We describe results from an application for relevance assessment in a setting related to multi-document summarization. For the task of characterizing given document collections by a short list of relevant terms, we have proposed the term utility function PxR. The measure is competitive to a variety of utility functions commonly used in text mining. Our function incorporates a user-definable par...

Let G = 〈G, ·〉 be a group definable in an o-minimal structure M. A subset H of G is G-definable if H is definable in the structure 〈G, ·〉 (while definable means definable in the structure M). Assume G has no Gdefinable proper subgroup of finite index. In this paper we prove that if G has no nontrivial abelian normal subgroup, then G is the direct product of G-definable subgroups H1, . . . ,Hk s...

Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps which are respectively increasing or decreasing to the same point is used to obtain a further equivalent criterion for Dedekind completeness of ordered fields: Every continuous one-to-one function defined on a closed bounded interval maps interior of that interval to the interior of the image. Next, it is shown that ov...

The concept of definability of physical fields in a set-theoretical foundation is introduced. A set theory is selected in which we get mathematics enough to produce a nonlinear sigma model. Quantization of the model requires only a null postulate and is then shown to be necessary and sufficient for definability in the theory. We obtain scale invariance and compactifica-tion of the spatial dimen...

Our informal description of the modularity of a definable set X of Morley dimension and degree 1 should have been: “There is no n-dimensional definable family of definable subsets of X × X with n ≥ 2, each of Morley dimension and Morley degree 1 and with pairwise intersections finite” in place of “There should be no infinite definable family of definable subsets of X×X , each of Morley dimensio...

We study the Vapnik-Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular we obtain uniform bounds on VC density of definable families in finite U-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

Fix an o-minimal expansion of the real exponential field that possesses smooth cell decomposition. We prove smooth approximation of definable differentiable functions with respect to a topology closely related to the Whitney topology. As a consequence we obtain a strong version of definable smooth separation of sets, which we use to prove that definable smooth manifolds are definably affine.

In [2] o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in W -definable groups – which are examples of locally definable spaces. We s...

Let G be a compact definable C∞ group and 2 ≤ r < ∞. Let X be a noncompact affine definable CG manifold and X1, . . . , Xk noncompact codimension one definable CG submanifolds of X such that X1, . . . , Xk are in general position in X and (X;X1, . . . , Xk) satisfies the frontier condition. We prove that (X;X1, . . . , Xk) admits a unique definable C∞G manifold structure (Y ;Y1, . . . , Yk).