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Displaying similar documents to “A theorem on generic intersections in an o-minimal structure”

On definably proper maps

Mário J. Edmundo, Marcello Mamino, Luca Prelli (2016)

Fundamenta Mathematicae

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In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of definable spaces. We give several other characterizations of definably proper, including one involving the existence of limits of definable types. We also...

Generic sets in definably compact groups

Ya'acov Peterzil, Anand Pillay (2007)

Fundamenta Mathematicae

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A subset X of a group G is called left genericif finitely many left translates of X cover G. Our main result is that if G is a definably compact group in an o-minimal structure and a definable X ⊆ G is not right generic then its complement is left generic. Among our additional results are (i) a new condition equivalent to definable compactness, (ii) the existence of a finitely additive invariant measure on definable sets in a definably compact group G in the case...

A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wiesław Pawłucki (2002)

Annales Polonici Mathematici

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A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such...

Minimality of non-σ-scattered orders

Tetsuya Ishiu, Justin Tatch Moore (2009)

Fundamenta Mathematicae

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We will characterize-under appropriate axiomatic assumptions-when a linear order is minimal with respect to not being a countable union of scattered suborders. We show that, assuming PFA⁺, the only linear orders which are minimal with respect to not being σ-scattered are either Countryman types or real types. We also outline a plausible approach to demonstrating the relative consistency of: There are no minimal non-σ-scattered linear orders. In the process of establishing these results,...

C 1 -minimal subsets of the circle

Dusa McDuff (1981)

Annales de l'institut Fourier

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Necessary conditions are found for a Cantor subset of the circle to be minimal for some C 1 -diffeomorphism. These conditions are not satisfied by the usual ternary Cantor set.

A first-order version of Pfaffian closure

Sergio Fratarcangeli (2008)

Fundamenta Mathematicae

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The purpose of this paper is to extend a theorem of Speissegger [J. Reine Angew. Math. 508 (1999)], which states that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Specifically, we display a collection of properties possessed by the real numbers that suffices for a version of the proof of this theorem to go through. The degree of flexibility revealed in this study permits the use of certain model-theoretic arguments for the first time, e.g. the compactness...

Two commuting maps without common minimal points

Tomasz Downarowicz (2011)

Colloquium Mathematicae

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We construct an example of two commuting homeomorphisms S, T of a compact metric space X such that the union of all minimal sets for S is disjoint from the union of all minimal sets for T. In other words, there are no common minimal points. This answers negatively a question posed in [C-L]. We remark that Furstenberg proved the existence of "doubly recurrent" points (see [F]). Not only are these points recurrent under both S and T, but they recur along the same sequence of powers. Our...

Minimal predictors in hat problems

Christopher S. Hardin, Alan D. Taylor (2010)

Fundamenta Mathematicae

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We consider a combinatorial problem related to guessing the values of a function at various points based on its values at certain other points, often presented by way of a hat-problem metaphor: there are a number of players who will have colored hats placed on their heads, and they wish to guess the colors of their own hats. A visibility relation specifies who can see which hats. This paper focuses on the existence of minimal predictors: strategies guaranteeing at least one player guesses...

Minimal Niven numbers

H. Fredricksen, E. J. Ionascu, F. Luca, P. Stănică (2008)

Acta Arithmetica

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Minimal sets of non-resonant torus homeomorphisms

Ferry Kwakkel (2011)

Fundamenta Mathematicae

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As was known to H. Poincaré, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms, that is, torus homeomorphisms isotopic to the identity for which the...