Resume. Nous etudions une membrane vibrante avec une distribution de densite dependant d'un petit parametre e, qui converge, lorsque e \ 0, vers une densite uniforme plus une masse ponctuelle a l'origine. Nous mettons en evidence l'existence de vibrations locales, au voisinage de l'origine, et globales de la membrane. L'etude asymptotique lorsque e \ 0 est effectuee a 1'aide de la methode des developpements asymptotiques raccordes.Abstract. We study a vibrating membrane with a distribution of density depending on e, which converges, as e \ 0, to a uniform density, plus a point mass at the origin. We establish local vibrations at the vicinity of the origin and global vibrations of the membrane. The asymptotic study for e \ 0 is performed using the method of matched asymptotic expansions.1. Introduction. We consider vibrating systems containing a small region, of diameter O(e), including the origin, where the density is very much higher than elsewhere. Quite different cases arise depending on the space dimension N and the order of magnitude of the ratio e~m of densities. Many studies are devoted to this problem of concentrated masses: E. Sanchez-Palencia [1], E. Sanchez-Palencia and H. Tchatat [2], H. Tchatat [3], O. A. Oleinik [4], In this paper we study the case N = 2 (i.e., the vibrating membrane) with m >2. Using the method of matched asymptotic expansions (see for instance [5] and [6]), we derive the structure of the eigenfunctions, which is not given by other methods. It appears that there are two kinds of eigenvibration:

Falling, and the fear of falling, is a serious health problem among the elderly. It often results in physical and mental injuries that have the potential to severely reduce their mobility, independence and overall quality of life. Nevertheless, the consequences of a fall can be largely diminished by providing fast assistance. These facts have lead to the development of several automatic fall detection systems. Recently, many researches have focused particularly on smartphone-based applications. In this paper, we study the capacity of smartphone built-in sensors to differentiate fall events from activities of daily living. We explore, in particular, the information provided by the accelerometer, magnetometer and gyroscope sensors. A collection of features is analyzed and the efficiency of different sensor output combinations is tested using experimental data. Based on these results, a new, simple, and reliable algorithm for fall detection is proposed. The proposed method is a threshold-based algorithm and is designed to require a low battery power consumption. The evaluation of the performance of the algorithm in collected data indicates 100 % for sensitivity and 93 % for specificity. Furthermore, evaluation conducted on a public dataset, for comparison with other existing smartphone-based fall detection algorithms, shows the high potential of the proposed method.

We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli-Navier equilibrium equations and reduced Maxwell-Gauss equations. Mathematics Subject Classifications (2000)74K10 · 74G10 · 78A30

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It is generally accepted that colorectal cancer is initiated in the small pits, called crypts, that line the colon. Normal crypts exhibit a regular pit pattern, similar in twodimensions to a U-shape, but aberrant crypts display different patterns, and in some cases show bifurcation. According to several medical articles, there is an interest in correlating pit patterns and the cellular kinetics, namely of proliferative and apoptotic cells, in colonic crypts. This paper proposes and implements a hybrid convection-diffusion-shape model for simulating and predicting what has been validated medically, with respect to some aberrant colonic crypt morphogenesis. The model demonstrates crypt fission, in which a single crypt starts dividing into two crypts, when there is an increase of proliferative cells. The overall model couples the cell movement and proliferation equations with the crypt geometry. It relies on classical continuum transport/mass conservation laws and the changes in the crypt shape are driven by the pressure exerted by the cells on the crypt wall. This pressure is related to the cell velocity by a Darcy-type law. Numerical Communicated by Gabriel Wittum. simulations are conducted and comparisons with the medical results are shown.