La conjecture de Zilber-Pink est une conjecture diophantienne concernant les intersections atypiques dans les variétés de Shimura mixtes. C’est une généralisation commune de la conjecture d’André-Oort et de la conjecture de Mordell-Lang. Le but de cette thèse est d’étudier Zilber-Pink. Plus concrètement, nous étudions la conjecture d’André-Oort, selon laquelle une sous-variété d’une variété de Shimura mixte est spéciale si son intersection avec l’ensemble des points spéciaux est dense, et la conjecture d’André-Pink-Zannier, selon laquelle une sous-variété d’une variété de Shimura mixte est faiblement spéciale si son intersection avec une orbite de Hecke généralisée est dense. Cette dernière conjecture généralise Mordell-Lang comme expliqué par Pink.Dans la méthode de Pila-Zannier, un point clef pour étudier la conjecture de Zilber-Pink est de démontrer le théorème d’Ax-Lindemann qui est une généralisation du théorème classique de Lindemann-Weierstrass dans un cadre fonctionnel. Un des résultats principaux de cette thèse est la démonstration du théorème d’Ax-Lindemann dans sa forme la plus générale, c’est- à-dire le théorème d’Ax-Lindemann mixte. Ceci généralise les résultats de Pila, Pila-Tsimerman, Ullmo-Yafaev et Klingler-Ullmo-Yafaev concernant Ax-Lindemann pour les variétés de Shimura pures.Un autre résultat de cette thèse est la démonstration de la conjecture d’André-Oort pour une grande collection de variétés de Shimura mixtes : in- conditionnellement pour une variété de Shimura mixte arbitraire dont la par- tie pure est une sous-variété de AN6 (par exemple les produits des familles universelles des variétés abéliennes de dimension 6 et le fibré de Poincaré sur A6) et sous GRH pour toutes les variétés de Shimura mixtes de type abélien. Ceci généralise des théorèmes connus de Klinger-Ullmo-Yafaev, Pila, Pila-Tsimerman et Ullmo pour les variétés de Shimura pures.Quant à la conjecture d’André-Pink-Zannier, nous démontrons plusieurs cas valables lorsque la variété de Shimura mixte ambiante est la famille universelle des variétés abéliennes. Tout d’abord nous démontrons l’intersection d’André-Oort et André-Pink-Zannier, c’est-à-dire que l’on étudie l’orbite de Hecke généralisée d’un point spécial. Ceci généralise des résultats d’Edixhoven-Yafaev et Klingler-Ullmo-Yafaev pour Ag. Nous prouvons ensuite la conjecture dans le cas suivant : une sous-variété d’un schéma abélien au dessus d’une courbe est faiblement spéciale si son intersection avec l’orbite de Hecke généralisée d’un point de torsion d’une fibre non CM est Zariski dense. Finalement pour une orbite de Hecke généralisée d’un point algébrique arbitraire, nous démontrons la conjecture pour toutes les courbes. Ces deux derniers cas généralisent des résultats de Habegger-Pila et Orr pour Ag.Dans toutes les démonstrations, la théorie o-minimale, en particulier le théorème de comptage de Pila-Wilkie, joue un rôle important. / The Zilber-Pink conjecture is a diophantine conjecture concerning unlikely intersections in mixed Shimura varieties. It is a common generalization of the André-Oort conjecture and the Mordell-Lang conjecture. This dissertation is aimed to study the Zilber-Pink conjecture. More concretely, we will study the André-Oort conjecture, which predicts that a subvariety of a mixed Shimura variety having dense intersection with the set of special points is special, and the André-Pink-Zannier conjecture which predicts that a subvariety of a mixed Shimura variety having dense intersection with a generalized Hecke orbit is weakly special. The latter conjecture generalizes the Mordell-Lang conjecture as explained by Pink.In the Pila-Zannier method, a key point to study the Zilber-Pink conjec- ture is to prove the Ax-Lindemann theorem, which is a generalization of the functional analogue of the classical Lindemann-Weierstrass theorem. One of the main results of this dissertation is to prove the Ax-Lindemann theorem in its most general form, i.e. the mixed Ax-Lindemann theorem. This generalizes results of Pila, Pila-Tsimerman, Ullmo-Yafaev and Klingler-Ullmo-Yafaev concerning the Ax-Lindemann theorem for pure Shimura varieties.Another main result of this dissertation is to prove the André-Oort conjecture for a large class of mixed Shimura varieties: unconditionally for any mixed Shimura variety whose pure part is a subvariety of AN6 (e.g. products of universal families of abelian varieties of dimension 6 and the Poincaré bundle over A6) and under GRH for all mixed Shimura varieties of abelian type. This generalizes existing theorems of Klinger-Ullmo-Yafaev, Pila, Pila-Tsimerman and Ullmo concerning pure Shimura varieties.As for the André-Pink-Zannier conjecture, we prove several cases when the ambient mixed Shimura variety is the universal family of abelian varieties. First we prove the overlap of André-Oort and André-Pink-Zannier, i.e. we study the generalized Hecke orbit of a special point. This generalizes results of Edixhoven-Yafaev and Klingler-Ullmo-Yafaev for Ag. Secondly we prove the conjecture in the following case: a subvariety of an abelian scheme over a curve is weakly special if its intersection with the generalized Hecke orbit of a torsion point of a non CM fiber is Zariski dense. Finally for the generalized Hecke orbit of an arbitrary algebraic point, we prove the conjecture for curves. These generalize existing results of Habegger-Pila and Orr for Ag.In all these proofs, the o-minimal theory, in particular the Pila-Wilkie counting theorems, plays an important role.

Le thème central de cette thèse est l'étude des résonances pour le problème intérieur en élastodynamique (géométries elliptique et sphéroïdale), et pour le problème de diffusion en acoustique (géométrie elliptique). On s'intéresse en particulier à la levée de dégénérescence des résonances liée à la brisure de symétrie de l'objet lors de la transition du disque circulaire vers le disque elliptique (2D), et de la sphère vers le sphéroïde (3D). Ce phénomène est étudié et interprété d'un point de vue théorique en prenant en compte les symétries de l'objet à l'aide de la théorie des groupes. Cette approche est complétée par une modélisation numérique et une partie expérimentale. En 2D, nous étudions le problème intérieur pour un disque elliptique élastique (étude des modes résonants) et le problème de la diffusion acoustique par des cylindres elliptiques élastiques. Ils sont traités à partir du formalisme modal combiné à la théorie des groupes dans le contexte vectoriel de l'élastodynamique. La levée de dégénérescence est observée théoriquement mais aussi expérimentalement en diffusion. La méthode simplifie considérablement le traitement numérique des problèmes étudiés, fournit une classification des résonances selon les 4 représentations irréductibles du groupe de symétrie C2v (associé à la géométrie elliptique) et donne une interprétation physique de la levée de dégénérescence en termes de brisure de symétrie. Une partie expérimentale en spectroscopie ultrasonore vient compléter l'étude théorique du problème de diffusion. Une série d'expériences en cuve est menée dans le cas de cylindres elliptiques de différentes excentricités en aluminium immergés dans l'eau, dans la bande de fréquence 0 ≤ kr ≤ 50, où kr est le nombre d'onde réduit dans le fluide. Les résultats expérimentaux présentent un très bon accord avec les résultats théoriques, la levée de dégénérescence est observée expérimentalement sur des fonctions de forme et mise en évidence sur des diagrammes angulaires. Le problème intérieur en 3D est traité expérimentalement à partir de la génération et la détection optiques d'ondes élastiques. Une série d'expérimentations sur des objets tridimensionnels (sphère, sphéroïdes oblates et prolates de différentes excentricités) en aluminium est réalisée. Ils sont mis en vibration par impacts laser et les mesures de vitesse et de fréquence s'effectuent par vibrométrie laser. On réalise ainsi une comparaison qualitative entre la théorie 2D et l'expérience 3D. Les mesures sont menées à la fois dans les domaines temporel et fréquentiel pour mettre en évidence la levée de dégénérescence d'une part, et l'onde de Rayleigh qui se propage sur la surface des objets d'autre part. Nous identifions deux trajets pour cette onde en géométrie sphéroïdale, l'un circulaire et l'autre elliptique.Enfin, dans le cadre des problèmes intérieurs 2D et 3D, on donne une interprétation en termes de rayons à travers la dualité entre le spectre des résonances et le spectre des longueurs des orbites périodiques (OPs), avec la mise en évidence du phénomène de conversion de mode et l'identification de l'onde de Rayleigh. Un phénomène, nouveau à notre connaissance, vient s'ajouter au phénomène de bifurcation de certaines orbites. Au cours de la déformation vers le disque elliptique, les orbites avec conversion de mode du disque circulaire se séparent en deux orbites dont les longueurs sont associées aux trajets minimal et maximal qu'elles parcourent. Cette observation s'interprète comme une conséquence du théorème de Fermat. Dans le cas du sphéroïde, on retrouve les orbites du disque circulaire dans le plan équatorial et celles du disque elliptique dans le plan méridien. Nous mettons également en évidence les pics associés aux deux trajets parcourus par l'onde de Rayleigh sur le spectre des OPs. / Resonances for the interior problem in elastodynamics and the acoustic scattering problem are considered in elliptical and spheroidal geometries. Interest is focused on the splitting up of resonances which occurs when the symmetry is broken in the transition from the circular disc to the elliptical one (2D), and from the sphere to the spheroid (3D). From the theoretical point of view, this physical phenomenon is studied and interpreted taking into account the symmetries of the object with the help of group theory. This approach is completed by a numerical modeling and an experimental part. As far as the two dimensional problems are concerned, the interior problem for an elliptical elastic disc (study of resonant modes) and the acoustic scattering problem for infinite elliptical elastic cylinders are studied combining modal formalism and group theory in the vectorial context of elastodynamics. The splitting up of resonances due to the circular symmetry breaking is not only theoretically observed but also experimentally for the scattering problem. The method significantly simplifies the numerical treatment of the problems studied, provides a full classification of resonances over the 4 irreducible representations of the symmetry group C2v (associated with the elliptical geometry) and gives a physical interpretation of the splitting up in terms of symmetry breaking of the symmetry group O(2) (invariance under rotation). An experimental part based on ultrasonic spectroscopy complements the theoretical study of the scattering problem. A series of tank experiments is carried out in the case of aluminum elliptical cylinders immersed in water, in the frequency range 0 ≤ kr ≤ 50, where kr is the reduced wave number in the fluid. The experimental results provide a very good agreement with the theoretical ones, the splitting up is observed on experimental form functions and the split resonant modes are identified on angular diagrams. The interior problem in 3D is studied by means of an experimental approach based on the optical generation and detection of elastic waves. A series of experiments is performed on three-dimensional objects in aluminium. These objects (sphere, prolate and oblate spheroids of various eccentricity) are excited by laser impacts, and the velocity and frequency measurements are carried out by laser vibrometry. Theory and experiments are qualitatively compared. The measurements are performed in both the frequency and time domains to highlight the splitting up of resonances on one hand, and the Rayleigh wave propagating on the surface of the objects on the other hand. We emphasize two paths for this surface wave in spheroidal geometry: a circular one in the equatorial plane and an elliptical one in the meridian plane. Finally, in the context of the interior problems in 2D and 3D, a physical interpretation of resonances in terms of geometrical paths is provided. Mode conversion is highlighted and the Rayleigh wave is identified on the periodic orbits lengths spectrum.In addition to the bifurcations of some periodic orbits, a phenomenon, new to our knowledge, appears. The orbits with mode conversion of the circular disc split in two orbits when the transition to the elliptic disc occurs. The lengths of these orbits are associated with the minimal and maximal travel paths. This observation is interpreted from Fermat's theorem.For the spheroid, orbits of the circular disc and those of the elliptical disc are recovered in the equatorial and meridian planes respectively. We also emphasize the peaks associated with the travel paths of Rayleigh wave in spheroidal geometry appearing on the periodic orbits spectrum.

Made available in DSpace on 2017-07-21T19:26:04Z (GMT). No. of bitstreams: 1 Rodrigo Frehse Pereira.pdf: 4666622 bytes, checksum: b2dcf2959eef9f7fd82301c2e45ac87f (MD5) Previous issue date: 2013-03-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Unstable dimension variability (UDV) is an extreme form of nonhyperbolicity. It is a structurally stable phenomenon, typical for high dimensional chaotic systems, which implies severe restrictions to shadowing of perturbed solutions. Perturbations are unavoidable in modelling Physical phenomena, since no system can be made completely isolated, states and parameters cannot be determined without uncertainties and any numeric approach to such models is affected by truncation and/or roundoff errors. Thus, the lack of shadowability in systems exhibiting UDV presents a challenge for modelling. Aiming to unveil the effect of perturbations a class of nonhyperbolic systems is studied. These systems present transversal unstable dimension variability (TUDV), which means the dynamics can be split in a skew direct product form, i. e. the phase space is decomposed in two components: a hyperbolic chaotic one, called longitudinal, and a nonhyperbolic transversal one. Moreover, in the absence of perturbations, the longitudinal component is a global attractor of the system. A prototype composed of two coupled piecewise-linear chaotic maps is presented in order to study the TUDV effects. This system has an invariant subspace S which characterizes the complete chaos synchronization and UDV, when present, is transversal to it. Taking advantage of (piecewise) linearity of the equations, an analytical method for unstable periodic orbits’ computation is presented. The set of all unstable periodic orbits (UPOs) is one of the building block of chaotic dynamics and its properties provide valuable informations about the asymptotic behaviour of the system as, for instance, the invariant natural measure. Therefore, the TUDV’s intensity is analytically studied by computing the contrast measure, which quantifies the difference between the statistical weights associated to UPOs with different unstable dimension. The effect of perturbations is modelled by the introduction of a small parameter mismatch, instead of noise addition, in order to keep the model’s determinism. Consequently, the characterization of dynamics by means of UPOs is still possible. It is shown the existence of a dense set G of UPOs outside the invariant subspace consistent with a chaotic repeller. When perturbation takes place, G merges with the set H of UPOs previously in S, given rise to a new nonhyperbolic stationary state. The analysis of G ∪H provides a topological explanation to the behaviour of systems with TUDV under perturbations. Moreover, the relation between the set of UPOs embedded in a chaotic attractor and its natural measure, proven only for hyperbolic systems, is successfully applied to this system: the error between the natural measure estimated both numerically and by means of UPOs is shown to be decreasing with p, the considered UPOs’ period. It is conjectured the coincidence between both in limit. Hence, a positive answer to reliability of numerical estimation to natural measure in nonhyperbolic systems via unstable dimension variability is presented. / A variabilidade da dimensão instável (VDI) é uma forma extrema de não-hiperbolicidade. É um fenômeno estruturalmente estável, típico para sistemas caóticos de alta dimensionalidade, que implica restrições severas ao sombreamento de soluções perturbadas. As perturbações¸ s são inevitáveis na modelagem de fenômenos fíısicos, uma vez que nenhum sistema pode ser isolado completamente, os estados e os parâmetros não podem ser determinados sem incertezas e qualquer abordagem numérica dos modelos é afetada por erros de arredondamento e/ou truncamento. Portanto, a falta da sombreabilidade em sistemas exibindo VDI apresenta um desafio à modelagem. Visando revelar os efeitos das perturbações, uma classe desses sistemas não hiperbó licos é estudada. Esses sistemas apresentam variabilidade da dimensão instável transversal (VDIT), significando que a dinâmica pode ser decomposta na forma de um produto direto assimétrico, i. e. o espação de fase é dividido em dois componentes: um hiperbólico e caótico, dito longitudinal, e um transversal e não-hiperbólico. Mais ainda, na ausência de perturbações, o componente longitudinal é um atrator global do sistema. Um protótipo composto de dois mapas ca´oticos lineares por partes acoplados é apresentado para o estudo dos efeitos da VDIT. Esse sistema possui um subespaço invariante S que caracteriza a sincronização completa de caos e a VDI, quando presente, é transversal a esse subespaço. Valendo-se da linearidade (por partes) das equações, um método analítico para o cálculo das órbitas periódicas instáveis é apresentado. O conjunto de todas as órbitas periódicas instáveis (OPIs) é um dos fundamentos da dinâmica caótica e suas propriedades fornecem informaões, valiosas sobre o comportamento assintótico do sistema como, por exemplo, a medida natural invariante. Assim, a intensidade da VDIT é estudada analiticamente pelo cálculo da medida de contraste, que quantifica a diferença entre o peso estatístico associado às OPIs com dimensão instável distintas. O efeito das perturbações é modelado pela introdução de um pequeno desvio nos parâmetros, ao invés da adição de ruído, a fim de manter o determinismo do modelo. Consequentemente, a caracterização da dinâmica em termos das OPIs ainda é possível. Demonstra-se a existência de um conjunto denso G de OPIs fora do subespaço invariante consistente com um repulsor caótico. Na presença de perturbações, G se funde com o conjunto H das OPIs previamente em S, dando origem a um novo estado estacionario não-hiperbólico. A análise de G ∪H fornece uma explicação topológica ao comportamento de sistemas com variabilidade da dimensão instável sob a açãoo de perturbações. Mais ainda, a relação entre o conjunto de OPIs imersas em um atrator caótico e sua medida natural, provada apenas para sistemas hiperbólicos, é aplicada com sucesso nesse sistema: mostra-se que o erro entre as medidas naturais estimadas numericamente e pelas OPIs é decrescente com p, o período das OPIs consideradas. Conjectura-se, portanto, a coincidência entre ambas no limite . Logo, apresenta-se uma resposta positiva à estimativa numérica da medida natural em sistemas não-hiperbólicos via variabilidade da dimensão instável.

Μελετάµε χωρικά εντοπισµένες και χρονικά περιοδικές λύσεις σε διακριτά συστήµατα που εκτείνονται σε µία χωρική διάσταση. Αυτού του είδους οι λύσεις είναι γνωστές µε τον όρο discrete breathers (DB) ή intrinsic localized modes (ILM). Στην ελληνική ϐιϐλιογραϕία, έχουν ονοµαστεί ∆ιακριτές Πνοές. Απαραίτητα χαρακτηριστικά για την εµϕάνιση τέτοιων λύσεων είναι η ύπαρξη ενός άνω φράγµατος του γραµµικού φάσµατος καθώς και η µη γραµµικότητα των εξισώσεων κίνησης, χαρακτηριστικά που συναντάµε σε πολλά φυσικά συστήµατα. Συγκεκριμένα, ασχολούµαστε µε πλέγµατα τύπου Klein Gordon και παρουσιάσουµε μια αποδείξη ύπαρξης τέτοιων λύσεων καθώς και αριθµητικά αποτελέσµατα µελετώντας παράλληλα την ευστάθεια των περιοδικών αυτών λύσεων µέσω της ϑεωρίας Floquet. Πέραν του κλασικού µοντέλου, όπου έχουµε αλληλεπιδράσεις πλησιέστερων γειτόνων, εισάγουµε επίσης ένα νέο µοντέλο µε αλληλεπιδράσεις µακράς εµβέλειας η οποία ελέγχεται µέσω µιας παράµετρου α και µελετάµε τις επιπτώσεις που έχει η μεταβολή του εύρους αλληλεπίδρασης στον χωρικό εντοπισµό και την ευστάθεια ενός DB. / We study time-periodic and spatially localized solutions in discrete dynamical systems describing Hamiltonian lattices in one spatial dimension. These solutions are called discrete breathers (DBs) or intrinsic localized modes (ILM). Necessary conditions for their occurrence are the boundedness of the spectrum of linear oscillations of the system as well as the nonlinearity of the equations of motion. More specifically, we focus on a Klein Gordon lattice and present an existence proof for such solutions, as well as numerical results revealing the stability (or instability) of DBs using Floquet theory. Besides reporting on the classical Klein Gordon model with nearest neighbor interactions, we also introduce long range interactions in our model, which are controlled by a parameter α and study the effect of varying the range of interactions on the spatial localization and the stability of a DB.

The main problem evaluated in this manuscript is the stabilization of periodic orbits of non-linear dynamical systems by use of feedback control. The goal of the control methods proposed in this work is to achieve a stable periodic oscillation. These control methods are applied to systems that present unstable periodic orbits in the state space, and the latter are the orbits to be stabilized.The methods proposed here are such that the resulting stable oscillation is obtained with low control effort, and the control signal is designed to converge to zero when the trajectory tends to the stabilized orbit. Local stability of the periodic orbits is analyzed by studying the stability of some linear time-periodic systems, using the Floquet stability theory. These linear systems are obtained by linearizing the trajectories in the vicinity of the periodic orbits.The control methods used for stabilization of periodic orbits here are the proportional feedback control, the delayed feedback control and the prediction-based feedback control. These methods are applied to discrete and continuous-time systems with the necessary modifications. The main contributions of the thesis are related to these methods, proposing an alternative control gain design, a new control law and related results.

On-orbit servicing (OOS) of space systems provides immense benefits by extending their lifetime, by reducing overall cost of space operations, and by adding flexibility to space missions. Refueling is an important aspect of OOS operations. The problem of determining the optimal strategy of refueling multiple satellites in a constellation, by expending minimum fuel during the orbital transfers, is challenging, and requires the solution of a large-scale optimization problem. The conventional notion about a refueling mission is to have a service vehicle visit all fuel-deficient satellites one by one and deliver fuel to them. A recently emerged concept, known as the peer-to-peer (P2P) strategy, is a distributed method of replenishing satellites with fuel. P2P strategy is an integral part of a mixed refueling strategy, in which a service vehicle delivers fuel to part (perhaps half) of the satellites in the constellation, and these satellites, in turn, engage in P2P maneuvers with the remaining satellites. During a P2P maneuver between a fuel-sufficient and a fuel-deficient satellite, one of them performs an orbital transfer to rendezvous with the other, exchanges fuel, and then returns back to its original orbital position. In terms of fuel expended during the refueling process, the mixed strategy outperforms the single service vehicle strategy, particularly with increasing number of satellites in the constellation. This dissertation looks at the problem of P2P refueling problem and proposes new extensions like the Cooperative P2P and Egalitarian P2P strategies. It presents an overview of the methodologies developed to determine the optimal set of orbital transfers required for cooperative and non-cooperative P2P refueling strategies. Results demonstrate that the proposed strategies help in reducing fuel expenditure during the refueling process.

Η διδακτορική διατριβή επικεντρώθηκε στη μελέτη νέων τεχνικών κρυπτογραφίας και κρυπτανάλυσης, αλλά και στην ανάπτυξη νέων πρωτοκόλλων για την ασφαλή ηλεκτρονική συγκέντρωση δεδομένων. Το πρώτο πρόβλημα το οποίο διερεύνησε η διατριβή ήταν η δυνατότητα εφαρμογής των μεθόδων Υπολογιστικής Νοημοσύνης στην κρυπτολογία. Στόχος ήταν η ανίχνευση των κρίσιμων σημείων κατά την εφαρμογή των μεθόδων αυτών στον πολύ απαιτητικό αυτό τομέα προβλημάτων και η μελέτη της αποτελεσματικότητας και της αποδοτικότητάς τους σε διάφορα προβλήματα κρυπτολογίας. Συνοψίζοντας, τα αποτελέσματα της διατριβής για την εφαρμογή μεθόδων Υπολογιστικής Νοημοσύνης στην κρυπτολογία υποδεικνύουν ότι παρά το γεγονός ότι η κατασκευή των αντικειμενικών συναρτήσεων είναι πολύ κρίσιμη για την αποδοτικότητα των μεθόδων, η Υπολογιστική Νοημοσύνη μπορεί να προσφέρει σημαντικά πλεονεκτήματα στον κλάδο αυτό όπως είναι η αυτοματοποίηση κάποιων διαδικασιών κρυπτανάλυσης ή κρυπτογράφησης, ο γρήγορος έλεγχος της σθεναρότητας νέων κρυπτοσυστημάτων αλλά και ο συνδυασμός τους με τυπικές μεθόδους που χρησιμοποιούνται μέχρι σήμερα για την αξιοποίηση της απλότητας και της αποδοτικότητάς τους. Το δεύτερο πρόβλημα που μελετάται στην διατριβή είναι η εφαρμογή μεθόδων αντίστροφης πολυωνυμικής παρεμβολής για την εύρεση της τιμής του διακριτού λογαρίθμου αλλά και του λογαρίθμου του Lucas. Για την μελέτη αυτή χρησιμοποιήθηκαν δύο υπολογιστικές μέθοδοι αντίστροφης πολυωνυμικής παρεμβολής, οι μέθοδοι Aitken και Neville, οι οποίες είναι κατασκευαστικές και επιτρέπουν την πρόσθεση νέων σημείων παρεμβολής για καλύτερη προσέγγιση του πολυωνύμου με μικρό υπολογιστικό κόστος. Η παρούσα μελέτη έδειξε ότι και με την προτεινόμενη μεθοδολογία το συνολικό κόστος υπολογισμού της τιμής των λογαρίθμων παραμένει υψηλό, ωστόσο η κατανομή των πολυωνύμων που έδωσαν την λύση των προβλημάτων δείχνει ότι η μεθοδολογία που χρησιμοποιήθηκε είτε εντόπισε την λύση στα πρώτα στάδια κατασκευής των πολυωνύμων είτε εντόπισε πολυώνυμα μικρού σχετικά βαθμού που προσεγγίζουν την αντίστοιχη λύση. Το τρίτο πρόβλημα που πραγματεύεται η παρούσα διατριβή είναι η δημιουργία νέων σθεναρών κρυπτοσυστημάτων με την χρήση μη-γραμμικών δυναμικών απεικονίσεων. Η αξιοποίηση των ιδιοτήτων του χάους στην κρυπτογραφία έχει αποτελέσει αντικείμενο μελέτης τα τελευταία χρόνια από τους ερευνητές λόγω της αποδεδειγμένης πολυπλοκότητας των συστημάτων του και των ιδιαίτερων στατιστικών ιδιοτήτων τους. Η διατριβή συνεισφέρει προτείνοντας ένα νέο συμμετρικό κρυπτοσύστημα που βασίζεται σε περιοδικές δυναμικές τροχιές και παρουσιάζει και τρεις τροποποιήσεις του που το καθιστούν ιδιαίτερα σθεναρό απέναντι στις συνήθεις κρυπταναλυτικές επιθέσεις. Δίνεται επίσης το υπολογιστικό κόστος κρυπτογράφησης και αποκρυπτογράφης του προτεινόμενου σχήματος και παρουσιάζονται πειραματικά αποτελέσματα που δείχνουν ότι η δομή των κρυπτογραφημάτων του κρυπτοσυστήματος δεν παρέχει πληροφορία για την ύπαρξη τυχόν μοτίβων στο αρχικό κείμενο. Τέλος, στην διατριβή αυτή προτείνονται δύο πρωτόκολλα για την ασφαλή ηλεκτρονική συγκέντρωση δεδομένων. Η συγκέντρωση δεδομένων από διαφορετικές βάσεις με ασφάλεια και ιδιωτικότητα θα ήταν σημαντική για την μελέτη των γνώσεων που ενυπάρχουν στα δεδομένα αυτά, με διάφορες μεθόδους εξόρυξης δεδομένων και ανάλυσης, καθώς οι γνώσεις αυτές ενδεχομένως δεν θα μπορούσαν να αποκαλυφθούν από την επιμέρους μελέτη των δεδομένων χωριστά από κάθε βάση. Τα δύο πρωτόκολλα που προτείνονται βασίζονται σε τροποποιήσεις πρωτοκόλλων ηλεκτρονικών εκλογών με τρόπο τέτοιο ώστε να ικανοποιούνται τα απαραίτητα κριτήρια ασφάλειας και ιδιωτικότητας που απαιτούνται για την συγκέντρωση των δεδομένων. Η βασική διαφορά των δύο πρωτοκόλλων είναι ότι στο ένα γίνεται χρήση έμπιστου τρίτου μέλους για την συγκέντρωση των δεδομένων, ενώ στο δεύτερο όχι. Και στις δύο περιπτώσεις, παρουσιάζεται ανάλυση της ασφάλειας των σχημάτων αλλά και της πολυπλοκότητάς τους αναφορικά με το υπολογιστικό τους κόστος. / In this PhD thesis we study problems of cryptography and cryptanalysis through Computational Intelligence methods and computational mathematics. Furthermore, we examine the establishment and security of new privacy preserving protocols for electronic data gathering. Part I is dedicated to the application of Computational Intelligence (CI) methods, namely Evolutionary Computation (EC) methods and Artificial Neural Networks (ANNs), for solving problems of cryptology. Initially, three problems of cryptanalysis are formulated as discrete optimization tasks and Evolutionary Computation methods are utilized to address them. The first conclusion derived by these experiments is that when EC methods are applied to cryptanalysis special attention must be paid to the design of the fitness function so as to include as much information as possible for the target problem. The second conclusion is that when EC methods (and CI methods in general) can be used as a quick practical assessment for the efficiency and the effectiveness of proposed cryptographic systems. We also apply EC methods for the cryptanalysis of Feistel ciphers and for designing strong Substitution boxes. The results show that the proposed methods are able to tackle theses problem efficiently and effectively with low cost and in automated way. Then, ANNs are employed for classical problems of cryptography as a measure of their robustness. The results show that although different topologies, training methods and formulation of the problems were tested, ANNs were able to obtain the solution of the problems at hand only for small values of their parameters. The performance of ANNs is also studied on the computation of a Boolean function derived from the use of elliptic curves in cryptographic applications. The results indicate that ANNs are able to adapt to the data presented with high accuracy, while their response to unknown data is slightly better than a random selection. Another important finding is that ANNs require a small amount of storage for the known patterns in contrast to the storage needed of the data itself. Finally, a theoretical study of the application of Ridge Polynomial Networks for the computation of the least significant bit of the discrete logarithm is presented. In Part II, computational mathematics are utilized for different cryptographic problems. Initially, we consider the Aitken and Neville inverse interpolation methods for a discrete exponential function and the Lucas logarithm function. The results indicate that the computational cost for addressing the problems through this approach is high; however interesting features regarding the degree of the resulting interpolation polynomials are reported. Next, a new symmetric key cryptosystem that exploits the idea of nonlinear mappings and their fixed points to encrypt information is presented. Furthermore, a measure of the quality of the keys used is introduced. The experimental results indicate that the proposed cryptosystem is efficient and secure to ciphertext-only attacks. Finally, three modifications of the basic cryptosystem that render it more robust are presented and efficiency issues are discussed. Finally, at Part III of the thesis, two protocols for privacy preserving electronic data gathering are proposed. The security requirements that must be met for data gathering with privacy are presented and then two protocols, based on electronic voting protocols, are analytically described. Security and complexity issues are also discussed.

Submitted by Josué Cardoso dos Santos (josuesantosunesp@gmail.com) on 2018-09-10T18:36:37Z No. of bitstreams: 1 Tese_Final_Josue_Cardoso_Santos.pdf: 78449557 bytes, checksum: 4515b9cb7cc346753f7e9682b8e037de (MD5) / Approved for entry into archive by Pamella Benevides Gonçalves null (pamella@feg.unesp.br) on 2018-09-10T18:47:05Z (GMT) No. of bitstreams: 1 santos_jc_dr_guara.pdf: 78449557 bytes, checksum: 4515b9cb7cc346753f7e9682b8e037de (MD5) / Made available in DSpace on 2018-09-10T18:47:05Z (GMT). No. of bitstreams: 1 santos_jc_dr_guara.pdf: 78449557 bytes, checksum: 4515b9cb7cc346753f7e9682b8e037de (MD5) Previous issue date: 2018-07-04 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Nowadays, despite the technological development experienced by science in general, a fact especially evident by the available powerful computer machines, the analytical and semi-analytical methods to study different space problems are still of great importance in the fields of astrodynamics and celestial mechanics. From the physical understanding of the motion of celestial bodies to the planing and designing of space missions, the use of mathematical models to deal with a very large number of contemporary problems plays a fundamental role in the progress of human knowledge. In this context, the present thesis presents the use of different mathematical techniques to deal with different various and current problems in astrodynamics and celestial mechanics. The studies developed throughout this work are applicable to both areas. The topics studied are the following ones: (1) The development of disturbing potentials using the double-averaging process, in order to be included in the Lagrange planetary which are numerically integrated to study features of orbits around Mercury and the Galilean moon Callisto; (2) The use of different perturbation integrals, techniques to identify and map different perturbations present in a planetary system, with focus on the analysis of systems of Giant planets with their massive moons; (3) The use of the concept of intermediary Hamiltonian and the use of a canonical transformation called elimination of the parallax, both to deal with binary systems in the context of the roto-orbital dynamics, this one as an approach of the fulltwo body problem; (4) An updated analysis of Gauss variational equations to study quasisatellite orbits around the Martian moon Phobos and with analytical predictions made after obtaining linear and averaged equations of motions. Therefore, this thesis intend not only to provide important analysis and results for each specific problem which it deals with along its pages, but also seeks to highlighting the merit and current relevance of different analytical and semi-analytical methods to be used in the fields of astrodynamics and celestial mechanics. Additionally, the author also hopes to offer an outcome of diverse interesting ideas and methods to be explored in future investigations in these research fields / Na atualidade, a despeito do desenvolvimento tecnológico experimentado pela ciência em geral, algo especialmente evidenciado por poderosas máquinas computacionais disponíveis, os métodos analíticos e semianalíticos para o estudo de diferentes problemas espaciais ainda são de grande importância nos campos de astrodinâmica e mecânica celeste. Desde a compreensão física do movimento de corpos celestes até ao planejamento e projeto de missões espaciais, o uso de modelos matemáticos para lidar com um grande número de problemas contemporâneos desempenha um papel fundamental no progresso do conhecimento humano. Neste contexto, a presente tese apresenta o uso de diferentes técnicas matemáticas para lidar com diversos e atuais problemas em astrodinâmica e mecânica celeste. Os estudos desenvolvidos ao longo deste trabalho são aplicáveis à ambas as áreas. Os tópicos estudados são os seguintes: (1) O desenvolvimento de potenciais perturbadores usando o processo de dupla média, de forma a serem incluídos nas equações planetárias de Lagrange que são integradas numericamente para estudar características de órbitas ao redor de Mercúrio e da lua galileana Calisto; (2) A utilização de diferentes integrais de perturbação, técnicas para identificar e mapear diferentes perturbações presentes em um sistema planetário, com foco na análise de sistemas de planetas gigantes com suas luas massivas; (3) A utilização do conceito de hamiltoniana intermediária e o uso de uma transformação canônica chamada eliminação da paralaxe, ambos para lidar com sistemas binários no contexto da dinâmica roto-orbital, essa sendo uma aproximação do problema completo de dois corpos; (3) Uma análise atualizada de equações variacionais de Gauss para o estudo de órbitas quasi-satélite ao redor da lua marciana Fobos e com predições analíticas realizadas após serem obtidas equações de movimento linearizadas e com média. Portanto, esta tese pretende não somente prover importantes análises e resultados para cada problema específico com os quais a mesma lida ao longo de suas páginas, mas também procura destacar o mérito e relevância atual de diferentes métodos analíticos e semianalíticos a serem utilizados nos campos de astrodinâmica e mecânica celeste. Adicionalmente, o autor também espera oferecer um produto de variadas ideias e métodos a serem explorados em futuras investigações nesses campos de pesquisa / 2013/26652-4 / 2015/18881-9

Orientadores: Patricio A. Letelier Sotomayor e Marcelo Moraes Guzzo / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-13T19:55:09Z (GMT). No. of bitstreams: 1 Castro_GianMachadode_D.pdf: 3217878 bytes, checksum: 48c026fc06d4c9e5db03014506ffc609 (MD5) Previous issue date: 2009 / Resumo: Nesta tese, estudamos o problema de um anel delgado de matéria de densidade constante com um buraco negro de Kerr em seu centro. Nosso objetivo foi resolver as equações de Einstein no vácuo com simetria axial para esse sistema gravitacional. Para fazer a sobreposição não-linear do anel com o buraco negro (BN), utilizamos o método de Belinsky e Zakharov (MBZ). Este método necessita de uma solução conhecida (solução semente) para gerar uma nova solução. Tomamos a aproximação da solução do anel em multipolos como solução semente. Como resultado, obtivemos a solução de um anel com o BN central. A expansão do anel em multipolos exige o truncamento da série. Esta aproximação introduz um erro em nossa solução. Realizamos o estudo do mesmo devido ao truncamento da série. Também estudamos a estabilidade de órbitas circulares equatoriais de partículas movendo-se ao redor do sistema anel-BN quanto a perturbações epicíclicas e verticais. Analisamos essas perturbações para os modelos de gravitação relativística e newtoniana. Como resultado, encon- tramos o efeito inesperado da duplicação das órbitas circulares de flotons para alguns valores de parâmetros relacionados com o anel e o BN, bem como zonas de estabilidade na região interna do anel. / Abstract: In this thesis, we will study the problem of a thin ring of matter of constant density with a central Kerr black hole. The aim of this work is to solve the Einstein equations in the vacuum with axial symmetry for that gravitational system. To do the nonlinear superposition of the ring with the black hole (BH), we used the Belinsky and Zakharov method (BZM). This method needs a known solution (called seed solution) to generate a new one. We took the Newtonian ring potential approximated by a multipolar expansion as seed solution. As result, we obtained the solution of a ring with a central BH. The ring multipolar expansion demands the truncation of the series. This approach introduces an error in our solution. Estimations of errors due to the truncation of the multipolar expansions are performed. We also studied the stability of equatorial circular orbits of particles moving around the system ring plus BH due to epicycle and vertical perturbations. We analyzed those perturbations for relativistic and Newtonian gravitational models. As result, we found the unexpected effect of the duplication of the photons circular orbits for certain values of parameters related with the ring and BH, as well as zones of stability in the inner area of the matter ring. / Doutorado / Relatividade e Gravitação / Doutor em Ciências

La détection et la caractérisation d'exoplanètes apparaissent clairement comme des thèmes centraux de l'observation astronomique pour les années à venir. Les projets spatiaux ou au sol sont nombreux (HARPS, CoRoT, Kepler, JWST, SPHERE...), mais les études théoriques visant à l'analyse et à la compréhension des données recueillies et à venir sont nécessaires. Durant cette thèse j'ai étudié divers processus physiques affectant la structure interne et l'évolution des planètes géantes, aussi bien au sein, qu'à l'extérieur de notre système solaire. J'ai notamment modélisé en détail: -L'impact de l'irradiation intense émise par l'étoile sur l'atmosphère d'une planète à faible distance orbitale, et l'effet induit sur l'évolution interne de cette planète. -Le couplage par dissipation de marée de l'évolution orbitale et thermique d'une planète interagissant avec sa proche étoile parente. -L'effet de la déformation due aux marées sur les paramètres observables d'une planète en transit grâce au suivi photométrique de son passage devant l'étoile. -L'incidence sur la structure et l'évolution d'une diminution de l'efficacité du transport de chaleur par convection due à un gradient d'éléments lourd dans l'enveloppe gazeuse d'une planète géante, conduisant au phénomène de convection double-diffusive. A travers l'étude des ces divers processus, j'ai développé différents modèles analytiques et codes numériques qui sont à la fois flexibles et robustes, et qui permettent maintenant d'étudier certaines propriétés des nouveaux objets substellaires détectés à mesure qu'ils sont découverts. / The detection and characterization of extrasolar planets clearly appears as one of the main goals of observational astronomy for the coming years. Space and ground project are numerous, but theoretical studies aimed at analyzing and understanding available and future data are needed. During this thesis, I study various physical processes affecting the internal structure and evolution of both solar, and extrasolar giant planets. In particular I investigate : -the impact of the intense stellar irradiation received by a close in planet on its subsequent internal evolution. This allows me to quantify the radius anomaly of bloated Hot Jupiters and to constrain their internal composition. -the tidal and centrifugal distortion of a fluid planet. By using both analytical and numerical models, I show how non-sphericity of the planet affects transit measurements, yielding an underestimation of its radius. -how the presence of double-diffusive convection caused by a heavy elements gradient in the gaseous envelope of a planet can decrease the efficiency of its internal heat transport, and affect its structure and evolution. -the coupling between the orbital and the thermal evolution of a planet arising from the strong star-planet tidal interaction. Subsequently, I find that tidal heating alone is not a viable explanation for the observed radius anomaly of transiting planets. Through these different studies, I developed various analytical models and numerical codes that are both flexible and robust, and which now allow one to study the properties of new extrasolar planets and brown dwarfs as they are discovered.