On the Ekeland-Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.Comment: v3: Final version, to appear in "Pacific J. Math.", 20 page

An extension theorem for regular functions of two quaternionic variables

For functions of two quaternionic variables that are regular in the sense of Fueter, we establish a result similar in spirit to the Hanges and Tr\`eves theorem. Namely, we show that a ball contained in the boundary of a domain is a propagator of regular extendability across the boundary.Comment: v3: Final version, to appear in "Journal of Mathematical Analysis and Applications", 10 pages, 1 figur

Time resolved tracking of a sound scatterer in a turbulent flow: non-stationary signal analysis and applications

It is known that ultrasound techniques yield non-intrusive measurements of hydrodynamic flows. For example, the study of the echoes produced by a large number of particle insonified by pulsed wavetrains has led to a now standard velocimetry technique. In this paper, we propose to extend the method to the continuous tracking of one single particle embedded in a complex flow. This gives a Lagrangian measurement of the fluid motion, which is of importance in mixing and turbulence studies. The method relies on the ability to resolve in time the Doppler shift of the sound scattered by the continuously insonfied particle. For this signal processing problem two classes of approaches are used: time-frequency analysis and parametric high resolution methods. In the first class we consider the spectrogram and reassigned spectrogram, and we apply it to detect the motion of a small bead settling in a fluid at rest. In more non-stationary turbulent flows where methods in the second class are more robust, we have adapted an Approximated Maximum Likelihood technique coupled with a generalized Kalman filter.Comment: 16 pages 9 figure

Loss of derivatives for systems of complex vector fields and sums of squares

We discuss, both for systems of complex vector fields and for sums of squares, the phenomenon discovered by Kohn of hypoellipticity with loss of derivatives.Comment: 13 page