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Resumen: Inverse spectral and scattering problems are a classical subject in mathematical physics. In this talk we present a particular variant: the inverse resonance problem in one dimension, particularly for the Schrödinger equation. In addition to the uniqueness question we investigate which information may be contained from finite noisy data. This is of interest, since, in practical settings, one cannot expect to obtain all the necessary data and, in any case, recovery algorithms cannot make use of all data even if they were available.
Resumen: Nodal patterns of oscillating membranes have been known for hundreds of years and are often demonstrated in undergraduate physics classes. They are usually called Chladni figures, although Robert Hooke demonstrated them at Royal Society two centuries before Chladni. Mathematically speaking, these figures are the nodal sets of eigenfunctions of the Laplace operator with Dirichlet boundary conditions on the corresponding domain (or manifold). In spite of them being known for quite a long time, the understanding of these patterns (e.g., how large the nodal set is, or how many nodal domains the pattern splits the membrane into) remains very incomplete. These patterns nowadays attract attention of leading mathematicians and physicists alike. The talk will provide a brief history of the subject and some recent results on the nodal domain count.
Resumen: The elementary units for human speech are known as phonemes, and the utterance of each phoneme is governed by a particular shape of the human vocal tract. In this talk, a mathematical description is presented for the shape of the vocal tract during the creation of each phoneme, which corresponds to a direct problem. Then, a corresponding inverse problem is analyzed; namely, the determination of the shape of the human vocal tract from the measurement of sound pressure associated with an uttered phoneme.
El Coloquio Interinstitucional de Análisis y sus Aplicaciones ha recibido apoyo del proyecto PAPIME PE100716.
