Through the application of layer potential techniques and Gohberg-Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.Mathematics Subject Classification (MSC2000): 35R30, 35C20.

Sampling information using timing is an approach that has received renewed attention in sampling theory. The question is how to map amplitude information into the timing domain. One such encoder, called time encoding machine, was introduced by Lazar and Tóth (2004 [23]) for the special case of band-limited functions. In this paper, we extend their result to a general framework including shift-invariant subspaces. We prove that time encoding machines may be considered as non-uniform sampling devices, where time locations are unknown a priori. Using this fact, we show that perfect representation and reconstruction of a signal with a time encoding machine is possible whenever this device satisfies some density property. We prove that this method is robust under timing quantization, and therefore can lead to the design of simple and energy efficient sampling devices.

The aim of this paper is to provide a mathematical and numerical framework for the analysis and design of bubble meta-screens. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on the acoustic wave propagation. In this paper, periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) is considered. It is shown that the structure behaves as an equivalent surface with Neumann boundary condition at the Minnaert resonant frequency which corresponds to a wavelength much greater than the size of the bubbles. Analytical formula for this resonance is derived. Numerical simulations confirm its accuracy and show how it depends on the ratio between the periodicity of the lattice, the size of the bubble, and the distance from the reflective surface. The results of this paper formally explain the super-absorption behavior observed in [V. Leroy et al., Phys. Rev. B, 2015].Mathematics Subject Classification (MSC2000): 35R30, 35C20.

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies a posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.The Homogeneous Electron Gas (HEG), where electrons are placed in a positively-charged uniform background, is a fundamental system in quantum physics and chemistry [1, 2]. In spite of its simplicity, it provides a good description of valence electrons in alcaline metals (e.g. in solid sodium [3]) and of the deep interior of white dwarfs [4,5]. It also plays a central role in the Local Density Approximation of Density Functional Theory [1], where it is used for deriving empirical functionals [6][7][8][9].The ground state of the HEG is highly correlated at low and intermediate densities. It was first predicted by Wigner that the particles form a BCC ferromagnetic crystal at small densities [10,11]. But correlation also plays an important role at high densities: The exact largedensity expansion of the correlation energy has a peculiar logarithm due to the long range of the Coulomb potential, which cannot be obtained from regular second-order perturbation theory [12][13][14][15].In principle, the correlation energy of the HEG is defined as the difference between the Hartree-Fock (HF) ground state energy and the true energy. However, many authors use instead the (paramagnetic) non-interacting Fermi Gas as a reference. This state indeed provides the first two terms of the large-density expansion of the HEG total energy [16]. But it is not the absolute ground state of the Hartree-Fock HEG. This was first suggested by Wigner [11] at high densities and then proved by Overhauser [17][18][19] who showed that the free Fermi Gas is unstable under the formation of spin or charge density waves. Recently, the phase diagram of the Hartree-Fock HEG has been studied numerically in great details [20][21][22][23]. It was discovered that the system is crystallized at all densities and that, at high densities, the electrons form an incommensurate lattice having more crystal sites than electrons [21,23]. Similar conclusions were reached in two space dimensions [23][24][25].These works naturally raise the question of determining the energy gain of the true HF ground state, compared to the free Fermi Gas. A too large deviation could affect the large-density expansion of the exact correlation en...