Given a random dynamical system T constructed from Jablonski transformations, consider its Perron-Frobenius operator P_T. We prove a weak form of the Lasota-Yorke inequality for P_T and thereby prove the existence of BV- invariant densities for T. Using the Spectral Decomposition Theorem we prove that the support of an invariant density is open a.e. and give conditions such that the invariant density for T is unique. We study the asymptotic behavior of the Markov operator P_T, especially when T has a unique absolutely continuous invariant measure (ACIM). Under the assumption of uniqueness, we obtain spectral stability in the sense of Keller. As an application, we can use Ulam's method to approximate the invariant density of P_T.

In this thesis, we investigate impulsive differential systems with delays of the form And more generally of the form The dissertation consists of five chapters. The first chapter serves as introduction, contains preliminary considerations and assertions that will be encountered in the sequel. In chapter 2, we construct the adjoint systems and obtain the variation of parameters formulas of the solutions in terms of fundamental matrices. The asymptotic behavior of solutions of systems satisfying the Perron condition is investigated in chapter 3. In chapter4, we give a result that characterizes the behavior of solutions in the case there is a bounded solution. Moreover, a necessary and sufficient condition for the existence of periodic solutions is obtained. In the last chapter, a series of consequences on the existence of periodic solutions of functionally equivlent impulsive systems with delays is established.

Recently there has been growing interest to characterize and reduce uncertainty in stochastic dynamical systems. This drive arises out of need to manage uncertainty in complex, high dimensional physical systems. Traditional techniques of uncertainty quantification (UQ) use local linearization of dynamics and assumes Gaussian probability evolution. But several difficulties arise when these UQ models are applied to real world problems, which, generally are nonlinear in nature. Hence, to improve performance, robust algorithms, which can work efficiently in a nonlinear non-Gaussian setting are desired. The main focus of this dissertation is to develop UQ algorithms for nonlinear systems, where uncertainty evolves in a non-Gaussian manner. The algorithms developed are then applied to state estimation of real-world systems. The first part of the dissertation focuses on using polynomial chaos (PC) for uncertainty propagation, and then achieving the estimation task by the use of higher order moment updates and Bayes rule. The second part mainly deals with Frobenius-Perron (FP) operator theory, how it can be used to propagate uncertainty in dynamical systems, and then using it to estimate states by the use of Bayesian update. Finally, a method to represent the process noise in a stochastic dynamical system using a nite term Karhunen-Loeve (KL) expansion is proposed. The uncertainty in the resulting approximated system is propagated using FP operator. The performance of the PC based estimation algorithms were compared with extended Kalman filter (EKF) and unscented Kalman filter (UKF), and the FP operator based techniques were compared with particle filters, when applied to a duffing oscillator system and hypersonic reentry of a vehicle in the atmosphere of Mars. It was found that the accuracy of the PC based estimators is higher than EKF or UKF and the FP operator based estimators were computationally superior to the particle filtering algorithms.

Ο ρόλος του εμπορίου στην οικονομία και τα οφέλη που επιφέρει είναι σχεδόν τόσο παλιά, όσο και η ίδια η οικονομική επιστήμη. Τις τελευταίες δεκαετίες ο ρόλος της απελευθέρωσης του εμπορίου και τα υποθετικά οφέλη που προκύπτουν από αυτό επεκτάθηκε και θεωρείται ένας από τους επιταχυντές της οικονομικής μεγέθυνσης, αλλά και πρωταρχική πηγή σύγκλισης των οικονομιών των κρατών. Μετά το 2ο παγκόσμιο πόλεμο η σταδιακή απελευθέρωση των αγορών και πρόσφατα το «δόγμα» της παγκοσμιοποίησης άνοιξε νέους ορίζοντες στο διεθνές εμπόριο και προώθησε την δημιουργία εμπορικών συνασπισμών μεταξύ των χωρών παγκοσμίως. Σε μια εν δυνάμει παγκόσμια αρένα συναλλαγών η γνώση για την επιρροή που μπορεί να ασκήσει στην προώθηση της οικονομικής μεγέθυνσης αποτελεί κρίσιμο ζήτημα. Παρά τη μεγάλη προσπάθεια που έχει αφιερωθεί στη μελέτη του ζητήματος, υπάρχουν λίγα πειστικά στοιχεία σχετικά με την επίδραση του εμπορικού ανοίγματος στο εισόδημα των κρατών και στη σύγκλιση της οικονομικής τους μεγέθυνσης. Η παρούσα διατριβή έχει ως θεματικό πυρήνα τη διερεύνηση της επίδρασης που είχε η Σύνοδος της Βαρκελώνης το 1995, στη σύγκλιση της οικονομικής μεγέθυνσης των χωρών της λεκάνης της Μεσογείου. Για τον έλεγχο της ύπαρξης της σύγκλισης αρχικά εφαρμόστηκαν οι έλεγχοι του Dickey-Fuller και Philips-Perron. Κατόπιν στα πλαίσια της ενδογενούς θεωρίας υιοθετήθηκε η ανάλυση με πάνελ παρατηρήσεις. Τα αποτελέσματα ανέδειξαν την παρουσία σύγκλισης μετά τη σύνοδο και με την μεταβλητή του εμπορικού ανοίγματος να συμβάλει σημαντικά. / The role of commerce in economy and the benefits incurred by it, are almost as old as the science of economy itself. Over the last decades, the role of free trade police and the potential benefits of it expanded and this considered to be one of the main factor of economic growth as well as the primary source of convergence of the economies of nations. After the second world war, the gradual free trade policy and recently the “dogma” of globalization opened new horizons in the international commerce and encouraged the establishment of commercial coalitions between countries worldwide. In an existing global arena of transactions, the insight on the influence it can exert in promoting the economic development, remains a crucial matter. In spite of the great effort given on the study of the issue, there is little convincing evidence for the influence of the trade gap concerning the income of nations and the convergence of the economic development. The present dissertation has its thematic core on the study of the influence which the Barcelona summit had in 1985 on the convergence of the economic growth of the Mediterranean countries. To ensure the existence of this convergence initially Dickey – Fuller’ s and Phillips – Perron’s proof techniques were applicable. Then, in the scope of the endogenous theory the analysis through panel observations was adopted. The results proved the existence of convergence after the summit, with the variable of the trade gap contributing significantly.

Este trabalho tem como objetivo provar a existência de gráficos mínimos em domínios do plano Será garantido através do Método de Perron a existência de um gráfico mínimo num dommio limitado do plano. Será também estudado o comportamento dessa solução na fronteira do seu domínio através do conceito de função barreira Serão provados três teoremas que garantem a existcncia de soluções do problema de Dirichlct para as mínimas em don11nios não corwe:xos c não compactos do plano com condições especiais de fronteira, sendo estes discutidn no trabalho de J Ripoll and F. Tomi [RT]. / This work has as objective to prove some existence thcorems Jor minimal graphs over planar domains lt will be guaranteed employ the Perron method one existence of one minimars graph in arbitrarv bounded domain. lt will be too studied this solution at the boundary through the concept o f barrier fi.mction We obtain three cxistcncc theorems to Dirichlct 's problem for non convex and non compacts domains having special boundary data being that results are containcd in the J. Ripoll's and F. Tomi 's works [RT].

Este trabalho tem como objetivo provar a existência de gráficos mínimos em domínios do plano Será garantido através do Método de Perron a existência de um gráfico mínimo num dommio limitado do plano. Será também estudado o comportamento dessa solução na fronteira do seu domínio através do conceito de função barreira Serão provados três teoremas que garantem a existcncia de soluções do problema de Dirichlct para as mínimas em don11nios não corwe:xos c não compactos do plano com condições especiais de fronteira, sendo estes discutidn no trabalho de J Ripoll and F. Tomi [RT]. / This work has as objective to prove some existence thcorems Jor minimal graphs over planar domains lt will be guaranteed employ the Perron method one existence of one minimars graph in arbitrarv bounded domain. lt will be too studied this solution at the boundary through the concept o f barrier fi.mction We obtain three cxistcncc theorems to Dirichlct 's problem for non convex and non compacts domains having special boundary data being that results are containcd in the J. Ripoll's and F. Tomi 's works [RT].

Este trabalho tem como objetivo provar a existência de gráficos mínimos em domínios do plano Será garantido através do Método de Perron a existência de um gráfico mínimo num dommio limitado do plano. Será também estudado o comportamento dessa solução na fronteira do seu domínio através do conceito de função barreira Serão provados três teoremas que garantem a existcncia de soluções do problema de Dirichlct para as mínimas em don11nios não corwe:xos c não compactos do plano com condições especiais de fronteira, sendo estes discutidn no trabalho de J Ripoll and F. Tomi [RT]. / This work has as objective to prove some existence thcorems Jor minimal graphs over planar domains lt will be guaranteed employ the Perron method one existence of one minimars graph in arbitrarv bounded domain. lt will be too studied this solution at the boundary through the concept o f barrier fi.mction We obtain three cxistcncc theorems to Dirichlct 's problem for non convex and non compacts domains having special boundary data being that results are containcd in the J. Ripoll's and F. Tomi 's works [RT].

This thesis aims to study a connection between indecomposable elements in the cubic fields and the Jacobi-Perron algorithm (JPA). JPA is a multidimensional generalization of the usual continued fractions algorithm. We work in the family of Ennola's cubic fields and we examine how the indecomposable elements are related to elements originating from this algorithm and whether some of these elements generate all indecomposable elements in the fields. We formulate conjectures on how to determine which elements will generate the indecomposable elements. We also prove some necessary conditions that have to hold for elements originating from this algorithm to generate indecomposable elements. 1

Con la implantación de un conjunto de políticas inspiradas en el Sumak Kawsay a partir del 2007, las expectativas de un cambio favorable en el Ecuador fueron bastante altas. En esta investigación se analiza si existió una diferencia significativa entre los períodos de vigencia de dichas políticas provenientes de los planes de desarrollo del Buen Vivir (2007-2019) y los períodos presidenciales previos, a partir de la dolarización (2000-2006). Para el análisis, se propone la construcción de dos indicadores compuestos: el primero basado en una propuesta de seis dimensiones del Plan Nacional de Desarrollo 2013-2017 y el segundo en tres dimensiones o “armonías” del Sumak Kawsay. Estos dos indicadores compuestos no incluyen el mismo conjunto de subindicadores, siendo el primero más amplio que el segundo. Para su construcción, se recurrió al Análisis de Componentes Principales. En el análisis se recurre a la prueba de Bai y Perron para detectar cambios estructurales significativos, se evalúa las medias de variaciones anuales entre los períodos generados y finalmente se aplica el análisis de Series Temporales Interrumpidas (ITSA) para determinar si hubo una mejora significativa durante la vigencia de los nuevos planes de desarrollo en comparación con las políticas de gobiernos anteriores. Desde la perspectiva del PIB per cápita, del IDH, del indicador compuesto de tres dimensiones (Sumak Kawsay) y del indicador compuesto de seis dimensiones (PNBV), los resultados indican que no existió el impacto socioeconómico esperado.

Les barrages en enrochement comptent parmi les des ouvrages les plus importants en génie civil. Ce sont des ouvrages critiques dont les exigences de performance sont très élevées. De plus, les normes parasismiques pour ces ouvrages ont évolué rapidement depuis quelques années ((Hydro- Québec, 2003; Gouv. du QC, 2002; Canadian Dam Association (CDA), 2007 (révisé en 2013); ICOLD, 2016; Federal Guidelines for Dam Safety (FEMA), 2005) et sont maintenant très sévères. Ces analyses sont très importantes puisqu’elles permettent de mieux définir le comportement d’un l’ouvrage face à un séisme et, par conséquent, de mieux évaluer sa sécurité. Cependant, ces méthodes sont complexes et leur mis en oeuvre pour des barrages existants constitue toujours un exercice délicat. L’Est du Canada se caractérise par une séismicité active mais modérée. Les secousses sismiques de l’Est canadien ont un contenu fréquentiel plus élevé que celui de l’Ouest américain. Cependant, les signaux de l’est canadien n’ont fait l’objet relativement que de peu d’études jusqu’à maintenant. Ce mémoire présente les résultats de l’étude dynamique unidimensionnelle (1D) et bidimensionnelle (2D) du comportement d’un barrage en enrochement sous sollicitations sismiques. L’ouvrage étudié est le barrage Denis-Perron dont la construction a été complétée en 1998. C’est un barrage en enrochement d’une hauteur de 171 mètres constitué d’un noyau en till. L’ouvrage ferme la rivière Ste-Marguerite sur une vallée étroite située au nord de Sept-Îles. Des enregistrements des accélérations en crête et à la base du barrage pour trois événements sismiques ont été utilisés dans les différentes simulations. Les analyses 1D ont été menées dans le domaine fréquentiel avec le logiciel ProShake. Les analyses 2D ont été menées dans le domaine temporel avec les logiciels QUAD4M, qui adopte le comportement linéaire-équivalent des sols, et FLAC, qui tient en compte la non-linéarité du comportement des sols. Pour les deux catégories d’analyse, les résultats ont été exploités dans les domaines spectral et temporel. Ces résultats ont démontré qu’une étude 1D sur ProShake avec une correction de la période prédominante selon la forme de la vallée sont nécessaire pour des études dynamiques préliminaires d’un barrage. Cependant, des études approfondies sur FLAC peuvent s’avérer essentiel pour l’évaluation des risques sismiques de ces ouvrages. / Rockfill dams are one of the most important structures in civil engineering. These are critical structures whose performance requirements are very high. In addition, the seismic standards for these structures have changed rapidly over the last few years (Hydro-Québec, 2003; Gouv.-du- Québec, 2002; Canadian Dam Association (CDA)) and currently are very sever. These analyzes are very important since they can define the behavior of structure when earthquakes happen, and consequently, evaluate its safety. However, these methods are complex and their application on existing dams is always a delicate exercise. Eastern Canada is characterized by an active but moderate seismicity. The earthquakes in Eastern Canada have a higher frequency content than the Western United States. Though, only a few studies have been made with signals from Eastern Canada. This thesis presents the results of the dynamic 1D and 2D behavior of a rockfill dam under seismic loading. The study takes into consideration the Denis-Perron dam activity, a rockfill dam of 171 meters height with a till core, which was finished constructing in 1998. This structure closes the Ste-Marguerite River on a narrow valley north of Sept-Îles. Recordings of accelerations at the top and at the base of the dam for three seismic events were used in different simulations. The 1D analyzes were conducted in the frequency domain with ProShake software. The 2D analyzes were conducted in the time domain with QUAD4M software, which adopts the linear behavior of soils, and FLAC, which takes into account the nonlinearity of soil behavior. For both analyses, the results were carried out in spectral and temporal domains. These results demonstrated that the 1D study on ProShake, with a correction of predominant period according to the shape of the valley, is necessary for preliminary dynamic studies of a dam. However, in depth studies, FLAC may be essential for seismic risk assessment of these structures