Most widely used special functions, such as orthogonal polynomials, Bessel Airy etc., are defined solutions to differential equations with polynomial coefficients. This class of functions is referred D-finite functions. There many symbolic algorithms (and implementations thereof) operate these objects exactly. Recently, we have extended this notion a more general that also allows for good handl...

In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or ln − exp definable sets). The idea is to study the cycles which are vanishing when we approach a special fiber. This also enables us to derive local metric invariant...

It is consistent that there is a partial order (P,≤) of size א1 such that every monotone function f : P → P is first order definable in (P,≤) It is an open problem whether there can be an infinite lattice L such that every monotone function from L to L is a polynomial. Kaiser and Sauer [KS] showed that such a lattice would have to be bounded, and cannot be countable. Sauer then asked the weaker...

In this paper we record some observations around groups definable in difference fields. We were motivated by a question of Zoe Chatzidakis as to whether any group definable in a model of ACFA is virtually definably embeddable in an algebraic group. We give a positive answer. Among the possibly “new” ingredients is the (stable) group configuration theorem in the ∗-definable category. The embedda...

Earlier work by several authors has focused on defining Boolean function classes by means of functional equations. In [10], it was shown that the classes of Boolean functions definable by functional equations coincide with initial segments of the quasi-ordered set (Ω,≤) made of the set Ω of Boolean functions, suitably quasi-ordered. Furthermore, the classes defined by finitely many equations co...

Let N be a structure definable in an o-minimal structureM and p ∈ SN (N), a complete N -1-type. If dimM(p) = 1 then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N -definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to...

The concept of definability of physical fields is introduced and a set-theoretical foundation for fields is proposed. In this foundation, we derive a nonlinear sigma model from a null postulate and then show that quantization of the model is necessary and sufficient for definability. We also obtain compactification of the spatial dimensions effectively and show its equivalence to quantization. ...

A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions ...