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Browsing by Author "Dias, N.C."

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  • Ordinary differential equations with point interactions: An inverse problema
    Publication . Dias, N.C.; C., Jorge; Prata, J.N.
    We study a class of linear ordinary differential equations (ODEs) with distributional coefficients. These equations are defined using an intrinsic multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with nonintersecting singular supports (Hörmander in The analysis of linear partial differential operators I, Springer, Berlin, 1983). We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler–Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.
    2019-03Newspaper article Open access Show more
  • Uncertainty principle via variational calculus on modulation spaces.
    Publication . Dias, N.C.; Luef, Frank; Prata, João .
    We approach uncertainty principles of Cowling-Price-Heis-enberg-type as a variational principle on modulation spaces. In our discussion we are naturally led to compact localization operators with symbols in modulation spaces. The optimal constant in these uncertainty principles is the smallest eigenvalue of the inverse of a compact localization operator. The Euler-Lagrange equations for the associated functional provide equations for the eigenfunctions of the smallest eigenvalue of these compact localization operators. As a by-product of our proofs we derive a generalization to mixed-norm spaces of an inequality for Wigner and Ambiguity functions due do Lieb.
    2022Contribution to journal Restricted access Show more