Multifractal analysis for the occupation measure of stable-like processes

In this article, we investigate the local behavior of the occupation measure µ of a class of real-valued Markov processes M, defined via a SDE. This (random) measure describes the time spent in each set A ⊂ R by the sample paths of M. We compute the multifractal spectrum of µ, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), where the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behavior. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes

Ultrasonic characterization and multiscale analysis for the evaluation of dental implant stability: a sensitivity study Biomedical Signal Processing and Control 42 (2018) 37-44

International audienceWith the aim of surgical success, the evaluation of dental implant long-term stability is an important task for dentists. About that, the complexity of the newly formed bone and the complex boundary conditions at the bone-implant interface induce the main difficulties. In this context, for the quantitative evaluation of primary and secondary stabilities of dental implants, ultrasound based techniques have already been proven to be effective. The microstructure, the mechanical properties and the geometry of the bone-implant system affect the ultrasonic response. The aim of this work is to extract relevant information about primary stability from the complex ultrasonic signal obtained from a probe screwed to the implant. To do this, signal processing based on multiscale analysis has been used. The comparison between experimental and numerical results has been carried out, and a correlation has been observed between the multifractal signature and the stability. Furthermore, a sensitivity study has shown that the variation of certain parameters (i.e. central frequency and trabecular bone density) does not lead to a change in the response

Caractérisation de la Réponse Ultrasonore d'Implant Dentaire : Simulation Numérique et Analyse des Signaux

International audienceThe long-term success of a dental implant is related to the properties of the bone-implant interface. It is important to follow the evolution of bone remodeling phenomena around the implant. To date, there is no satisfactory method for tracking physiological and mechanical properties of this area, and it is difficult for clinicians to qualitatively and quantitatively assess the stability of a dental implant. In this context, methods based on ultrasound wave propagation were already successfully used by our group, in the qualitative and quantitative evaluation of primary and secondary stability of dental implants. In this study we perform numerical simulations, using the finite element method, of wave propagation in a dental implant inserted into bone. To simplify the calculations, an axisymmetric geometry is considered. Given the importance of monitoring of peri-prosthetic area, particular attention is given to the boundary conditions between the implant and the bone. The numerical results are compared with those from experimental tests. These results, numerical and experimental, are then analysed with signal processing tools based on multifractal methods. Analysis of the first results shows that these methods are potentially efficient in this case because they can explore and exploit the multi-scale structure of the signal

Pointwise regularity of parameterized affine zipper fractal curves

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets of the pointwise Hölder exponent for a subinterval of the spectrum. We give an equivalent characterization for the existence of regular pointwise Hölder exponent for Lebesgue almost every point. In this case, we extend the multifractal analysis to the full spectrum. In particular, we apply our results for the de Rham’s curve

Multifractal properties of typical convex functions

We study the singularity (multifractal) spectrum of continuous convex functions defined on (Formula presented.). Let (Formula presented.) be the set of points at which f has a pointwise exponent equal to h. We first obtain general upper bounds for the Hausdorff dimension of these sets (Formula presented.), for all convex functions f and all (Formula presented.). We prove that for typical/generic (in the sense of Baire) continuous convex functions (Formula presented.), one has (Formula presented.) for all (Formula presented.) and in addition, we obtain that the set (Formula presented.) is empty if (Formula presented.). Also, when f is typical, the boundary of (Formula presented.) belongs to (Formula presented.). © 2017 Springer-Verlag GmbH Austria, part of Springer Natur

Improved stability criteria for sampled-data systems with input saturations

International audienceIn this chapter, the design of either the controller or the network is ad- dressed for sampled-data systems with input saturation. Using modified sector conditions, an adequate looped functional, and the Wirtinger-based integral inequality, quasi-LMI conditions, with a scalar parameter to tune, are proposed in the regional (or local) context for both design problems. The associated convex optimizations are briefly described. Finally some examples show the efficiency of the methods with respect to existing results

Delay-Dependent Reciprocally Convex Combination Lemma for the Stability Analysis of Systems with a Fast-Varying Delay

International audienceThis chapter deals with the stability analysis of linear systems subject to fast-varying delays. The main result is the derivation of a delay-dependent reciprocally convex lemma allowing a notable reduction of the conservatism of the resulting stability conditions with the introduction of a reasonable number of decision variables. Several examples are studied to show the potential of the proposed method