Related papers: Pathological examples of structures with o-minimal…

In this short note we give and discuss a general multilinear expression of the structure function of an arbitrary semicoherent system in terms of its minimal path and cut sets. We also examine the link between the number of minimal path and…

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as Andr\'e-Oort conjecture. Among the many tools developed in…

We compute the cardinality $\mathfrak n_{\dim}(\mathcal M)$ of the sets of dimension functions on the ordered structures $\mathcal M$. The inequality $\mathfrak n_{\dim}(\mathcal M) \leq 1$ holds if $\mathcal M$ is a d-minimal expansion of…

We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of…

Let R be an o-minimal field with a proper convex subring V. We axiomatize the class of all structures (R,V) such that k_ind, the corresponding residue field with structure induced from R via the residue map, is o-minimal. More precisely, in…

For a weakly mixing bounded rank-one construction the disjointness of its powers is proved. For non-rigid constructions we get minimal self-joinings. Examples of non-mixing rank one actions with explicit weak closure are proposed.

Let $\mathcal F=(F, +. \cdot, <, 0, 1, \dots)$ be a definably complete locally o-minimal expansion of an ordered field. We demonstrate the existence of definable quotients of definable sets by definable equivalence relations when several…

We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite…

This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of the frontier of A is strictly smaller than the dimension of A itself, and that A has a…

Let I be a dense linear order with a left endpoint but no right endpoint. We consider the lattice L(I) of finite unions of closed intervals of I. This lattice arises naturally in the setting of o-minimality, as these are precisely the…

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…

A note connecting arguments scattered in the extant literature proving that, in any o-minimal expansion of the real field, a definable family of sets has the property that the set of parameters corresponding to finite-volume fibers is…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

We introduce an explicit combinatorial characterization of the minimal model ${\cal O}_{\infty}$ of the coloured operad ${\cal O}$ encoding non-symmetric operads. In our description of ${\cal O}_{\infty}$, the spaces of operations are…

In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets. As an…

We construct an open set $\Omega\subset\mathbb{R}^N$ on which an eigenvalue problem for the $p-$Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann…

A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups with only finitely many convex subgroups…

Non--minimal $q$-deformations are defined. Their role in the explicit construction of the matrix elements of the generators of ${\cal U}_{q}(SO(5))$ on suitably parametrized bases are exhibited. The implications are discussed.

We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…

We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed…