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Bernhard Riemann, the ear, and an atom of consciousness
(2021)SAM Research ReportWhy did Bernhard Riemann (1826–1866), arguably the most original mathematician of his generation, spend the last year of life investigating the mechanism of hearing? Fighting tuberculosis and the hostility of eminent scientists such as Hermann Helmholtz, he appeared to forsake mathematics to prosecute a case close to his heart. Only sketchy pages from his last paper remain, but here we assemble some significant clues and triangulate from ...Report -
Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes
(2021)SAM Research ReportReport -
Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies
(2021)SAM Research ReportAs a continuation of the previous works [13, 4, 15], this paper provides several contributions to the mathematical analysis of subwavelength resonances in a high-contrast medium containing N acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of a N dimensional linear system, and characterizes resonant frequencies as the solutions to a N-dimensional ...Report -
On the validity of the tight-binding method for describing systems of subwavelength resonators
(2021)SAM Research ReportThe goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We firstly study how the capacitance matrix formalism, both when the material parameters are static ...Report -
Close-to-touching acoustic subwavelength resonators: eigenfrequency separation and gradient blow-up
(2020)SAM Research ReportIn this paper, we study the behaviour of the coupled subwavelength resonant modes when two high-contrast acoustic resonators are brought close together. We consider the case of spherical resonators and use bispherical coordinates to derive explicit representations for the capacitance coefficients which, we show, capture the system's resonant behaviour at leading order. We prove that the pair of resonators has two subwavelength resonant ...Report -
Subwavelength resonances in 1D high-contrast acoustic media
(2022)SAM Research ReportWe propose a mathematical theory of acoustic wave scattering in one-dimensional finite high-contrast media. The system considered is constituted of a finite alternance of high-contrast segments of arbitrary lengths and interdistances, called the ``resonators'', and a background medium. We prove the existence of subwavelength resonances, which are the counterparts of the well-known Minnaert resonances in 3D systems. Our main contribution ...Report -
Transmission properties of space-time modulated metamaterials
(2022)SAM Research ReportWe prove the possibility of achieving exponentially growing wave propagation in space-time modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence of k-gaps in the band structures of space-time modulated systems of subwavelength resonators.Report -
Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials
(2021)SAM Research ReportOur aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one ...Report -
Super-resolution in recovering embedded electromagnetic sources in high contrast media
(2020)SAM Research ReportThe purpose of this work is to provide a rigorous mathematical analysis of the expected super-resolution phenomenon in the time-reversal imaging of electromagnetic (EM) radiating sources embedded in a high contrast medium. It is known that the resolution limit is essentially determined by the sharpness of the imaginary part of the EM Green's tensor for the associated background. We first establish the close connection between the resolution ...Report -
Mathematical modelling of plasmonic strain sensors
(2020)SAM Research ReportWe provide a mathematical analysis for a metasurface constructed of plasmonic nanoparticles mounted periodically on the surface of a microcapsule. We derive an effective transmission condition, which exhibits resonances depending on the inter-particle distance. When the microcapsule is deformed, the resonances are shifted. We fully characterise the dependence of these resonances on the deformation of the microcapsule, enabling the detection ...Report