A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak

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Real closed exponential fields

Paola D&amp;amp;#039;Aquino, Julia F. Knight, Salma Kuhlmann, Karen Lange (2012)

Fundamenta Mathematicae

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Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In 1993, he outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre’s construction and then analyze the complexity. Ressayre’s construction is canonical once we fix the real closed exponential field R, a residue field section k, and a well ordering...

Weierstrass division theorem in definable $C^{\infty}$ germs in a polynomially bounded o-minimal structure

Abdelhafed Elkhadiri, Hassan Sfouli (2006)

Annales Polonici Mathematici

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We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{\infty}$ functions.

O-minimal version of Whitney's extension theorem

Krzysztof Kurdyka, Wiesław Pawłucki (2014)

Studia Mathematica

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This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic $^{p}$ -Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a $^{p}$ -function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a...

Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

Krzysztof Jan Nowak (2009)

Annales Polonici Mathematici

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This paper investigates the geometry of the expansion $_{Q}$ of the real field ℝ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study non-standard models of the universal diagram T of $_{Q}$ in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass...

O-minimal fields with standard part map

Jana Maříková (2010)

Fundamenta Mathematicae

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Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V → k. Let $k_{i n d}$ be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in $k_{i n d}$ and conditions on (R,V) which imply o-minimality of $k_{i n d}$ . We also show that if R is ω-saturated and V is the convex hull of ℚ in R, then the sets definable in $k_{i n d}$ are exactly the standard parts of the sets definable in (R,V).

Definable stratification satisfying the Whitney property with exponent 1

Beata Kocel-Cynk (2007)

Annales Polonici Mathematici

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We prove that for a finite collection of sets $A ₁, . . ., A_{s} \subset ℝ^{k + n}$ definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto $ℝ^{k}$ satisfy the Whitney property with exponent 1.

The minimal resultant locus

Robert Rumely (2015)

Acta Arithmetica

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Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $γ \mapsto o r d (R e s (φ^{γ}))$ factors through a function $o r d R e s_{φ} (\cdot)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P ¹_{K}$ , or on a segment, and the minimal resultant locus is contained in the tree in $P ¹_{K}$ spanned by the fixed points...

Expansions of o-minimal structures by sparse sets

Harvey Friedman, Chris Miller (2001)

Fundamenta Mathematicae

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Given an o-minimal expansion ℜ of the ordered additive group of real numbers and E ⊆ ℝ, we consider the extent to which basic metric and topological properties of subsets of ℝ definable in the expansion (ℜ,E) are inherited by the subsets of ℝ definable in certain expansions of (ℜ,E). In particular, suppose that $f (E^{m})$ has no interior for each m ∈ ℕ and $f : ℝ^{m} \to ℝ$ definable in ℜ, and that every subset of ℝ definable in (ℜ,E) has interior or is nowhere dense. Then every subset of ℝ definable in (ℜ,(S))...

Exponential sums of the form $\sum χ {(x)}^{a x} ζ_{m}^{b x}$

S. Gurak (2007)

Acta Arithmetica

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A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

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A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a ₁, . . ., a_{h}$ and negative terms $b ₁, . . ., b_{k}$ . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $σ ⁺ = \sum_{i = 1}^{h} a_{i} = - \sum_{j = 1}^{k} b_{j}$ . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive...

The sum of digits of $⌊ n^{c} ⌋$

Johannes F. Morgenbesser (2011)

Acta Arithmetica

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Constructibility in Ackermann's set theory

C. Alkor

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CONTENTSIntroduction......................... 5Section I. Preliminaries............ 6 § 1. Notation..................... 6 § 2. Ackermann’s set theory and some extensions................. 7 § 3. Absoluteness............................................... 8 § 4. Ordinals................................................... 9 § 5. Reflection principles...................................... 10Section 2. The usual notion of constructibility.............. 11 § 1. General considerations about...

Density of Morse functions on sets definable in o-minimal structures

Ta Lê Loi (2006)

Annales Polonici Mathematici

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We present a tameness property of sets definable in o-minimal structures by showing that Morse functions on a definable closed set form a dense and open subset in the space of definable $C^{p}$ functions endowed with the Whitney topology.

A note on generalized projections in c₀

Beata Deręgowska, Barbara Lewandowska (2014)

Annales Polonici Mathematici

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Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by $_{V} (X, Z) : = {P \in (X, Z) : P |}_{V} = i d$ . Now let X = c₀ or $l_{\infty}^{m}$ , Z:= kerf for some f ∈ X* and $V : = Z \cap l ⁿ_{\infty}$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and...

A characterization of a minimal submanifold in $R^{n + p}$

Themis Koufogiorgos (1983)

Annales Polonici Mathematici

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A uniform dichotomy for generic $SL (2, ℝ)$ cocycles over a minimal base

Artur Avila, Jairo Bochi (2007)

Bulletin de la Société Mathématique de France

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We consider continuous $SL (2, ℝ)$ -cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

Some notes on the composite $G$ -valuations

Angeliki Kontolatou (1994)

Archivum Mathematicum

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In analogy with the notion of the composite semi-valuations, we define the composite $G$ -valuation $v$ from two other $G$ -valuations $w$ and $u$ . We consider a lexicographically exact sequence $(a, β) : A_{u} \to B_{v} \to C_{w}$ and the composite $G$ -valuation $v$ of a field $K$ with value group $B_{v}$ . If the assigned to $v$ set $R_{v} = {x \in K / v (x) \geq 0$ or $v (x)$ non comparable to $0}$ is a local ring, then a $G$ -valuation $w$ of $K$ into $C_{w}$ is defined with its assigned set $R_{w}$ a local ring, as well as another $G$ -valuation $u$ of a residue field is defined with $G$ -value group $A_{u}$ . ...

Elliptic functions, area integrals and the exponential square class on B₁(0) ⊆ ℝⁿ, n > 2

Caroline Sweezy (2004)

Studia Mathematica

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For two strictly elliptic operators L₀ and L₁ on the unit ball in ℝⁿ, whose coefficients have a difference function that satisfies a Carleson-type condition, it is shown that a pointwise comparison concerning Lusin area integrals is valid. This result is used to prove that if L₁u₁ = 0 in B₁(0) and $S u ₁ \in L^{\infty} (S^{n - 1})$ then ${u ₁ |}_{S^{n - 1}} = f$ lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and $S u ₀ \in L^{\infty}$ implies ${u ₀ |}_{S^{n - 1}}$ is in the exponential square class; here S is the Lusin area integral. The exponential...

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

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Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let $^{\infty}$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits $^{\infty}$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a $^{\infty}$ function f:Rⁿ → R. This implies $^{\infty}$ approximation of definable continuous functions and gluing of $^{\infty}$ functions defined on closed definable sets.

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this article, we study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give...

Minimal actions of homeomorphism groups

Yonatan Gutman (2008)

Fundamenta Mathematicae

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Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ \subset 2^{2^{X}}$ , the space of maximal chains in $2^{X}$ , equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{H o m e o (X)}$ , the universal minimal space of Homeo(X), is not transitive...