Note:

The distance to uncontrollability of a controllable linear system is a measure of the degree of perturbation a system can undergo and remain controllable. The definition of the distance to uncontrollability leads to a non-convex optimization problem in two variables. In 2000 Gu proposed the first polynomial time algorithm to compute this distance. This algorithm relies heavily on efficient eigenvalue solvers. In this work we examine two alternative algorithms that result in linear matrix inequalities. For the first algorithm, proposed by Ebihara et. al., a semidefinite programming problem is derived via the Kalman-Yakubovich-Popov (KYP) lemma. The dual formulation is also considered and leads to rank conditions for exactness verification of the approximation. For the second algorithm, by Dumitrescu, Şicleru and Ştefan, a semidefinite programming problem is derived using a sum-of-squares relaxation of an associated matrix-polynomial and the associated Gram matrix parameterization. In both cases the optimization problems are solved using primal-dual-interior point methods that retain positive semidefiniteness at each iteration. Numerical results are presented to compare the three algorithms for a number of benchmark examples. In addition, we also consider a system that results from a finite element discretization of the one-dimensional advection-diffusion equation. Here our objective is to test these algorithms for larger problems that originate in PDE-control. / Master of Science

A new upwind scheme is developed for solving the three-dimensional Euler equations on unstructured tetrahedral meshes. The method yields solution accuracy and efficiency comparable to that currently available from similar structured-grid codes. The key to achieving this result is a novel cell reconstruction process which is based on an analytical formulation for computing solution gradients within tetrahedral cells. Prior methodology requires the application of cumbersome numerical procedures to evaluate surface integrals around the cell volume. The result is that higher-order differences can now be constructed more efficiently to attain computational times per cell comparable to those of structured codes. The underlying philosophy employed in constructing the basic flow solver is to draw on proven structured-grid technology whenever possible in order to reduce risk. Thus, spatial discretization is accomplished by a cell-centered finite-volume formulation using flux-difference splitting. Solutions are advanced in time by a 3- stage Runge-Kutta time-stepping scheme with convergence accelerated to steady state by local time stepping and implicit residual smoothing. The flow solver operates at a speed of 34 microseconds per cell per cycle on a CRAY-2S supercomputer and requires 64 words of memory per cell. Transonic solutions are presented for a broad class of configurations to demonstrate the accuracy, speed, and robustness of the new scheme. Solutions are shown for the ONERA M6 wing, the Boeing 747-200 configuration, a low-wing transport configuration, a high-speed civil transport configuration, and the space shuttle ascent configuration. Computed surface pressure-coefficient distributions on the ONERA M6 wing are compared with structured-grid results as well as experimental data to quantify the accuracy. A further assessment of grid sensitivity and the effect of convergence acceleration parameters is also included for this configuration. The more complex configurations serve to demonstrate the robustness and efficiency of the new method and its potential for performing routine aerodynamic analysis of full aircraft configurations. For example, the basic transonic flow features are well captured on the space shuttle ascent configuration with only 7 megawords of memory and 142 minutes of CRAY-YMP run time. / Ph. D.

Σκοπός της εργασίας είναι η παρουσίαση των εξισώσεων κίνησης του στερεού σώματος και η μελέτη δύο σημαντικών επιλύσιμων περιπτώσεων κίνησης στρόβου (Lagrange, Euler) . Στo πρώτο κεφάλαιο περιγράφουμε την κίνηση ενός στερεού σώματος χρησιμοποιώντας την ομάδα στροφών. Αποδεικνύουμε το θεώρημα Chasles το οποίο μας δείχνει πως η μετακίνηση ενός στερεού μπορεί να αποσυντεθεί σε περιστροφή γύρω από έναν άξονα και μεταφορά πάνω σε αυτόν. Στη συνέχεια σκοπός μας είναι η κατανόηση της γωνιακής ταχύτητας ενός στερεού σώματος. Σημαντικό ρόλο σε αυτή την πορεία παίζει τόσο το αδρανειακό όσο και το ενσωματωμένο στο στερεό σύστημα αναφοράς. Έπειτα δίνονται οι ορισμοί της ενέργειας, της στροφορμής, της ροπής και οι εκφράσεις τους συναρτήσει γνωστών πλέον εννοιών από τα προηγούμενα. Το κεφάλαιο ολοκληρώνεται με την Δυναμική που έχει ως αντικείμενο μελέτης και έρευνας τη κίνηση των σωμάτων υπό την επίδραση δυνάμεων, και καταλήγει στην παρουσίαση των εξισώσεων Euler. Στο δεύτερο κεφάλαιο στρέφουμε το ενδιαφέρον μας στις εφαρμογές και παρουσιάζουμε την επίλυση δύο σημαντικών προβλημάτων της μηχανικής: η κίνηση ενός συμμετρικού στρόβου που κινείται υπό την επίδραση του βάρους του έχοντας ένα σταθερό σημείο (ο στρόβος του Lagrange) και η κίνηση ενός στερεού που κινείται χωρίς την επίδραση εξωτερικών ροπών (ο στρόβος του Euler). Οι λύσεις εκφράζονται μέσω Ελλειπτικών Συναρτήσεων. Τέλος, στο τρίτο κεφάλαιο παρατίθενται σχόλια στις εργασίες των Holmes - Marsden και των Heijden - Yagasaki που αφορούν την ύπαρξη χαοτικής συμπεριφοράς στην διαταραγμένη περίπτωση Lagrange, που αναφέρεται σε σχεδόν συμμετρικό στρόβο. / Σκοπός της εργασίας είναι η παρουσίαση των εξισώσεων κίνησης του στερεού σώματος και η μελέτη δύο σημαντικών επιλύσιμων περιπτώσεων κίνησης στρόβου (Lagrange, Euler) . Στo πρώτο κεφάλαιο περιγράφουμε την κίνηση ενός στερεού σώματος χρησιμοποιώντας την ομάδα στροφών. Αποδεικνύουμε το θεώρημα Chasles το οποίο μας δείχνει πως η μετακίνηση ενός στερεού μπορεί να αποσυντεθεί σε περιστροφή γύρω από έναν άξονα και μεταφορά πάνω σε αυτόν. Στη συνέχεια σκοπός μας είναι η κατανόηση της γωνιακής ταχύτητας ενός στερεού σώματος. Σημαντικό ρόλο σε αυτή την πορεία παίζει τόσο το αδρανειακό όσο και το ενσωματωμένο στο στερεό σύστημα αναφοράς. Έπειτα δίνονται οι ορισμοί της ενέργειας, της στροφορμής, της ροπής και οι εκφράσεις τους συναρτήσει γνωστών πλέον εννοιών από τα προηγούμενα. Το κεφάλαιο ολοκληρώνεται με την Δυναμική που έχει ως αντικείμενο μελέτης και έρευνας τη κίνηση των σωμάτων υπό την επίδραση δυνάμεων, και καταλήγει στην παρουσίαση των εξισώσεων Euler. Στο δεύτερο κεφάλαιο στρέφουμε το ενδιαφέρον μας στις εφαρμογές και παρουσιάζουμε την επίλυση δύο σημαντικών προβλημάτων της μηχανικής: η κίνηση ενός συμμετρικού στρόβου που κινείται υπό την επίδραση του βάρους του έχοντας ένα σταθερό σημείο (ο στρόβος του Lagrange) και η κίνηση ενός στερεού που κινείται χωρίς την επίδραση εξωτερικών ροπών (ο στρόβος του Euler). Οι λύσεις εκφράζονται μέσω Ελλειπτικών Συναρτήσεων. Τέλος, στο τρίτο κεφάλαιο παρατίθενται σχόλια στις εργασίες των Holmes - Marsden και των Heijden - Yagasaki που αφορούν την ύπαρξη χαοτικής συμπεριφοράς στην διαταραγμένη περίπτωση Lagrange, που αναφέρεται σε σχεδόν συμμετρικό στρόβο.

Cette thèse est consacrée à l’étude de familles à un paramètre de coproduits sur lesfonctions symétriques et leurs analogues non commutatifs. On montre en introduisant une base appropriée qu’une famille à un paramètre d’algèbres de Hopf introduite par Foissy interpole entre l’algèbre de Faà di Bruno et l’algèbre de Farahat-Higman. Les constantes de structure dans cette base sont des déformations des constantes de structures de l’algèbre de Farahat-Higman dans la base des projections des classes de conjugaison. On obtient pour ces constantes de structure déformées un analogue des formules de Macdonald. Foissy a également introduit un analogue non commutatif de cette famille d’algèbres de Hopf, qui interpole entre l’algèbre de Hopf des fonctions symétriques non commutatives et l’algèbre de Faà di Bruno non commutative. Après avoir donné une nouvelle interprétation combinatoire de la formule de Brouder-Frabetti-Krattenthaler pour l’antipode de l’algèbre de Faà di Bruno non commutative, qui est une forme de la formule d’inversion de Lagrange non commutative, on donne une déformation à un paramètre de cette formule. Plus précisément, on obtient une formule explicite pour l’antipode de la déformation de Foissy dans sa version non commutative. On donne aussi d’autres propriétés combinatoires de l’algèbre de Faà di Bruno non commutative et d’autres résultats permettant d’étudier les deux familles d’algèbre de Hopf de Foissy. Ainsi, on généralise par exemple d’autres formes de la formule d’inversion de Lagrange non commutative en donnant d’autres formules qui calculent l’antipode de la deuxième déformation. / This thesis is devoted to study one-parameter families of coproducts on symmetric functionsand their noncommutative analogues. We show, by introducing an appropriate basis,that a one-parameter family of Hopf algebras introduced by Foissy interpolates between theFa`a di Bruno algebra and the Farahat-Higman algebra. The structure constants in this basisare deformations of the structure constants of the Farahat-Higman algebra in the basis ofprojections of conjugacy classes. For these deformed structure constants, we obtain an analogueof the Macdonald formulas.Foissy has also introduced a noncommutative analogue of this family of Hopf algebras. Itinterpolates between the Hopf algebra of noncommutative symmetric functions and the noncommutativeFa`a di Bruno algebra. First, we give a new combinatorial interpretation ofthe Brouder-Frabetti-Krattenthaler formula for the antipode of the noncommutative Fa`a diBruno algebra, that is a form of the noncommutative Lagrange inversion formula. Then, wegive a one-parameter deformation of this formula. Namely, it is an explicit formula for theantipode of the noncommutative family.We also give other combinatorial properties of the noncommutative Fa`a di Bruno algebra,and other results about the families of Hopf algebras of Foissy. In this way, we generalize otherforms of the noncommutative Lagrange inversion formula. Namely, we give other formulasfor the antipode of the noncommutative family.

Nous étudions le problème d'unicité, de l'interpolation faible et de la convergence de la série d'interpolation de Lagrange dans les espaces de Fock pondérés par des poids radiaux. Nous étudions aussi les suites d'échatillonnage, d'interpolation et les bases de Riesz dans les petit espaces de Fock. / We study the uniqueness sets, the weak interpolation sets, and convergence of the Lagrange interpolation series in radial weighted Fock spaces. We study also sampling, interpolation and Riesz bases in small radial weighted Fock spaces

Neste trabalho, estudamos algumas inversões parciais do teorema de Dirichlet-Lagrange, essencialmente estendendo os resultados em dois graus de liberdade de Garcia e Tal (2003) para algumas situações em $R^$. Mais precisamente, um dos objetivos é mostrar, no contexto da mecânica lagrangeana, que se há um split da energia potencial em uma parte no plano cujo jato $k$ mostra que ela não tem mínimo no ponto de equilíbrio e existe o jato $k-1$ do seu gradiente, e a outra em $R^$ que tenha mínimo no ponto de equilíbrio, este é instável. A instabilidade do ponto de equilíbrio em estudo é provada mostrando a existência de uma trajetória assintótica ao mesmo. Para isso, apresentamos um resultado inicial para lagrangeanos com uma forma bem específica e, a seguir, mostramos que a classe de lagrangeanos que descrevemos acima pode ser levada a esta forma, através de uma adequada mudança de coordenadas espaciais. Além disso, consideramos a extensão desses resultados a sistemas com forças giroscópicas. / In this work, we study some partial inversions of the Lagrange-Dirichlet theorem, extending the results in two degrees of freedom of Garcia and Tal (2003) for some other situations in $\\mathbb^$. More precisely, one of our objectives is to show, in the context of lagrangian mechanics, that if there is a splitting of the potential energy in one part in the plane which its $k$-jet shows that it does not have a minimum in the equilibrium and there exists the $(k-1)$-jet of its gradient, and the other part in $\\mathbb^$ has a minimum in the equilibrium, then the equilibrium point is unstable. Instability of the equilibrium point is shown by proving the existence of an assymptotic trajectory to it. For this purpose, first it is proven a result for lagrangians with a specific form and, next, we show that the class of lagrangians we are interested in can be transformed into this specific form by a subtle change of spatial coordinates. Finally, we consider the extension of this results to systems with gyroscopic forces.

Neste trabalho, estudamos algumas inversões parciais do teorema de Dirichlet-Lagrange, essencialmente estendendo os resultados em dois graus de liberdade de Garcia e Tal (2003) para algumas situações em $R^$. Mais precisamente, um dos objetivos é mostrar, no contexto da mecânica lagrangeana, que se há um split da energia potencial em uma parte no plano cujo jato $k$ mostra que ela não tem mínimo no ponto de equilíbrio e existe o jato $k-1$ do seu gradiente, e a outra em $R^$ que tenha mínimo no ponto de equilíbrio, este é instável. A instabilidade do ponto de equilíbrio em estudo é provada mostrando a existência de uma trajetória assintótica ao mesmo. Para isso, apresentamos um resultado inicial para lagrangeanos com uma forma bem específica e, a seguir, mostramos que a classe de lagrangeanos que descrevemos acima pode ser levada a esta forma, através de uma adequada mudança de coordenadas espaciais. Além disso, consideramos a extensão desses resultados a sistemas com forças giroscópicas. / In this work, we study some partial inversions of the Lagrange-Dirichlet theorem, extending the results in two degrees of freedom of Garcia and Tal (2003) for some other situations in $\\mathbb^$. More precisely, one of our objectives is to show, in the context of lagrangian mechanics, that if there is a splitting of the potential energy in one part in the plane which its $k$-jet shows that it does not have a minimum in the equilibrium and there exists the $(k-1)$-jet of its gradient, and the other part in $\\mathbb^$ has a minimum in the equilibrium, then the equilibrium point is unstable. Instability of the equilibrium point is shown by proving the existence of an assymptotic trajectory to it. For this purpose, first it is proven a result for lagrangians with a specific form and, next, we show that the class of lagrangians we are interested in can be transformed into this specific form by a subtle change of spatial coordinates. Finally, we consider the extension of this results to systems with gyroscopic forces.

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-05-03T18:30:03Z No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-05-04T14:06:15Z (GMT) No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-05-04T14:06:23Z (GMT) No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) / Made available in DSpace on 2017-05-04T14:10:34Z (GMT). No. of bitstreams: 1 DissAISC.pdf: 2959521 bytes, checksum: 8f70ee52c92314238d50ce361fc05981 (MD5) Previous issue date: 2016-10-21 / Não recebi financiamento / In this work we explain an elegant and accessible proof of the Fundamental Theorem of Algebra using the Lagrange Multipliers method. We believe this will be a valuable resource not only to Mathematics students, but also to students in related areas, as the Lagrange Multipliers method that lies at the heart of the proof is widely taught. / Neste trabalho expomos uma demonstração acessível e elegante do Teorema Fundamental da Álgebra utilizando o método dos multiplicadores de Lagrange. Acreditamos que este trabalho seria uma fonte valiosa não são para estudantes de Matemática, mas também para estudantes de áreas relacionadas, uma vez que o método dos multiplicadores de Lagrange é amplamente ensinado em cursos de exatas.

Orientador: Renato Pavanello / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-18T00:16:08Z (GMT). No. of bitstreams: 1 Bispo_RafaelSantana_M.pdf: 3924997 bytes, checksum: eedf2e79312457d2dfe0c4ce2418dfe4 (MD5) Previous issue date: 2011 / Resumo: Atualmente, existe uma grande tendência no incremento da produção de energia elétrica através de fontes renováveis. Em especial, a geração de energia elétrica produzida através de parques eólicos tem sido bastante adotada. O projeto desses equipamentos envolve a modelagem dinâmica acoplada solo-fluido-estrutura que pode ser estudada usando-se a formulação particionada, onde o problema da interação entre os meios é tratado de maneira iterativa. Nesse tipo de técnica, é possível que modelos fisicamente heterogêneos, chamados de partições, possam utilizar diferentes técnicas de discretização, como por exemplo o domínio do fluido ser baseado em uma formulação de Elementos de Contorno e o domínio estrutura baseado em uma formulação em Elementos Finitos. Neste trabalho, é realizado um estudo dinâmico de turbinas eólicas, utilizando tratamento particionado e Multiplicadores de Lagrange afim de se obter as frequências características e as curvas de resposta em frequência do sistema em análise. A discretização do problema é realizada através do Método dos Elementos Finitos (FEM) utilizando elemento de pórtico e quadrilateral de Wilson. Desta forma, a resolução de problemas de interação, utilizando a formulação particionada, é estudada com a finalidade de avaliar a convergência e a viabilidade da técnica em problemas harmônicos estruturais / Abstract: Currently, there is a great tendency in increasing the production of electricity through renewable sources. In this context, the generation of electric energy produced by wind farms has been widely adopted. The design of these devices involves the dynamic modeling of coupling fluid-structure-soil that can be studied using the partitioned formulation, where the problem of interaction between the parties is iterative manner. In this type of technique, it is possible that physically heterogeneous models, called partitions, can use different discretization techniques, such as the domain of fluid is based on a formulation of boundary element or based on a finite element formulation. In this paper a harmonic simulation of wind turbines, using partitioned treatment and Lagrange multipliers is studied in order to obtain the characteristic frequencies and frequency response function of the system under analysis. The discretization of the problem is performed using the Finite Element Method (FEM), as well as beam elements and the quadrilateral Wilson element. Thus, the resolution of elastodynamics problems, using the partitioned formulation is studied with the aim of assessing the feasibility and convergence of this technique, applied to dynamic harmonic analysis / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica