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A hierarchy in the family of real surjective functions

Mar Fenoy-Muñoz, José Luis Gámez-Merino, Gustavo A. Muñoz-Fernández, Eva Sáez-Maestro (2017)

Open Mathematics

This expository paper focuses on the study of extreme surjective functions in ℝℝ. We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The...

An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem

Frank Oertel (2015)

Dependence Modeling

We revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.

An o-minimal structure which does not admit C ∞ cellular decomposition

Olivier Le Gal, Jean-Philippe Rolin (2009)

Annales de l’institut Fourier

We present an example of an o-minimal structure which does not admit C ∞ cellular decomposition. To this end, we construct a function H whose germ at the origin admits a C k representative for each integer k , but no C ∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜 n ( H ) induced by H . The o-minimality of the structure generated by H is deduced from this quasianalyticity property.

Analyse 2-microlocale et développementen série de chirps d'une fonction de Riemann et de ses généralisations

Daniel Boichu (1994)

Colloquium Mathematicae

En dimension 1 on analyse la fonction irrégulière r ( x ) = ∑ n = 1 ∞ n - p s i n ( n p x ) (p entier ≥ 2) en un point x 0 de dérivabilité (π est un tel point) et on démontre que le terme d’erreur est un chirp de classe (1 + 1/(2p-2), 1/(p-1), (p-1)/p). La fonction r(x) est dans l’espace 2-microlocal C x 0 s , s ' si et seulement si s+s’ ≤ 1 - 1/p et ps+s’≤ p - 1/2. En dimension 2, on obtient en (π,π) l’existence d’un plan tangent pour la surface z = ∑ m , n = 1 ∞ ( m 2 + n 2 ) - γ s i n ( m 2 x + n 2 y ) dès que γ>1.

Characterization of compact subsets of curves with ω-continuous derivatives

Marcin Pilipczuk (2011)

Fundamenta Mathematicae

We give a characterization of compact subsets of finite unions of disjoint finite-length curves in ℝⁿ with ω-continuous derivative and without self-intersections. Intuitively, our condition can be formulated as follows: there exists a finite set of regular curves covering a compact set K iff every triple of points of K behaves like a triple of points of a regular curve. This work was inspired by theorems by Jones, Okikiolu, Schul and others that characterize compact subsets of...

Comportement local moyen de la fonction de Brjuno

Michel Balazard, Bruno Martin (2012)

Fundamenta Mathematicae

We describe the average behaviour of the Brjuno function Φ in the neighbourhood of any given point of the unit interval. In particular, we show that the Lebesgue set of Φ is the set of Brjuno numbers and we find the asymptotic behaviour of the modulus of continuity of the integral of Φ.

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