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On the geometric properties of Vandermonde's mapping and on the problem of moments

  • V. P. Kostov (a1)

Synopsis

In this paper we prove that the domain of hyperbolicity of the polynomial xn + λ2nn−23xn−3+ … + λniϵR intersected by the half-space λ2 ≧ – 1, has the property of Whitney, i.e., every two points of this set can be connected by a piecewise-smooth curve belonging to it, whose length is ≦C times greater than the euclidian distance between the points, where the constant C does not depend on the choice of the points. Parallel with this, we show that the values x1≦x2≦…≦xn of a random variable are uniquely determined by the corresponding probabilities and by thefirst n moments.

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1Alexandrov, P. S. and Pasynkov, B. A.. Introduction in the theory of dimension (Moscow: Science, 1973—in Russian).
2Arnol'd, V. I.. Hyperbolic polynomials and Vandermonde's mapping. Functional Anal. Appl. 20 (1986), 5253 (in Russian).
3Ball, J. M.. Differentiability properties of symmetric and isotopic functions. Duke Math. J. 51 (1984), 699728.
4Barbangon, G.. A propos de thèoréme de Newton pour les fonctions de classe C n et d'une generalisation de la notion de multiplicateur rugueux. Ann. Fac. Sci. Phnom Penh (1969).
5Guivental', A. B.. Moments of random variables and equivariant Morse lemma. Russian Math. Surveys 42 (1987), 275276 (translation of Uspekhi Mat. Nauk 42 (1987), 221–222).

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