Percorrer por autor "Dias, C. Nuno"
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- A metaplectic perspective of the uncertainty principle in the Linear Canonical Transform domainPublication . Dias, C. Nuno; Gosson, Maurice de; Prata, João Nuno; ElsevierWe derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
- On Orthogonal Projections of Symplectic BallsPublication . Dias, C. Nuno; Gosson, A. Maurice; Prata, N. JoãoWe study the orthogonal projections of symplectic balls in R2n on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov’s non-squeezing theorem. We use a conceptually simpler approachwhere the Schur complement of a matrix plays a central role. An application to the partial traces of density matrices is given.
- Vibration modes of the Euler–Bernoulli beam equation with singularitiesPublication . Dias, C. Nuno; Jorge, Cristina; Prata, João NunoWe consider the time dependent Euler–Bernoulli beam equation with discontinuous and singular coeffi-cients. Using an extension of the Hörmander product of distributions with non-intersecting singular supports (L. Hörmander, 1983 [25]), we obtain an explicit formulation of the differential problem which is strictly defined within the space of Schwartz distributions. We determine the general structure of its separable solu-tions and prove existence, uniqueness and regularity results under quite general conditions. This formalism is used to study the dynamics of an Euler–Bernoulli beam model with discontinuous flexural stiffness and structural cracks. We consider the cases of simply supported and clamped-clamped boundary conditions and study the relation between the characteristic frequencies of the beam and the position, magnitude and struc-ture of the singularities in the flexural stiffness. Our results are compared with some recent formulations of the same problem.