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A gradient inequality at infinity for tame functions.

Didier D'Acunto, Vincent Grandjean (2005)

Revista Matemática Complutense

Let f be a C1 function defined over Rn and definable in a given o-minimal structure M expanding the real field. We prove here a gradient-like inequality at infinity in a neighborhood of an asymptotic critical value c. When f is C2 we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level.

Affinely invariant symmetry sets

Peter Giblin (2008)

Banach Center Publications

The classical medial axis and symmetry set of a smooth simple plane curve M, depending as they do on circles bitangent to M, are invariant under euclidean transformations. This article surveys the various ways in which the construction has been adapted to be invariant under affine transformations. They include affine distance and area constructions, and also the 'centre symmetry set' which generalizes central symmetry. A connexion is also made with the tricentre set of a convex plane curve, which...

Bifurcations in symplectic space

G. Ishikawa, S. Janeczko (2008)

Banach Center Publications

In this paper we take new steps in the theory of symplectic and isotropic bifurcations, by solving the classification problem under a natural equivalence in several typical cases. Moreover we define the notion of coisotropic varieties and formulate also the coisotropic bifurcation problem. We consider several symplectic invariants of isotropic and coisotropic varieties, providing illustrative examples in the simplest non-trivial cases.

Invertible polynomial mappings via Newton non-degeneracy

Ying Chen, Luis Renato G. Dias, Kiyoshi Takeuchi, Mihai Tibăr (2014)

Annales de l’institut Fourier

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

On asymptotic critical values and the Rabier Theorem

Zbigniew Jelonek (2004)

Banach Center Publications

Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let f = ( f , . . . , f m ) : X k m be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function...

On families of trajectories of an analytic gradient vector field

Adam Dzedzej, Zbigniew Szafraniec (2005)

Annales Polonici Mathematici

For an analytic function f:ℝⁿ,0 → ℝ,0 having a critical point at the origin, we describe the topological properties of the partition of the family of trajectories of the gradient equation ẋ = ∇f(x) attracted by the origin, given by characteristic exponents and asymptotic critical values.

On second order Thom-Boardman singularities

László M. Fehér, Balázs Kőműves (2006)

Fundamenta Mathematicae

We derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities Σ i , j . The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.

Singular open book structures from real mappings

Raimundo Araújo dos Santos, Ying Chen, Mihai Tibăr (2013)

Open Mathematics

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.

Some quantitative results in singularity theory

Y. Yomdin (2005)

Annales Polonici Mathematici

The classical singularity theory deals with singularities of various mathematical objects: curves and surfaces, mappings, solutions of differential equations, etc. In particular, singularity theory treats the tasks of recognition, description and classification of singularities in each of these cases. In many applications of singularity theory it is important to sharpen its basic results, making them "quantitative", i.e. providing explicit and effectively computable estimates for all the important...

Stability modulo singular sets

J. Iglesias, A. Portela, A. Rovella (2009)

Fundamenta Mathematicae

A new concept of stability, closely related to that of structural stability, is introduced and applied to the study of C¹ endomorphisms with singularities. A map that is stable in this sense is conjugate to each perturbation that is equivalent to it in a geometric sense. It is shown that this kind of stability implies Axiom A and Ω-stability, and that every critical point is wandering. A partial converse is also shown, providing new examples of C³ structurally stable maps.

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