Abstract. The statistical properties of acoustic signals re ected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident u p o n i t . The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equations that characterize multiply scattered signals observed at the surface of the layered medium. The results of extensive n umerical simulations are presented, illustrating the scope of the theory. A n umber of inverse problems for randomly layered media are also formulated where we recover large scale properties of the sound speed pro le from the statistics of re ected signals.

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Abstract. This essentially numerical study, sets out to investigate various geometrical properties of exact boundary controllability o f t h e w ave equation when the control is applied on a part of the boundary. R elationships between the geometry of the domain, the geometry of the controlled boundary, the time needed to control and the energy of the control are dealt with. A new norm of the control and an energetic cost factor are introduced. These quantities enable a detailed appraisal of the numerical solutions obtained and the detection of trapped rays. Contents

BackgroundThe use of the desirability function approach combined with the response surface methodology (RSM), also called Desirability Optimization Methodology (DOM), has been successfully applied to solve medical, chemical, and technological questions. It is particularly efficient for the determination of the optimal conditions in natural or industrial processes involving different factors leading to the antagonist responses.ObjectivesSurprisingly, DOM has never been applied to the research programs devoted to the study of plant responses to the complex environmental changes, and thus to biotechnological questions.Materials and MethodsIn this article, DOM is used to study the response of Datura stramonium hairy roots (HRs), obtained by genetic transformation with Agrobacterium rhizogenes A4 strain, subjected to the jasmonate treatments.ResultsAntagonist effects on the growth and tropane alkaloid biosynthesis are confirmed. With a limited number of experimental conditions, it is shown that 0.06 mM jasmonic acid (JA) applied for 24 h leads to an optimal compromise. Hyoscyamine levels increase by up to 290% after 24 h and this treatment does not significantly inhibit biomass growth.ConclusionsIt is thus demonstrated that the use of DOM can efficiently - with a minimized number of replicates - leads to the optimization of the biotechnological processes.