We prove that the roots of a definable C∞ curve of monic hyperbolic polynomials admit a definable C∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (R,+, ·). Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of Cp (for p ∈ N) arrangements of the roots in both ...

Let F be a collection of holomorphic functions and let R(PR(F)) denote the reduct of the structure Ran to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in F . We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure R(...

We study the definability of certain subgroups of a group G that does not have the independence property. If a (type) definable subset X of an elementary extension G of G has property P , we call its trace X∩G over G an externally (type) definable P set. We show the following. Centralisers of subsets of G are externally definable subgroups. Cores of externally definable subgroups and iterated c...

We introduce a class of neighbourhood frames for graded modal logic using an operation from Kripke to frames. This is shown be first-order definable but not modally definable. also obtain new definition bisimulation by modifying the monotonic bisimulation.

We initiate a systematic study of the convolution operation on Keisler measures, generalizing work Newelski in case types. Adapting results Glicksberg, we show that supports definable and finitely satisfiable (or just definable, assuming NIP) measures are nice semigroups, classify idempotent stable groups as invariant type-definable subgroups. establish left-continuity map NIP theories, use it ...

Let G be a compact definable C group, X a compact affine definable CG manifold and 2 ≤ r < ∞. We prove that the set of equivariant definable Morse functions on X whose loci are finite unions of nondegenerate critical orbits is open and dense in the set of G invariant definable C functions with respect to the definable C topology.

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies e...