11 research outputs found

    On the measure of polynomials attaining maxima on a vertex

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    We calculate the probability that a k-homogeneous polynomial in n variables attain a local maximum on a vertex in terms of the “sharpness” of the vertex, and then study the dependence of this measure on the growth of dimension and degree. We find that the behavior of vertices with orthogonal edges is markedly different to that of sharper vertices. If the degree k grows with the dimension n, the probability that a polynomial attain a local maximum tends to 1/2, but for orthogonal edges the growth-rate of k must be larger than nlnn, while for sharper vertices a growth-rate larger than lnn will suffice.Fil: Pinasco, Damian. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; Argentin

    A probabilistic Approach to polynomial Inequalities

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    We define a probability measure on the space of polynomials over Rn in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities. Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of n 1 tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.Fil: Pinasco, Damian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Algebras of functions with prescribed radii of boundedness and the spectra of H(u)

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    We study the spectra of algebras of holomorphic functions with prescribed radii of boundedness, and use these results to study the τω and τδ spectra of H(U), where U is an open subset of a non-separable Banach space. We construct τδ continuous characters on H(U) which are not evaluations at points of U. We also discuss subsets of ℓ∞ which are bounding for H(U), U⊂ℓ∞.Fil: García, Domingo. Universidad de Valencia; EspañaFil: Maestre, Manuel. Universidad de Valencia; EspañaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    To what extent does the dual Banach space E′ determine the polynomials over E?

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    We show that under conditions of regularity, if E′ is isomorphic to F′, then the spaces of homogeneous polynomials over E and F are isomorphic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral, extendible) are shown to be isomorphic in full generality.Fil: Lassalle, Silvia Beatriz. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad de San Andrés; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Tchakaloff’s theorem and k-integral polynomials in Banach spaces

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    Tchakaloff’s theorem gives a quadrature formula for polynomials of a given degree with respect to a compactly supported positive measure which is absolutely continuous with respect to Lebesgue measure. We study the validity of two possible analogues of Tchakaloff’s theorem in an infinite-dimensional Banach space E: a weak form valid when E has a Schauder basis, and a stronger form requiring conditions on the support of the measure as well as on the space E.Fil: Pinasco, Damian. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; Argentin

    Linear and bilinear operators and their zero-sets

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    We study the extent to which several classical results relating linear or multilinear forms and their zero-sets can be generalised to linear or bilinear operators with values in Rn. We find some analogues of the classical theorems, and also some restrictions.Fil: Pinasco, Damian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Algebras of functions with prescribed radii of boundedness and the spectra of H(U)

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    We study the spectra of algebras of holomorphic functions with prescribed radii of boundedness, and use these results to study the τω and τδ spectra of H(U), where U is an open subset of a non-separable Banach space. We construct τδ continuous characters on H(U) which are not evaluations at points of U. We also discuss subsets of ℓ∞ which are bounding for H(U), U⊂ℓ∞.Fil: García, Domingo. Universidad de Valencia; EspañaFil: Maestre, Manuel. Universidad de Valencia; EspañaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Orthant probabilities and the attainment of maxima on a vertex of a simplex

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    We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree k>4, the probability that a k-homogeneous polynomial in n variables attains a local constrained maximum on a vertex of the n-dimensional simplex tends to one as the dimension n grows. The bounds we obtain for the orthant probabilities are tight up to log⁡(n) factors.Fil: Pinasco, Damian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; ArgentinaFil: Smucler, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; ArgentinaFil: Zalduendo, Ignacio Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella; Argentin

    The Spectra of Algebras of Group-Symmetric Functions

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    In the study of the spectra of algebras of holomorphic functions on a Banach space E, the bidual E″ has a central role, and the spectrum is often shown to be locally homeomorphic to E″. In this paper we consider the problem of spectra of subalgebras invariant under the action of a group (functions f such that f ○ g = f). It is natural to attempt a characterization in terms of the space of orbits E″/ obtained from E″ through the action of the group, so we pursue this approach here and introduce an analytic structure on the spectrum in some situations. In other situations we encounter some obstacles: in some cases, the lack of structure of E″/ itself; in others, problems of weak continuity and non-approximability of functions in the algebra. We also define a convolution operation related to the spectrum.Fil: García, Domingo. Universidad de Valencia; EspañaFil: Maestre Vera, Manuel. Universidad de Valencia; EspañaFil: Zalduendo, Ignacio Martin. Universidad Torcuato Di Tella. Instituto "Torcuato Di Tella"; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

    Orthogonally additive holomorphic functions of bounded type over C(K)

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    It is known that all k-homogeneous orthogonally additive polynomials P over C(K) are of the form P(x)= ∫Kxkdμ. Thus, x → xk factors all orthogonally additive polynomials through some linear form μ. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic function of bounded type f over C(K) is of the form f(x)=∫Kh(x)dμ for some μ and holomorphic h : C (K) → L1(μ) of bounded type.Fil: Carando, Daniel Germán. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Lassalle, Silvia Beatriz. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Zalduendo, Ignacio Martin. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentin
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