O-minimal Expansion Research Articles

A Superlinearly Convergent Smoothing Newton Continuation Algorithm for Variational Inequalities over Definable Sets

In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li [SIAM J. Optim., 23 (2013), pp. 745--769] to extend the smoothing Newton continuation algorithm of Hayashi, Yamashita, and Fukushima [SIAM J. Optim., 15 (2005), pp. 593--615] to variational inequalities over general closed convex sets $X$. We prove that when the underlying barrier has a gradient map that is definable in some o-minimal structure, the iterates generated converge superlinearly to a solution of the variational inequality. We further prove that if $X$ is proper and definable in the o-minimal structure $\mathbb{R}_{\mathrm{an}}^{\mathbb{R}_{\mathrm{alg}}}$, then the gradient map of its universal barrier is definable in the o-minimal expansion $\mathbb{R}_{\mathrm{an}, \exp}$. Finally, we consider the application of the algorithm to complementarity problems over epigraphs of matrix operator norm and nuclear norm and present preliminary numerical results.

Topological dynamics for groups definable in real closed field

We study the definable topological dynamics of groups definable in an o-minimal expansion of an arbitrary real closed field M. For a definable group G which admits a compact-torsion-free decomposition G=HK, we give a description of the minimal subflow and Ellis group of the universal definable G(M)-flow SG,ext(M). This Ellis group is isomorphic to NG(H)∩K(R), which extends the result of G. Jagiella from [7]. We also consider SL(2,M) as an example, explaining the difference between the universal definable SL(2,R)-flow, SG(R) and the universal definable G(M)-flow, SG,ext(M) for an arbitrary model M≻R.

Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation

AbstractThis article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.

Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation

AbstractThis article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.

Tame properties of sets and functions definable in weakly o-minimal structures

Let \({{\mathcal{M}}=(M, <, \ldots )}\) be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in \({{\mathcal{M}}}\) which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in \({{\mathcal{M}}}\) satisfy an extended version of IVP. After introducing a weak version of definable connectedness in \({{\mathcal{M}}}\) , we prove that strong cells in \({{\mathcal{M}}}\) are weakly definably connected, so every set definable in \({{\mathcal{M}}}\) is a finite union of its weakly definably connected components, provided that \({{\mathcal{M}}}\) has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in \({{\mathcal{M}}}\) and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph (NDG) which was examined for definable functions in (Dolich et al. in Trans. Am. Math. Soc. 362:1371–1411, 2010) and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure \({{\mathcal{M}}}\) having cell decomposition, satisfies NDG, i.e. every definable function in \({{\mathcal{M}}}\) has no dense graph.

Coverings by open cells

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.

On smooth locally o-minimal functions

We study the $\mathcal{C}^{\infty}$-smooth functions which are locally definable in an o-minimal expansion of the real exponential field with some additional smoothness conditions. Here, the local definability generalizes the subanalytic setting to more transcendental sets and functions. The focus is set on the locally definable diffeomorphisms between manifolds, for which we prove analogies to classical differential geometric results. Moreover, we investigate the relation between classical diffeomorphy and locally definable diffeomorphy.

Approximation of o-minimal maps satisfying a Lipschitz condition

Consider an o-minimal expansion of the real field. We show that definable Lipschitz continuous maps can be definably fine approximated by definable continuously differentiable Lipschitz maps whose Lipschitz constant is close to that of the original map.

Connected components of groups and o-minimal expansions of real closed fields

Connected components of groups and o-minimal expansions of real closed fields

The universal covering map in o-minimal expansions of groups

In this paper we study locally definable manifolds and we prove: (i) the existence of universal locally definable covering maps; (ii) invariance results for locally definable covering maps, o-minimal fundamental groups and fundamental groupoids; (iii) monodromy equivalence for locally constant o-minimal sheaves; (iv) classification results for locally definable covering maps; (v) o-minimal Hurewicz and Seifert–van Kampen theorems.

O-minimal homotopy and generalized (co)homology

This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes.

Discrete subgroups of locally definable groups

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group \(G\) in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of \(G\). Given a locally definable connected group \(G\) (not necessarily definably generated), we prove that the \(n\)-torsion subgroup of \(G\) is finite and that every zero-dimensional compatible subgroup of \(G\) has finite rank. Under a convexity hypothesis, we show that every zero-dimensional compatible subgroup of \(G\) is finitely generated.

Some Results and Problems on Complex Germs with Definable Mittag–Leffler Stars

Working in an o-minimal expansion of the real field, we investigate when a germ (around zero, say) of a complex analytic function has a definable analytic continuation to its Mittag–Leffler star. As an application we show that any algebro-logarithmic function that is complex analytic in a neighborhood of the origin in C has an analytic continuation to all but finitely many points in C.

Elementary equivalence of lattices of open sets definable in o-minimal expansions of real closed fields

Elementary equivalence of lattices of open sets definable in o-minimal expansions of real closed fields

Definable smoothing of continuous functions

Let $\mathbf{R}$ be an o-minimal expansion of a real closed field. Given definable continuous functions $f:U\rightarrow R$ and $\epsilon:U\rightarrow(0,+\infty)$, where $U$ is an open subset of $R^{n}$, we construct a definable $C^{m}$-function $g:U\to R$ with $\vert g(x)-f(x)\vert <\epsilon(x)$ for all $x\in U$. Moreover, we show that if $f$ is uniformly continuous, then $g$ can also chosen to be uniformly continuous.

Hausdorff limits of Rolle leaves

Let \({\mathcal{R}}\) be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T ∞-limits over \({\mathcal{R}}\), that do not in general fall under the scope of Marker and Steinhorn’s definability-of-types theorem. We prove that if \({\mathcal{R}}\) admits analytic cell decomposition, then every T ∞-limit over \({\mathcal{R}}\) is definable in the pfaffian closure of \({\mathcal{R}}\).

First order tameness of measures

We develop a general framework for measure theory and integration theory that is compatible with o-minimality. Therefore the following natural definitions are introduced. Given are an o-minimal structure M and a Borel measure μ on some Rn. We say that μ is M-compatible if there is an o-minimal expansion of M such that for every parameterized family of subsets of Rn that is definable in M the corresponding family of μ-measures is definable in this o-minimal expansion. We say that μ is M-tame if there is an o-minimal expansion of M such that for every parameterized family of functions on Rn that is definable in M the corresponding family of integrals with respect to μ is definable in this o-minimal expansion. We substantiate these definitions with existing and many new examples. We investigate the Lebesgue measure in their light. We prove definable versions of fundamental results such as the theorem of Radon–Nikodym, the Lebesgue decomposition theorem and the Riesz representation theorem. They allow us to describe explicitly compatible and tame measures and to classify them in dimension one.

Locally o-minimal structures

In this paper we study (strongly) locally o-minimal structures. We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (R, +, <).

Definable groups as homomorphic images of semi-linear and field-definable groups

We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their locally definable covers. As a corollary, we prove the compact domination conjecture in o-minimal expansions of ordered groups.

Local analysis for semi-bounded groups

An o-minimal expansion $\mathcal{M}=\langle M, <, +, 0, \dots\rangle$ of an ordered group is called <i>semi-bounded</i> if it does not expand a real closed field. Possibly, it defines a real closed field with bounded domain $I\subseteq M$. Let us call