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The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Satoshi Koike, Laurentiu Paunescu (2009)

Annales de l’institut Fourier

Let A ⊂ ℝ n be a set-germ at 0 ∈ ℝ n such that 0 ∈ A ¯ . We say that r ∈ S n - 1 is a direction of A at 0 ∈ ℝ n if there is a sequence of points { x i } ⊂ A ∖ { 0 } tending to 0 ∈ ℝ n such that x i ∥ x i ∥ → r as i → ∞ . Let D ( A ) denote the set of all directions of A at 0 ∈ ℝ n .Let A , B ⊂ ℝ n be subanalytic set-germs at 0 ∈ ℝ n such that 0 ∈ A ¯ ∩ B ¯ . We study the problem of whether the dimension of the common direction set, dim ( D ( A ) ∩ D ( B ) ) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular if two subanalytic...

The index of analytic vector fields and Newton polyhedra

Carles Bivià-Ausina (2003)

Fundamenta Mathematicae

We prove that if f:(ℝⁿ,0) → (ℝⁿ,0) is an analytic map germ such that f - 1 ( 0 ) = 0 and f satisfies a certain non-degeneracy condition with respect to a Newton polyhedron Γ₊ ⊆ ℝⁿ, then the index of f only depends on the principal parts of f with respect to the compact faces of Γ₊. In particular, we obtain a known result on the index of semi-weighted-homogeneous map germs. We also discuss non-degenerate vector fields in the sense of Khovanskiĭand special applications of our results to planar analytic vector fields....

The L²-invariants and Morse numbers

Vladimir V. Sharko (2009)

Banach Center Publications

We study the homotopy invariants of free cochain complexes and Hilbert complexes. These invariants are applied to calculation of exact values of Morse numbers of smooth manifolds.

Thom polynomials and Schur functions: the singularities I 2 , 2 ( - )

Piotr Pragacz (2007)

Annales de l’institut Fourier

We give the Thom polynomials for the singularities I 2 , 2 associated with maps ( ℂ • , 0 ) → ( ℂ • + k , 0 ) with parameter k ≥ 0 . Our computations combine the characterization of Thom polynomials via the “method of restriction equations” of Rimanyi et al. with the techniques of Schur functions.

Thom polynomials and Schur functions: the singularities I I I 2 , 3 ( - )

Özer Öztürk (2010)

Annales Polonici Mathematici

We give a closed formula for the Thom polynomials of the singularities I I I 2 , 3 ( - ) in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the “method of restriction equations” of Rimányi et al. with the techniques of Schur functions.

Currently displaying 121 – 140 of 148