Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-10-04T20:06:24Z No. of bitstreams: 1 RaimundaSilva.pdf: 3007003 bytes, checksum: 6d2583372b22c581e239bf77c0d1338e (MD5) / Made available in DSpace on 2017-10-04T20:06:24Z (GMT). No. of bitstreams: 1 RaimundaSilva.pdf: 3007003 bytes, checksum: 6d2583372b22c581e239bf77c0d1338e (MD5) Previous issue date: 2017-03-06 / Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e Tecnológico do Maranhão / ABSTRACT: Soil is the habitat for a number of living organisms that perform essential functions to the ecosystem. The present work aimed to determine the edaphic diversity in large groups under different uses and management of the soil in Cerrado Biome. The study was developed in the city of Mata Roma (3º 70 '80.88' 'S and 43º 18' 71.27 '' W), in the eastern region of Maranhão state, Brazil. It were installed 130 pitfall traps in five areas with different management (millet, soybean, maize, eucalyptus, and pasture) and two reference areas with natural vegetation with different uses (anthropized Cerrado and preserved Cerrado). The traps remained in the field for a period of seven days, after this, the contents were maintained in plastic bottles and taken to the laboratory, where they were sampled and identified in large groups (orders and family). After identification, the biodiversity indexes were determined: (Shanon index, Pielou, Average and total richness and abundance). The data were analyzed using descriptive statistics and multivariate techniques using group dissimilarity. The geostatistical analysis was evaluated by a semivariogram, adjusted to a geostatistical, spherical, gaussian or exponential model. The multifractality was analyzed by the current method, in successive segments of different sizes of 2k , k=0 a k= 7 in the range of q = +10 to q = -10. 20,995 arthropods were collected throughout the study. The highest abundance was found for millet (9,974 individuals), and the lowest abundance values were reported for soybean (222) and maize (824), respectively. The highest biodiversity index is reported for the soybean area (2.69), although there is less abundance, in this area, the groups are evenly distributed due to the homogeneous management in the study area. The main axis in the analysis of the main components (PCA) explained 50.9% of the correlation of the groups with the sampled areas. The dendrogram had demonstrated that the area of soybean and maize are similar and had isolated the area of millet with the most dissimilar in relation to the others. The use and management of the soil in the study areas determine the occurrence of soil arthropods in function of food availability. For the areas of millet, maize, eucalyptus, anthropized Cerrado and pasture the Shanon diversity index obtained pure nugget effect. For the areas of millet, maize, anthropized Cerrado and pasture, the total diversity index was adjusted to the gaussian model. Only for the areas of soybean and pasture the staggered semivariograms showed similarity in the spatial variability of indexes, indicating that they behave similarly. The multifractality generated generalized dimension, D0, for all the indexes in the millet area, with invariant values, D0 = 1.000 ± 0.000. The singularity spectra were curves in concave parables with greater or smaller asymmetry for all areas sampled. In general, the fauna of soil presented spatial variability and multifractal parameters. / ABSTRACT: Soil is the habitat for a number of living organisms that perform essential functions to the ecosystem. The present work aimed to determine the edaphic diversity in large groups under different uses and management of the soil in Cerrado Biome. The study was developed in the city of Mata Roma (3º 70 '80.88' 'S and 43º 18' 71.27 '' W), in the eastern region of the State of Maranhão, Brazil. Were installed 130 pitfall traps in five areas with different management (Millet, Soybean, Maize, Eucalyptus, and Pasture) and two reference areas with natural vegetation with different uses (anthropized Cerrado and preserved Cerrado). The traps remained in the field for a period of seven days, after this, the contents were maintained in plastic bottles and taken to the laboratory, where they were sampled and identified in large groups (orders and family). After identification, the biodiversity indexes were determined: (Shanon index, Pielou, Average and total richness and abundance). The data were analyzed using descriptive statistics and multivariate techniques using group dissimilarity. 20,995 arthropods were collected throughout the study. The highest abundance was found for millet (9,974 individuals), and the lowest abundance values were reported for soybean (222) and maize (824), respectively. The highest biodiversity index is reported for the soybean area (2.69), although there is less abundance, in this area, the groups are evenly distributed due to the homogeneous management in the study area. The main axis in the analysis of the main components (PCA) explained 50.9% of the correlation of the groups with the sampled areas. The dendrogram had demonstrated that the area of soybean and maize are similar and had isolated the area of millet with the most dissimilar in relation to the others. The use and management of the soil in the study areas determine the occurrence of soil arthropods in function of food availability.

This thesis examines scaling properties of petrol and diesel prices in the Czech Republic and a crude oil price over the period from January 2004 to February 2013. Using generalised Hurst exponent and multifractal detrended fluctuation analysis techniques we find out that crude oil market is efficient, do not contain long memory and the returns exhibit monofractal behaviour. On the other hand, petrol and diesel markets in the Czech Republic are not efficient, because their returns contain long-range dependence in autocorrelations and exhibit multifractal behaviour caused mostly by fat-tailed distribution. Thus, fuels can be modelled by complex methods like Markov switching multifractal model. JEL Classification C15, C16, C46 Keywords petrol, diesel, crude oil, long memory, multifrac- tality, GHE, MF-DFA Author's e-mail martin.baletka@ies-prague.org Supervisor's e-mail kristoufek@ies-prague.org Abstrakt Tato práce zkoumá škálování cen benzínu a motorové nafty v České repub- lice a ceny ropy na datech v období od ledna 2004 do února 2013. Použitím metod zobecněného Hurstova exponentu a multifraktální detrendované fluk- tuační analýzy jsme zjistili, že trh s ropou je efektivní, bez přítomnosti dlouhé paměti v autokorelacích a výnosy na trhu s ropou vykazují monofraktální...

For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily self-similar. Their dimensional properties and measures have been studied in terms of the sequence $a$. In this thesis, we extend these results to a more general collection of Cantor sets. We study their Hausdorff and packing measures, and compare the size of Cantor sets with the more refined notion of dimension partitions. The properties of these Cantor sets in relation to the collection of cut-out sets are then considered. The multifractal spectrum of $\mathbf{p}$-Cantor measures on these Cantor sets are also computed. We then focus on the special case of homogeneous Cantor sets and obtain a more accurate estimate of their exact measures. Finally, we prove the $L^p$-improving property of the $\mathbf{p}$-Cantor measure on a homogeneous Cantor set as a convolution operator.

An important problem in Physics concerns the study of stochastic processes and fluctuations away from the mean of dynamical variables. In a wide range of systems, some of the observed variables have a macroscopic quality, in the sense that they represent averages or sums over time or space of "microscopic" quantities. When long-range memory or correlation effects do not play a significant role, then the necessary and sufficient conditions for the Central Limit Theorem to hold can become satisfied. Quite often, the second moments of the studied dynamical variable do not diverge, hence in many important instances, the fluctuations of many systems follow Gaussian statistics. On the other hand, complex systems generate some variabilities that often deviate them from Gaussian statistics. Here, we focus on two properties related to Gaussian fluctuations: (i) monofractality and (ii) homoscedasticity. Specifically, we first address the general question about the nature of the relationship between multifractality and heteroscedasticity. We applied multifractal detrended fluctuation analysis to a nonstationary high frequency financial time series obtained from currency markets. As a second test, we applied the technique to the audio time series of Beethoven's fifth symphony. We obtained results suggesting that heteroscedasticity can cause or increase multifractality. We also investigate in greater detail the convergence to the homoskedastic and monofractal Gaussian regime, using the mathematical formalism of Lévy sections, as previously applied to time series. We report several conclusions related to these questions and discuss the generality of these results in the context of the physics of complex systems. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Um importante problema em Física está relacionado ao estudo de processos estocásticos e flutuações de variáveis dinâmicas. Em uma variedade de sistemas, algumas das variáveis observadas têm uma qualidade macroscópica, no sentido de que elas representam a média ou a soma sobre o espaço ou tempo de quantidades microscópicas. Quando efeitos de memória de longo alcance ou correlação não desempenharem um papel significativo, então as condições necessárias e suficientes para a validade do Teorema do Limite Central podem ser satisfeitas. Frequentemente o segundo momento da variável em questão não diverge. Consequentemente em muitos exemplos importantes, as flutuações de muitos sistemas seguem uma estatística Gaussiana. Em contraste, sistemas complexos geram flutuações que muitas vezes os desviam da estatística Gaussiana. Aqui, nós focamos em duas propriedades relacionadas à flutuações Gaussianas: (i) monofractalidade e (ii) homocedasticidade. Especificamente, discutimos primeiro a questão geral sobre a natureza da relação entre multifractalidade e heterocedasticidade. Aplicamos a multifractal detrended fluctuations analysis a uma série temporal financeira não estacionária e de alta freqüência referente à taxa cambial. Como um segundo teste, aplicamos a mesma técnica de análise para a série de áudio da quinta sinfonia de Beethoven. Obtivemos resultados que indicam que a heterocedasticidade pode causar ou aumentar a multifractalidade. Também investigamos em detalhes a convergência para o regime homocedástico e monofratal Gaussiano usando o método matemático de seções de Lévy, como previamente aplicado a séries temporais. Apresentamos conclusões relacionadas a estes questionamentos e discutimos a generalidade destes resultados no contexto da Física de sistemas complexos.

Submitted by Mario BC (mario@bc.ufrpe.br) on 2017-04-19T13:01:07Z No. of bitstreams: 1 Jader da Silva Jale.pdf: 9845063 bytes, checksum: a4099ac7d2600e6b68a1349ce0d9bc53 (MD5) / Made available in DSpace on 2017-04-19T13:01:07Z (GMT). No. of bitstreams: 1 Jader da Silva Jale.pdf: 9845063 bytes, checksum: a4099ac7d2600e6b68a1349ce0d9bc53 (MD5) Previous issue date: 2015-06-01 / The growth of the world economy, driven by emerging countries, especially China, has generated signi cant changes in the commodities market since 2002. The commodity prices have shown a signi cant increase, reecting the erce conditions of supply and demand for these products, driven by the climatic phenomena that have negatively afected the supply, and by the demand growth rate. The global nancial crisis began in the US market, and eventually turned out the worst global nancial crisis since 1929 (the break of the New York Stock Exchange). The bankruptcy of Lehman Brothers investment bank on September 15, 2008 marks the transformation of the international nancial crisis, after which in Brazil there was a great reduction of international credit, accompanied by a sharp increase of the dollar exchange rate. Considering that the agricultural sector is of fundamental importance to the economic health, being a major investor in environmental and rural technologies, Brazil can not succumb to the idea of a slowdown in this sector, as in 2008 the Brazilian agribusiness represented 36.7% of exports, generating 37% of jobs, and 28% of gross domestic product (GDP). This work investigates the returns asynchrony and the behavior of the cross-correlations for six agricultural Brazilian agribusiness commodities, for the period prior to the global nancial crisis (2006-2009), and after the crisis (2010-2014). The Cross-Sample Entropy method was used for quantifying the asynchrony among the commodity returns series. In addition, the methods Multifractal Detrended Cross-Correlation Analysis (MF-DCCA), Multifractal Detrended Fluctuation Analysis (MF-DFA) and Detrended Cross-Correlation Analysis (DCCA) were used to investigate cross-correlations and auto correlations in the returns series. The results of multifractal analysis show that for all time series, the multifractality decreased after the global nancial crisis, indicating smaller range of the scale invariant fluctuations, except for Cotton, which exhibits precisely the opposite behavior. Based on the obtained results, it can be concluded that the multifractal analysis and the complexity analysis can be useful in the studies of the dynamics of the Brazilian agribusiness, given its importance within the global economic scenario, for adoption of monetary and scal policies by the responsible economic agents, or by the federal government. / O crescimento da economia mundial, impulsionado por países emergentes, principalmente a China, gerou mudanças relevantes no mercado de commodities a partir de 2002. Observou-se uma mudança nos preços das commodities, que mostraram uma elevação expressiva, mostrando condições acirradas entre oferta e demanda desses produtos, impulsionadas pela existência de problemas climáticos que afetaram negativamente a oferta e pelo ritmo de crescimento da demanda. A crise financeira mundial iniciou-se no mercado americano e acabou se tornando a pior crise financeira mundial desde 1929 (quebra da bolsa de Nova York). A falência do banco de investimento Lehman Brothers no dia 15 de setembro de 2008 marca a transformação da crise financeira internacional, e após isso, ocorre uma grande redução do crédito internacional e o dólar dispara no Brasil. Considerando que o setor agrícola é de fundamental importância para a sanidade econômica e por ser um grande investidor em tecnologias ambiental e rural, o Brasil não pode sucumbir a idéia uma desaceleração neste setor, pois o agronegócio brasileiro representou, em 2008, 36.7% das exportações brasileiras, geração de 37% dos empregos e 28% do Produto Interno Bruto (PIB). Neste trabalho investigou-se a assincronia, a transferência de informação e o comportamento das correlações cruzadas dos retornos de seis commodities agrícolas do agronegócio brasileiro, para os períodos anteriores (2006-2009) e posteriores a crise financeira mundial (2010-2014). Utilizou-se o método Cross-Sample Entropy para quantificar a assincronia entre todas as séries de retornos das commodities. Utilizou-se os métodos Multifractal Detrended Cross-Correlation Analysis (MF-DCCA), Multifractal Detrended Fluctuation Analysis (MF-DFA) e Detrended Cross-Correlation Analysis (DCCA) para investigar correlações cruzadas e auto correlações. Os resultados da análise multifractal mostram que para todas as séries temporais, a multifractalidade diminuiu após a crise financeira mundial, indicando menor variedade do tamanho das flutuações que apresentam invariância de escala, exceto o algodão, que apresentou comportamento contrário. Com base nos resultados obtidos, pode-se concluir que a análise multifractal e a análise de complexidade podem ser úteis nos estudos da dinâmica do agronegócio brasileiro, dada a sua importância, diante do cenário econômico mundial seja para adoção de políticas monetárias e fiscal dos órgãos responsáveis, agentes econômicos ou pelo governo federal.

Nous considérons le problème d’un électron sur des pavages quasipériodiques en une et deux dimensions. Nous introduisons tout d’abord les pavages quasipériodiques d’un point de vue géométrique, et défendons en particulier l’idée que ces pavages sont les pavages apériodiques les plus proches de la périodicité. Nous concentrant plus particulièrement sur l’un des pavages quasipériodiques les plus simples, la chaîne de Fibonacci, nous montrons à l’aide d’un groupe de renormalisation que la multifractalité des états électroniques découle directement de l’invariance d’échelle de la chaîne. Élargissant ensuite notre champ d’étude à un ensemble de chaînes quasipériodiques, nous nous intéressons au théorème de label des gaps, qui décrit comment la géométrie d’une chaîne donnée contraint les valeurs que peut prendre la densité d’états intégrée dans les gaps du spectre électronique. Plus précisément, nous nous intéressons à la façon dont l’énoncé de ce théorème est modifié lorsque l’on considère une séquence d’approximants périodiques approchant une chaîne quasipériodique. Enfin, nous montrons comment des champs de hauteurs géométriques peuvent être utilisés pour construire des états électroniques exacts sur des pavages en une et deux dimensions. Ces états sont robustes aux perturbations du hamiltonien, sous réserve que ces dernières respectent les symétries du pavage sous-jacent. Nous relions les dimensions fractales de ces états à la distribution de probabilités des hauteurs, que nous calculons de façon exacte. Dans le cas des chaînes quasipériodiques, nous montrons que la conductivité suit une loi d’échelle de la taille de l’échantillon, dont l’exposant est relié à cette même distribution de probabilités. / We consider the problem of a single electron on one and two-dimensional quasiperiodic tilings. We first introduce quasiperiodic tilings from a geometrical point of view, and point out that among aperiodic tilings, they are the closest to being periodic. Focusing on one of the simplest one-dimensional quasiperiodic tilings, the Fibonacci chain, we show, with the help of a renormalization group analysis, that the multifractality of the electronic states is a direct consequence of the scale invariance of the chain. Considering now a broader class of quasiperiodic chains, we study the gap labeling theorem, which relates the geometry of a given chain to the set of values the integrated density of states can take in the gaps of the electronic spectrum. More precisely, we study how this theorem is modified when considering a sequence of approximant chains approaching a quasiperiodic one. Finally, we show how geometrical height fields can be used to construct exact eigenstates on one and two-dimensional quasiperiodic tilings. These states are robust to perturbations of the Hamiltonian, provided that they respect the symmetries of the underlying tiling. These states are critical, and we relate their fractal dimensions to the probability distribution of the height field, which we compute exactly. In the case of quasiperiodic chains, we show that the conductivity follows a scaling law, with an exponent given by the same probability distribution.

No description available.

In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

Cette thèse se concentre sur la caractérisation ultrasonore de l’interphase os-implant. Cette région est une zone de transition où a lieu le processus d’ostéointégration (i.e. le processus de guérison du tissu entourant l’implant). Donc, cette interphase a un rôle crucial dans l’ancrage à long-terme de l’implant, puisqu’elle dépend de la quantité ainsi que la qualité du tissu osseux environnant. Ensuite, en plus d’être un milieu complexe en remodelage continu, l’os néoformé présente une nature multi échelle et qui évolue dans le temps. Toutes ces motivations rendent la caractérisation de l’interphase os-implant critique et difficile. Dans ce contexte, les méthodes ultrasonores sont largement utilisées aujourd’hui dans le domaine clinique pour leur capacité de donner des informations sur les propriétés biomécaniques du tissu osseux. Compte tenu de ces éléments, dans le but de caractériser les propriétés mécaniques et microstructurales de l’interphase os-implant à travers des méthodes ultrasonores, il est important de développer et valider des modèles mécaniques ainsi que de méthodes de traitement du signal. A cause de la complexité du problème, afin de décrire avec précision le tissu environnant à l’implant, il est d’abord essentiel une modélisation fiable du tissu osseux. Pour cela, on étudie l’interaction entre une onde ultrasonore et le tissu osseux, en considérant aussi les effets dues à la microstructure. Pour ce faire, un modèle continu généralisé a été utilisé. Dans ce contexte, un test de transmission/réflexion réalisé sur un échantillon poroélastique immergé dans un fluide a renforcé la fiabilité du modèle. Les champs de pression réfléchi et transmis sont influencés par les paramètres de la microstructure. De plus, les résultats issus de l’analyse de dispersion sont en accord avec ceux observés dans les expériences pour les échantillons poroélastiques. Après, le problème a été compliqué en considérant une interphase qui se situe entre l’os et l’implant. Ainsi, on peut gérer la complexité ajoutée par la présence du tissu néoformé. Comme on l’a déjà mentionné, une difficulté additionnelle est représentée par le fait que l’interphase est un milieu hétérogène, un mélange de phases solides et fluides dont les propriétés évoluent avec le temps. Donc, afin de modéliser l’interaction des ondes ultrasonores avec une interphase, on a considéré dans le modèle une couche très fine avec des propriétés élastiques et inertielles. En partant de ça, on a étudié les effets des propriétés de réflexion d’une transition entre un milieu homogène et un milieu microstructuré. De même, il a aussi été étudié la caractérisation du milieu via des techniques avances de traitement du signal. En particulier, la réponse dynamique due à l’excitation ultrasonore du système os-implant a été analysée à travers une approche multifractale. Une première analyse basée sur les coefficients des ondelettes a montré une signature multifractale pour les signaux dérivants des simulations et aussi des expériences. Ensuite, une étude de sensibilité a aussi montré que la variation des paramètres tels que la fréquence centrale et la densité de l’os trabéculaire ne contribue pas à un changement dans la réponse. L’originalité réside dans le fait qu’il s’agit d’un des premiers efforts d’exploiter l’approche multifractale dans la propagation ultrasonore dans un milieu hétérogène / This thesis focus on the ultrasonic characterization of bone-implant interphase. This region is a transition zone where the osteointegration process (i.e. the healing process of the tissues surrounding the implant) takes place. Thus, this interphase is of crucial importance in the long-term anchorage of the implant, since it depends on the quantity and quality of the surrounding bone tissue. However, other than being a complex medium in constant remodeling, the newly formed bone presents a multiscale and time evolving nature. All these reasons make the characterization of the bone-implant interphase critical and difficult. In this context, ultrasound methods are nowadays widely used in the clinic field because of their ability to give information about the biomechanical properties of bone tissue. On this basis, with the aim of characterizing the mechanical and microstructural properties of the bone-implant interphase by ultrasound methods, it is important to develop and validate mechanical models and signal processing methods. Due to the complexity of the problem, in order to precisely describe the bone tissue surrounding the implant, first an accurate modelling of bone tissue is essential. Thus, the interaction between an ultrasonic wave and bone tissue has been investigated by also taking into account the effects dues to the microstructure. To do this, a generalized continuum modelling has been used. In this context, a transmission/reflection test performed on a poroelastic sample dipped in a fluid enhanced the reliability of the model. The reflected and transmitted pressure fields result to be affected by the microstructure parameters and the results coming from the dispersion analysis are in agreement with those observed in experiments for poroelastic specimens. Then, the problem has been complicated by considering the interphase taking place between the bone and the implant. In this way, we could handle the complexity added by the presence of the newly formed tissue. As already said, the fact that this interphase is a heterogeneous medium, a mixture of both solid and fluid phases whose properties evolve with time is an additional difficulty. Thus, in order to model the interaction of ultrasonic waves with this interphase, a thin layer with elastic and inertial properties has been considered in the model. The effects on the reflection properties of a transition between a homogeneous and a microstructured continuum have been investigated.Therefore, the characterization of the medium also via advanced signal processing techniques is investigated. In particular, the dynamic response due to the ultrasonic excitation of the bone-implant system is analyzed through the multifractal approach. A first analysis based on the wavelet coefficients pointed out a multifractal signature for the signals from both simulations and experiences. Then, a sensitivity study has also shown that the variation of parameters such as central frequency and trabecular bone density does not lead to a change in the response. The originality lies in the fact that it is one of the early efforts to exploit the multifractal approach in the ultrasonic propagation inside a heterogeneous medium