > t∗, z1 > z∗ 1}) the bivariate distribution at (t∗, z∗ 1) (Ak...

Theorem 1.1 (Hrushovski’s Theorem 3.5). Let G be a definable group, X a definable subset, and G̃ be the ∨-definable group generated by X. Suppose all this data is defined over a model M . Suppose μ is an M-invariant translation-invariant ideal on subsets of G̃. Let q be a wide type over M , contained in G̃. Identify q with its ∧-definable set of realizations. Suppose (F) there exist two realizatio...

In this paper we prove that if κ is a cardinal in L[0]], then there is an inner model M such that M |= (Vκ,∈) has no elementary end extension. In particular if 0] exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than א1 of uncountable cofinality in L[0]] is Mahlo in every strict inner model of L[0]].

The following assertion is an exercise in Lecture 4. Let ψ be a complete sentence in Lω1,ω in a countable language L. Then there is a countable language L ′ extending L and a first order L′-theory T such that reduct is a 1-1 map from the atomic models of T onto the models of ψ. So in particular, any complete sentence of Lω1,ω can be replaced (for spectrum purposes) by considering the atomic mod...