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Equivalence of differentiable functions, rational functions and polynomials

Masahito Shiota (1982)

Annales de l'institut Fourier

We show under some assumptions that a differentiable function can be transformed globally to a polynomial or a rational function by some diffeomorphism. One of the assumptions is that the function is proper, the number of critical points is finite, and the Milnor number of the germ at each critical point is finite.

Étude de la classification topologique des fonctions unimodales

Michel Cosnard (1985)

Annales de l'institut Fourier

À l’aide de la théorie des itinéraires et des suites de tricotage, nous étudions la conjugaison topologique des fonctions unimodales. Nous introduisons la notion de conjugaison macroscopique, caractérisée par l’égalité des suites de tricotage. Puis nous présentons un théorème de classification des fonctions unimodales. Pour illustrer ces résultats, nous montrons que l’ensemble des solutions de l’équation de Feigenbaum contient une infinité de classes topologiques.

Fonctions composées différentiables : cas algébrique

Jean-Claude Tougeron (1980)

Annales de l'institut Fourier

Soit f un morphisme propre et de Nash d’un ouvert Ω de R n dans un ouvert Ω ' de R p . Nous démontrons que l’image par f * de l’algèbre C ( Ω ' ) des fonctions réelles C dans Ω ' est fermée dans C ( Ω ) munie de sa topologie habituelle d’espace de Fréchet. Ce résultat généralise, dans le cas algébrique, un résultat de G. Glaeser sur les fonctions composées différentiables.

Fractal-classic interpolants

M. A. Navascués, M. V. Sebastián (2009)

Banach Center Publications

The methodology of fractal interpolation is very useful for processing experimental signals in order to extract their characteristics of complexity. We go further and prove that the Iterated Function System involved may also be used to obtain new approximants that are close to classical ones. In this work a classical function and a fractal function are combined to construct a new interpolant. The fractal function is first defined as a perturbation of a classical mapping. The additional condition...

Fractional Derivatives in Spaces of Generalized Functions

Stojanović, Mirjana (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 46Fxx, 58C05 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe generalize the two forms of the fractional derivatives (in Riemann-Liouville and Caputo sense) to spaces of generalized functions using appropriate techniques such as the multiplication of absolutely continuous function by the Heaviside function, and the analytical continuation. As an application, we give the two forms of the fractional derivatives of discontinuous functions in spaces of distributions.

Generators for algebras dense in L p -spaces

Alexander J. Izzo, Bo Li (2013)

Studia Mathematica

For various L p -spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in L p ( μ ) . For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...

Implicit function theorem for locally blow-analytic functions

Laurentiu Paunescu (2001)

Annales de l’institut Fourier

In this paper we prove the implicit function theorem for locally blow-analytic functions, and as an interesting application of using blow-analytic homeomorphisms, we describe a very easy way to resolve singularities of analytic curves.

La géométrie différentielle dans la catégorie P L

Howard Osborn (1973)

Annales de l'institut Fourier

La catégorie des fibrés vectoriels sur les variétés M linéaires par morceaux se plonge dans une catégorie des classes d’équivalence [ I ] de faisceaux I de modules sur les faisceaux A ( M ) de germes des fonctions lissables, et on construit les classes p ( [ I ] ) H 4 * ( M ; R ) de Pontrjagin, vérifiant des axiomes habituels. Chaque variété M possède un objet tangent [ ξ ( M ) ] dans cette catégorie, et p ( [ ξ ( M ) ] ) est la classe totale de Pontrjagin associée à M .

Le théorème de préparation différentiable en classe p

Guy Lassalle (1973)

Annales de l'institut Fourier

Exposé succinct d’une démonstration du théorème de division pour les fonctions p fois continûment différentiables ( p ) , donnant pour les classes du quotient et du reste les meilleurs résultats possibles lorsque p est fini.

Morse-Bott functions with two critical values on a surface

Irina Gelbukh (2021)

Czechoslovak Mathematical Journal

We study Morse-Bott functions with two critical values (equivalently, nonconstant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb...

Natural pseudodistances between closed surfaces

Pietro Donatini, Patrizio Frosini (2007)

Journal of the European Mathematical Society

Let us consider two closed surfaces , 𝒩 of class C 1 and two functions ϕ : , ψ : 𝒩 of class C 1 , called measuring functions. The natural pseudodistance d between the pairs ( , ) , ( 𝒩 , ψ ) is defined as the infimum of Θ ( f ) : = max P | ϕ ( P ) ψ ( f ( P ) ) | as f varies in the set of all homeomorphisms from onto 𝒩 . In this paper we prove that the natural pseudodistance equals either | c 1 c 2 | , 1 2 | c 1 c 2 | , or 1 3 | c 1 c 2 | , where c 1 and c 2 are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and critical values...

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