Les broncho-pneumopathies chroniques obstructives (BPCO) constituent un groupe de maladies des poumons caractérisées par le blocage du passage de l’air, rendant la respiration de plus en plus difficile. L’emphysème et la bronchite chronique sont les deux principales affections parmi les BPCO, mais les BPCO peuvent également être provoquées par les dégâts causés par des bronchites chroniques asthmatiques. L’emphysème pulmonaire est une maladie pulmonaire caractérisée par l’élargissement des espaces aériens distaux en amont des bronchioles terminales non respiratoires, accompagné de la destruction des parois alvéolaires. Ces modifications du parenchyme pulmonaire sont pathognomoniques de l’emphysème. La bronchite chronique est une forme de bronchite caractérisée par une production excessive d’expectoration, entraînant l’apparition d’une toux chronique et l’obstruction des voies respiratoires. Dans toutes ces affections, les dégâts causés aux voies respiratoires finissent par affecter les échanges gazeux dans les poumons. L’emphysème est généralement diagnostiqué de façon indirecte, sur la base d’un examen clinique, d’explorations de la fonction respiratoire (EFR), et d’une évaluation visuelle subjective des scanners des tomodensitogrammes. Ces tests présentent une valeur limitée dans les cas d’emphysème léger ou modéré. La présente étude aborde la possibilité d’appliquer une démarche d’analyse non linéaire à la répartition de la densité de l’air au sein de l’arbre des voies respiratoires des poumons à un quelconque niveau des ramifications. Les images sources de tomodensitométrie (TDM) du poumon sont traitées en deux phases, afin de produire un coefficient fractal de répartition de la densité de l’air. Au cours de la première phase, les valeurs brutes de pixel des images sources correspondant à toutes les densités d’air possibles sont traitées par un outil logiciel, mis au point pour construire une image cible. On y parvient par suppression en cascade des éléments indésirables (SCEI) : une étape de prétraitement dans l’analyse de l’image source. Celle-ci permet d’identifier les valeurs de densité d’air au sein de l’arbre des voies respiratoires, tout en éliminant toutes les valeurs non relatives à la densité de l’air. La seconde phase consiste en une réduction itérative de la résolution (RIR). Chaque réduction de la résolution produit un nouvel histogramme. Chaque histogramme ainsi produit comporte un certain nombre de pics, chacun d’entre eux correspondant à un ensemble de densités d’air. La courbe mettant en relation chaque réduction de la résolution avec le nombre de pics correspondant, obtenus à la résolution concernée, est tracée. Ceci permet de calculer la dimension fractale par une régression linéaire sur un graphique log – log. / Chronic Obstructive Pulmonary Disease (COPD) refers to a group of lung diseases that block airflow and make it increasingly difficult for you to breathe. Emphysema and chronic bronchitis are the two main conditions that make up COPD, but COPD can also refer to damage caused by chronic asthmatic bronchitis. Pulmonary emphysema is defined as a lung disease characterized by “abnormal enlargement of the air spaces distal to the terminal, non-respiratory bronchiole, accompanied by destructive changes of the alveolar walls”. These lung parenchymal changes are pathognomonic for emphysema. Chronic bronchitis is a form of bronchitis characterized by excess production of sputum leading to a chronic cough and obstruction of air flow. In all cases, damage to your airways eventually interferes with the exchange of oxygen and carbon dioxide in your lungs. Habitual techniques of emphysema’s diagnosis are based on indirect features, such as clinical examination; Pulmonary Function Tests (PFT) and subjective visual evaluation of CT scans. These tests are of limited value in assessing mild to moderate emphysema. The presented work discusses the possibility of applying a nonlinear analysis approach on air density distribution within lung airways tree at any level of branching. Computed Tomography (CT) source images of the lung are subjected to two phases of treatment in order to produce a fractal coefficient of the air density distribution. In the first phase, raw pixel values from source images, corresponding to all possible air densities, are processed by a software tool, developed in order to, construct a product image. This is done through Cascading Elimination of Unwanted Elements (CEUE): a preprocessing analysis step of the source image. It identifies values of air density within the airways tree, while eliminating all non-air-density values. Then, during the second phase, in an iterative manner, a process of Resolution Diminution Iterations (RDI) takes place. Every resolution reduction produces a new resultant histogram. A resultant histogram is composed of a number of peaks, each of which corresponding to a cluster of air densities. A curve is plotted for each resolution reduction versus the number of peaks counted at this particular resolution. It permits the calculation of the fractal dimension from the regression slope of log-log power law plot.

Dans ce texte, Valérie Barry tente de : - construire une intelligibilité des réussites et difficultés d'élèves que le système éducatif français définit comme étant porteurs de troubles importants des fonctions cognitives, en situation collective d'apprentissage mathématique. Elle pose trois hypothèses de recherche : 1) une approche systémique de l'activité mentale des élèves favorise la révélation de causalités complexes et interactionnelles des phénomènes d'enseignement/apprentissage ; 2) la traduction des observables de l'activité mentale des élèves en besoins d'apprentissage permet d'investir, au plan de la réflexion, un espace potentiel propice à l'adhésion au postulat d'éducabilité ; 3) une modélisation fractale des besoins d'apprentissage aide le chercheur élucider ceux-ci, dans la construction d'interprétations génératrices d'apprentissages pour les élèves. La dialectisation de la théorie et de l'action s'inscrit dans une recherche action clinique, laquelle postule l'importance de la prise en compte de la subjectivité des protagonistes de la recherche dans la construction d'une intelligibilité des phénomènes observés. La recherche-action a eu lieu au cours de l'année universitaire 2006-2007, dans un institut médico-éducatif, auprès de quatre élèves en situation de construction du nombre et des concepts comparatifs. L'analyse a permis de valider les hypothèses de recherche et d'identifier des démarches enseignantes potentiellement efficaces/efficiences / In this text, Valérie Barry attempts to: - build intelligibility of difficulties and successes, in a collective situation of mathematics learning, of children defined by the French education system as pupils with important cognitive disorders. She makes three research hypothesis: 1) a systemic approach of pupils' mental activity promotes the revelation of interactional and complex causalities of learning/teaching phenomena ; 2) translating manifestations of pupils' mental activity into educational needs makes it possible to invest a potential space that supports the educability postulate ; 3) A fractal modeling of educational needs helps the researcher to elucidate these needs, and to build interpretations that promote pupils' learning. The dialectisation of theory and action takes place in a clinical action research, which postulates that it is fundamental to take the subjectivity of the protagonists into consideration when building intelligibility of phenomena. The action research was conducted during the university year 2006-2007, in a Medical and Educational Centre, and concerned four pupils that learned numbers and comparative concepts. The analysed has revealed the validity of the three hypothesis, and potentially efficient/effective teaching methods

Cette thèse est une étude théorique d’un système d’équations de Cahn-Hilliard/Allen-Cahn couplées qui représente un mélange binaire en séparation de phase. Le but principal de l’étude est le comportement asymptotique des solutions en termes d’attracteurs exponentiels/globaux. Pour cette raison, l’existence et l’unicité de la solution sont étudiées tout d’abord. Une des principales applications de ce modèle d’équations est la cristallographie.Dans la première partie de la thèse, on examine le modèle proposé avec des conditions de type Dirichlet sur le bord et une non linéarité régulière de type polynomial : on réussit à trouver un attracteur exponentiel et par conséquence un attracteur global de dimension finie. Une non linéarité singulière de type logarithmique est ensuite prise dans la deuxième partie, cette fonction étant approchée par une suite de fonctions régulières et l’existence d’un attracteur global est démontrée sous des conditions au bord de type Dirichlet.Enfin, dans la dernière partie, le système est couplé avec une équation pour la température: suivant la loi de Fourrier premièrement, puis la loi de type III de la thermo-élasticité. Dans les deux cas, la dynamique de l’équation est étudiée et un attracteur exponentiel est trouvé malgré la difficulté créée par l’équation hyperbolique dans le deuxième cas. / This thesis is a theoretical study of a coupled system of equations of Cahn-Hilliard and Allen-Cahn that represents phase separation of binary alloys. The main goal of this study is to investigate the asymptotic behavior of the solution in terms of exponential/global attractors. For this reason, the existence and unicity of the solution are first studied. One of the most important applications of this proposed model of equations is crystallography. In the first part of the thesis, the system is studied with boundary conditions of Dirichlet type and a regular nonlinearity (a polynomial). There, we prove the existence of an exponential attractor that leads to the existence of a global attractor of finite dimension. Then, a singular nonlinearity (a logarithmic potential) is considered in the second part. This function is approximated by a sequence of regular ones and a global attractor is found.At the end, the system of equations is coupled with temperature: with the Fourrier law in the first case, then with the type III law (in the context of thermoelasticity) in the second case. The dynamics of the equations are studied and the existence of an exponential attractor is obtained.

<p></p><p></p><p></p><p>In urbanized river basins, sanitary wastewater and urban runoff (non-sanitary water) from urban agglomerations drain to complex engineered networks, are treated at centralized wastewater treatment plants (WWTPs) and discharged to river networks. Discharge from multiple WWTPs distributed in urbanized river basins contributes to impairments of river water-quality and aquatic ecosystem integrity. The size and location of WWTPs are determined by spatial patterns of population in urban agglomerations within a river basin. Economic and engineering constraints determine the combination of wastewater treatment technologies used to meet required environmental regulatory standards for treated wastewater discharged to river networks. Thus, it is necessary to understand the natural-human-engineered networks as coupled systems, to characterize their interrelations, and to understand emergent spatiotemporal patterns and scaling of geochemical and ecological responses. </p><br><p></p><p></p><p>My PhD research involved data-model synthesis, using publicly available data and application of well-established network analysis/modeling synthesis approaches. I present the scope and specific subjects of my PhD project by employing the <i>Drivers-Pressures-Status-Impacts-Responses</i> (<i>DPSIR</i>) framework. The defined research scope is organized as three main themes: (1) River network and urban drainage networks (<i>Foundation</i>-<i>Pathway of Pressures</i>); (2) River network, human population, and WWTPs (<i>Foundation</i>-<i>Drivers</i>-<i>Pathway of Pressures</i>); and (3) Nutrient loads and their impacts at reach- and basin-scales (<i>Pressures</i>-<i>Impacts</i>).</p><br><p></p><p></p><p>Three inter-related research topics are: (1) the similarities and differences in scaling and topology of engineered urban drainage networks (UDNs) in two cities, and UDN evolution over decades; (2) the scaling and spatial organization of three attributes: human population (POP), population equivalents (PE; the aggregated population served by each WWTP), and the number/sizes of WWTPs using geo-referenced data for WWTPs in three large urbanized basins in Germany; and (3) the scaling of nutrient loads (P and N) discharged from ~845 WWTPs (five class-sizes) in urbanized Weser River basin in Germany, and likely water-quality impacts from point- and diffuse- nutrient sources. </p><br><p></p><p></p><p>I investigate the UDN scaling using two power-law scaling characteristics widely employed for river networks: (1) Hack’s law (length-area power-law relationship), and (2) exceedance probability distribution of upstream contributing area. For the smallest UDNs, length-area scales linearly, but power-law scaling emerges as the UDNs grow. While area-exceedance plots for river networks are abruptly truncated, those for UDNs display exponential tempering. The tempering parameter decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power-law exponent for mature UDNs tends to be larger than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and non-random branching.</p><br><p></p><p></p><p>In this study, I also examine the spatial patterns of POP, PE, and WWTPs from two perspectives by employing fractal river networks as structural platforms: spatial hierarchy (stream order) and patterns along longitudinal flow paths (width function). I propose three dimensionless scaling indices to quantify: (1) human settlement preferences by stream order, (2) non-sanitary flow contribution to total wastewater treated at WWTPs, and (3) degree of centralization in WWTPs locations. I select as case studies three large urbanized river basins (Weser, Elbe, and Rhine), home to about 70% of the population in Germany. Across the three river basins, the study shows scale-invariant distributions for each of the three attributes with stream order, quantified using extended Horton scaling ratios; a weak downstream clustering of POP in the three basins. Variations in PE clustering among different class-sizes of WWTPs reflect the size, number, and locations of urban agglomerations in these catchments. <b></b></p><br><p></p><p></p><p>WWTP effluents have impacts on hydrologic attributes and water quality of receiving river bodies at the reach- and basin-scales. I analyze the adverse impacts of WWTP discharges for the Weser River basin (Germany), at two steady river discharge conditions (median flow; low-flow). This study shows that significant variability in treated wastewater discharge within and among different five class-sizes WWTPs, and variability of river discharge within the stream order <3, contribute to large variations in capacity to dilute WWTP nutrient loads. For the median flow, reach-scale water quality impairment assessed by nutrient concentration is likely at 136 (~16%) locations for P and 15 locations (~2%) for N. About 90% of the impaired locations are the stream order < 3. At basin-scale analysis, considering in stream uptake resulted 225 (~27%) P-impaired streams, which was ~5% reduction from considering only dilution. This result suggests the dominant role of dilution in the Weser River basin. Under the low flow conditions, water quality impaired locations are likely double than the median flow status for the analyses. This study for the Weser River basin reveals that the role of in-stream uptake diminishes along the flow paths, while dilution in larger streams (4≤ stream order ≤7) minimizes the impact of WWTP loads. </p><br><p></p><p></p><p>Furthermore, I investigate eutrophication risk from spatially heterogeneous diffuse- and point-source P loads in the Weser River basin, using the basin-scale network model with in-stream losses (nutrient uptake).Considering long-term shifts in P loads for three representative periods, my analysis shows that P loads from diffuse-sources, mainly from agricultural areas, played a dominant role in contributing to eutrophication risk since 2000s, because of ~87% reduction of point-source P loads compared to 1980s through the implementation of the EU WFD. Nevertheless, point-sources discharged to smaller streams (stream order < 3) pose amplification effects on water quality impairment, consistent with the reach-scale analyses only for WWTPs effluents. Comparing to the long-term water quality monitoring data, I demonstrate that point-sources loads are the primary contributors for eutrophication in smaller streams, whereas diffuse-source loads mainly from agricultural areas address eutrophication in larger streams. The results are reflective of spatial patterns of WWTPs and land cover in the Weser River basin.</p><br><p></p><p></p><p>Through data-model synthesis, I identify the characteristics of the coupled natural (rivers) – humans – engineered (urban drainage infrastructure) systems (CNHES), inspired by analogy, coexistence, and causality across the coupled networks in urbanized river basins. The quantitative measures and the basin-scale network model presented in my PhD project could extend to other large urbanized basins for better understanding the spatial distribution patterns of the CNHES and the resultant impacts on river water-quality impairment.</p><p><br></p><p></p>

La connaissance de la dynamique des matières en suspension (MES) et des turbidités en milieu côtier est un élément clé pour comprendre les écosystèmes marins. Les processus de floculation/défloculation représentent des mécanismes essentiels contrôlant la dynamique des caractéristiques physiques des MES et, par conséquent, leur devenir dans l’environnement. Cette thèse a pour objectif de mieux comprendre la caractérisation des MES et leur variabilité de l’échelle tidale à l’échelle annuelle en mer côtière, en réponse aux différents forçages hydrodynamiques, hydrologiques et biologiques ayant lieu à l’interface entre estuaire et baie. À cette fin, des capteurs optiques et acoustiques ont été déployés in situ, lors de 6 campagnes en mer réalisées en 2016, en vue de caractériser et de quantifier les MES dans la colonne d’eau, à la sortie de l’embouchure de l’estuaire de Seine (France). Une analyse critique des mesures granulométriques issues du LISST-100X est proposée. Un travail de quantification des incertitudes de mesure associées à l’estimation de la concentration en MES a mis en avant le risqué lié à un rinçage insuffisant des filtres et a conduit à définir une masse minimum à filtrer, de l'ordre de 10 mg, quelle que soit la concentration. Ce travail a notamment permis de proposer une méthode d’estimation d’un volume optimal de filtration basée sur une mesure préalable de turbidité de référence. Les mesures à haute fréquence ont montré que la dynamique des caractéristiques physiques des MES en termes de concentration et de taille médiane sont contrôlées par le cycle advection-floculation-sédimentation-remise en suspension. Ces résultats ont permis de hiérarchiser les paramètres de contrôle des processus de floculation. La turbulence a été identifiée comme le principal paramètre de contrôle à l’échelle tidale et semi-lunaire, la taille médiane maximale des flocs étant inversement corrélée à l’intensité des courants. À l’échelle saisonnière, la variation des caractéristiques des MES (taille, densité, vitesse de chute) est liée à la variabilité du contenu en matière organique (MO), avec une floculation favorisée quand le taux de MO particulaire augmente. Ces flocs plus grands et moins denses sont également plus résistants à la fragmentation induite par cisaillement turbulent. Ces travaux ont également permis de discuter le concept de structuration fractale des flocs.La dynamique de la dimension fractale, déterminée en combinant des données in situ de concentration en MES et de distribution en classe de taille, peut traduire des variations de composition des MES, mais peut aussi résulter d'incertitudes de mesure associées aux instruments. Ces dernières sont discutées dans cette thèse. La variabilité saisonnière des caractéristiques des MES est plus prononcée au large que dans la zone sous influence de l’estuaire de Seine. À partir des observations ponctuelles en Baie de Seine et d’une série de mesure long terme dans la zone côtière belge acquise par le laboratoire RBINS, les méthodes acoustiques et optiques sont combinées. Ces dernières ont mis en avant une augmentation de la rétrodiffusion acoustique lorsque la densité moyenne des flocs diminue. / The knowledge of suspended particulate matter (SPM) and turbidity dynamics in coastal waters is essential for studying marine ecosystems. Flocculation/deflocculation processes are crucial mechanisms controlling the dynamics of SPM physical characteristics and thus, the fate of these SPM in the environment. This PhD thesis focuses on the SPM characteristics and their dynamics in coastal waters, from tidal to annual scales, in response to hydrodynamic, hydrological and biological forcing that take place at the interface between estuaries and coastal seas. To this end, optical and acoustic sensors were deployed in situ through 6 field campaigns in 2016, in order to characterize and quantify SPM in the water column, near the mouth of the Seine estuary (France). A critical analysis of LISST-100X measurements in coastal waters is presented. The quantification of uncertainties on SPM concentration measurements is investigated: it highlights the crucial effect of salt retention, and the need for a minimum mass to filter, around 10 mg. A method for estimating an optimal filtration volume based on a reference turbidity measurement is proposed. High frequency measurements show that the dynamics of SPM and median diameter are controlled by the advection-flocculationsedimentation-resuspension cycle.These results allow to classify the factors controlling flocculation processes. Turbulence is identified as the main factor at the semi-diurnal and semi-lunar tidal scales, as the maximum median size of flocs decreases when the tidal currents intensify.At the seasonal scale, the variation of SPM characteristics (size, density, settling velocity) is correlated to the variability of the organic matter (OM) content: in particular, flocculation is enhanced by an increase of the particular OM fraction. The resulting larger and lesser dense flocs are also more resistant to the fragmentation induced by shear. This work also investigates the fractal approach applied to flocs. The fractal dimension variability, calculated by combining in situ data of SPM concentration and particle size distribution, can be associated to variations of the SPM composition, but can also result from uncertainties linked to instrument limitations. The latter are discussed in this PhD thesis. The seasonal variability of SPM characteristics is more pronounced offshore than at the mouth of the Seine estuary. From short-term observations in the Seine Bay and from long-term series in the Belgian coastal zone provided by the RBINS, optical turbidity and acoustic backscatter measurements are combined. They highlight an increase of the acoustic backscatter intensity when mean floc density.

Les avalanches de débris, qui résultent du démantèlement des flancs des édifices volcaniques et montagneux, sont des écoulements granulaires rapides et dangereux dont le monteur est la gravité et qui présentent des distances de transport extrêmement importantes. La dynamique de leur mise en place et leurs mécanismes de transport permettant cette très grande mobilité sont des phénomènes qui demeurent encore mal compris. De nombreux modèles existent pour expliquer la grande mobilité des avalanches de débris et incluent des processus basés sur la lubrification ou la fluidification de la masse granulaire mais également sur le phénomène de désintégration dynamique des éléments. Cependant la grande majorité des modèles proposés souffre du manque d’observations de terrain et de quantification de l’évolution des matériaux au cours de leur transport au sein de la masse granulaire. Afin d’identifier les principaux mécanismes de transport des avalanches de débris, nous proposons dans ce travail une étude de terrain détaillée de dépôts d’avalanches de débris volcaniques qui résultent du démantèlement d’un volcan bouclier océanique, le Piton des Neiges (île de La Réunion, océan Indien). L’approche est couplée à un examen morphométrique (dimension fractale et circularité), exoscopique et granulométrique des particules présentes dans les dépôts. Elle est complétée par l’examen de la fabrique des dépôts basée sur l’anisotropie de la susceptibilité magnétique (ASM). Les données obtenues nous permettent de mettre en évidence une évolution de la dynamique de transport et de mise en place des dépôts d’avalanches de débris depuis les zones sources jusqu’aux domaines de dépôt distaux. On montre également que la désintégration dynamique et le gonflement dispersif qui l’accompagne opèrent tout au long du transport et à toutes les échelles au-dessus d’une limite inférieure de broyage à 500 μm. En dessous de cette limite, la réduction granulométrique résulte uniquement de processus d’attrition par friction entre les particules. La grande mobilité des avalanches de débris pourrait ainsi être expliquée par l’effet combiné de la libération d’énergie élastique par la désintégration dynamique des particules > 500μm et par une réduction de la friction interne à la matrice liée aux interactions dispersives des particules fines (< 500 μm). L’ensemble des données permettent également de préciser les directions de transport et l’ampleur des avalanches de débris liées aux déstabilisations du massif du Piton des Neiges. / Debris avalanches, resulting from flank collapses that shape volcanic and mountainous edifices are rapidand dangerous gravity-driven granular flows that travel long run out distances. The dynamic and the transport mechanisms behind this high mobility remain poorly understood. The numerous models proposed to explain this high mobility include processes based on lubrication or fluidisation of the granular mass of the flow body, but also the dynamic disintegration of the transported particles. To date,all these proposed mechanisms lack observational support and quantification of the state of the particles of the granular mass during the transport. To identify the main transport mechanisms, we propose here detailed field studies of volcanic debris avalanches deposits resulting of flank-collapse events on an oceanic shield volcano, the Piton des Neiges (La Réunion Island, Indian Ocean). This study has been combined with morphometric (fractal dimension and circularity), exoscopic and grain-size analyses. Moreover, the fabric of the deposits has been investigated by with the characteristics of the anisotropy of the magnetic susceptibility (ASM). From these data we highlight a proximal to distal evolution of the debris avalanches transport and emplacement dynamics. We demonstrate that syn-transport dynamic disintegration continuously operates with the distance from the source down to a grinding limit of 500μm. Below this limit, the particle size reduction exclusively results from the attrition of the particles by frictional interactions. Thus, the exceptional mobility of debris avalanches may be explained by thecombined effect of elastic energy release during the dynamic disintegration of the larger clasts (> 500μm) and frictional reduction within the matrix due to the dispersive interactions between the finer particles (< 500 μm). All these data also allow to specify the transport direction and the approximate size of the debris avalanches related to the Piton des Neiges destabilisations.

本文採用傅利葉轉換與小波轉換以探討非線性現象：長期相依的碎形結構與混沌現象。藉由傅利葉轉換與小波轉換兩種研究方法，所得到臺灣股票市場加權股價指數的實證結論如下：1.藉由傅利葉轉換所得到的H值為0.4632；藉由小波轉換所得到的H值為0.4750。這兩種研究方法皆顯示臺灣股票市場具有負的長期相依的碎形結構。2.藉由傅利葉轉換的研究方法，臺灣股票市場加權股價指數的頻譜由初始向下與寬的連續的頻帶所組成；臺灣股票市場加權股價指數的自我相關函數則隨著時間差距的增加而遞減。此顯示臺灣股票市場具有混沌現象。3.小波轉換可以檢測出臺灣股票市場加權股價指數的奇異之處，並且指出存有一能說明臺灣股票市場碎形結構的複雜性的機制。藉由以上的實證結論，可以得知臺灣股票市場具有反持續性的碎形結構，股票價格的變動來自於臺灣股票市場尺度上的自我相似性。即使如此，由於混沌不可預測性的本質，使得股票價格的預測似乎是不可能的。 / The Fourier transform and the wavelet transform are utilized in this research to explore the nonlinear phenomena: the fractal structure of long trem dependence and the phenomenon of chaos. 　　In terms of the two research methods of the Fourier transform and the wavelet transform, the empirical conclusions of the Taiwan stock exchange weighted stock index are derived as follows: 　　1. The $H$ value of the research method of the Fourier transform is 0.4632; the $H$ value of the research method of the wavelet transform is 0.4750. The two research methods show that the Taiwan stock market has a fractal structure of negative long term dependence. 　　2. In terms of the research method of the Fourier transform, the power spectrum of the Taiwan stock exchange weighted stock index consists of initially downward and wide continuous band of frequencies; the autocorrelation function of the Taiwan stock exchange weighted stock index decreases as the time lag increases. These observations show that there exists the phenomenon of chaos in the Taiwan stock market. 　　3. The wavelet transform can detect out the singularities of the Taiwan stock exchange weighted stock index and can point out the heirarchy that illustrates the complexity of the fractal sturcture in the Taiwan stock market. 　　By the above empirical conclusions, there exists the antipersistent fractal structure in the Taiwan stock market. The variations of stock prices result from the self-similarity of the scales of the Taiwan stock market. Even so, the prediction of stock prices seems very impossible as a result of the unpredictability of chaotic nature.

<p>Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.</p><p>In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.</p><p>A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.</p>

Most materials including concrete have pre-existing flaws or defects. The fracture energy of concrete is a basic material property needed to understand fracture initiation and propagation in concrete. Whether fracture energy is size dependent or not is being discussed world over. Strictly the fracture energy if taken as a material property should be constant, and should be independent of the method of measurement, test methods, specimen shapes and sizes. A computational study on simulation of fracture in concrete using two dimensional lattice models is presented. A comparison is made with acoustic emission (AE) events with the number of fractured elements. A three-point bend specimen (TPB) is modeled using regular triangular lattice network. It was observed that the number of fractured elements increases near the peak load and beyond the peak load. AE events also increase rapidly beyond the peak load. Singular Fractal Functions (S.F.F) has been employed to interpret the size effect of quasi-brittle materials like concrete. The usual size dependent fracture energy of High Strength Concrete (HSC) beam is reported. The results are presented which are obtained directly from the experiments related to size effect in concrete carried out in the Structural engineering laboratory, Department of Civil engineering, IISc. Various fracture parameters studied in this experimental program are (a) Nominal strength N (b) Fracture energy, Gf (c)Fracture toughness, KIc, (c) Crack mouth opening displacement, CMOD (d) Size effect on the strength of concrete. Three-point-bend (TPB) specimen was chosen for the experimental study. Six different concrete mixes viz. A-mix, B-Mix, C-mix, D-Mix, E-mix, F-Mix were used. Acoustic Emission (AE) experiments are conducted to relate acoustic emission energy to fracture energy. It is interesting to note that both acoustic emission energy and fracture energy have similar characteristics. The advantage of the above relationship is that now it is possible to evaluate fracture energy by non-destructive testing methods. The b-value analysis of AE was carried out to study the damage in concrete structures. The Guttengberg-Richter relation for frequency versus magnitude can be applied to the AE method to study the scaling of the amplitude distribution of the acoustic emission waves generated during the cracking process in the test specimen at laboratory or in engineering structures. In the next part of this chapter b-value at various stages of damage of a reinforced concrete beam are obtained experimentally under typical cyclic loadings. The b-values at different levels of damage are tabulated. As fracture is size dependent, it may not be very useful unless its size dependency is eliminated. An effort is made to obtain size independent fracture energy by a hybrid technique.

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.