Let X be a simply connected, compact Riemannian symmetric space. We can represent X as the homogeneous space U/K, where U is a simply connected compact Lie group, and K is the fixed point set of an involution θ of U. Let G be the complexification of U. We consider the intersections of the image of the Cartan embedding Φ : U/K → U ⊂ G : uK → uu⁻ᶿ with the strata of the Birkhoff (or triangular, or LDU) decomposition G = ⫫(w∈W) ∑(G/w), ∑(G/w) = N⁻wHN⁺ relative to a θ-stable decomposition of the Lie algebra, g = n⁻ ⊕h ⊕ n⁺. For a generic element g in this intersection, g ∈ Φ(U/K) ∩ ∑(G/1), this yields a unique triangular factorization g = ldu. Our main contribution is to produce explicit formulas for the diagonal term d in classical cases, using Cayley coordinates (this choice of coordinate is motivated by considerations beyond sheer convenience). These formulas have several applications: 1) we can compute π₀(Φ(U/K) \ ∩ ∑(G/1) ) explicitly; 2) we can compute ʃ(Φ(U/K))ᵃΦ^-iλ (where ᵃΦ is the positive part of d) using elementary techniques in rank 1 cases; 3) they are useful in explicitly calculating Evens-Lu Poisson structures on U=K (see [Caine(2006)]). Our set-up involves choosing specific representations of the various u in su(n;C) that are compatible with θ; that is, θ fixes each of the subspaces n⁻; h; and n⁺ which, in our setup, always consist of strictly lower triangular, diagonal, and strictly upper triangular matrices, respectively. The formulas contain determinants such as det(1 + X), where X is in ip, the -1-eigenspace of θ acting on the Lie algebra u. Due to the relatively sparse nature of these matrices, these determinants are often easily calculable, and we illustrate this with many examples.

L’objectif de ce travail est d’établir des résultats de rigidité spectrale pour des familles C1 d’opérateurs (pseudo-)différentiels elliptiques auto-adjoints Pt, t ϵ [0, ẟ] sur une variété lisse compacte M sans bord de dimension n ≥ 2. Dans les deux premiers chapitres, on étudie des hamiltoniens proches d’un hamiltonien intégrable qui est non dégénéré au sens de Kolmogorov (Système KAM). On y construit une forme normale de Birkhoff au voisinage de chaque tore KAM ayant une fréquence diophantienne. Dans les chapitres 3 et 4 on établit une forme normale de Birkfoff quantique afin de construire des familles C1 de quasi-modes. Ces dernières permettent de relier les propriétés spectrales de Pt aux propriétés dynamiques des tores KAM. Les deux derniers chapitres proposent des applications en lien avec la transformée de Radon ainsi qu’une étude sur les surfaces de rotation. / The aim of this work is to obtain spectral rigidity results for C1 families of elliptic self-adjoint (pseudo-)differential operators Pt, t ϵ [0, ẟ], on a smooth closed manifold M of dimension n ≥ 2. In the first two chapters, we investigate Hamiltonians close to a given integrable Hamiltonian which is non-degenerate in the sense of Kolmogorov (KAM system). This allows us to obtain a Birkhoff normal form in a neighborhood of any KAM tori with a Diophantine frequency. In the third and fourth chapters, we construct a quantum Birkhoff normal form and obtain C1 families of quasimodes. Using the quasi-modes, we establish a connection between the spectral properties of Pt and the dynamical properties of the KAM tori. The last two chapters provide applications of these results to the Radon transform and the surfaces of revolution.

Results from combinatorial topology have shown that certain combinatorial lemmas are equivalent to certain topologocal fixed point theorems. For example, Sperner’s lemma about labelings of triangulated simplices is equivalent to the fixed point theorem of Brouwer. Moreover, since Sperner’s lemma has a constructive proof, its equivalence to the Brouwer fixed point theorem provides a constructive method for actually finding the fixed points rather than just stating their existence. The goal of this research project is to develop a combinatorial analogue for the Poincare ́-Birkhoff fixed point theorem.

Στην παρούσα εργασία παρουσιάζεται μία μέθοδος πλήρους διακριτοποίησης (χωρικής και χρονικής) για την εξίσωση των Korteweg και de Vries. H μέθοδος αυτή μελετήθηκε από τον J. Schiff στην εργασία Loop groups and discrete KdV equations και στηρίζεται στην διάσπαση Birkhoff σε κατάλληλη ομάδα βρόχων για την εύρεση του ζεύγους Lax. Για τις προκύπτουσες εξισώσεις μερικών διαφορών κατασκευάζονται μετασχηματισμοί Backlund μέσω της ίδιας μεθόδου, οι οποίοι, στην συνέχεια, χρησιμοποιούνται για την εύρεση σολιτονικών λύσεων. Ειδικότερα, μία από τις διακριτοποιήσεις έχει άμεσο ("φυσικό") συνεχές όριο την εξίσωση potential KdV. Σε κάθε περίπτωση διακριτοποίησης, κατασκευάζονται σολιτονικές λύσεις, οι οποίες συγκρίνονται με αυτές της συνεχούς περίπτωσης και εξετάζονται ως προς την σολιτονική αλληλεπίδραση. / In this thesis, we present a method of full discretization (both spatial and temporal coordinates are discretized) for the Korteweg and de Vries' equation. This method was studied by J. Schiff in his paper Loop groups and discrete KdV equations. The procedure is based on Birkhoff decomposition in an appropriate loop group in order to derive a Lax representation. For the resulting partial difference equations, we construct Backlund transformations via the same method, which are used to generate soliton solutions. In particular, one discretization has the potential KdV equation as a standard (natural) continuum limit. In both cases, soliton solutions are produced and compared with those of the continuous case. Finally, we study their soliton interaction.

<p>This thesis consists of two papers.</p><p>The first paper is a study of the structure of the k-assignment polytope, whose vertices are the <em>m x n</em> (0; 1)-matrices with exactly <em>k</em> 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. Two equivalent representations of the faces are given, one as (0; 1)-matrices and one as ear decompositions of bipartite graphs. These tools are used to describe properties of the polytope, especially a complete description of the cover relation in the face lattice of the polytope and an exact expression for the diameter.</p><p>The second paper studies the edge-product space <em>Є(X)</em> for trees on <em>X</em>. This space is generated by the set of edge-weighted finite trees on <em>X</em>, and arises by multiplying the weights of edges on paths in trees. These spaces are closely connected to tree-indexed Markov processes in molecular evolutionary biology. It is known that <em>Є(X)</em> has a natural <em>CW</em>-complex structure, and a combinatorial description of the associated face poset exists which is a poset <em>S(X)</em> of <em>X</em>-forests. In this paper it is shown that the edge-product space is a regular cell complex. One important part in showing that is to conclude that all intervals <em>[Ô, Г], Г </em>Є<em> S(X),</em> have recursive coatom orderings.</p> / Report code: LiU-TEK-LIC-2004:46.

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This thesis consists of two papers. The first paper is a study of the structure of the k-assignment polytope, whose vertices are the m x n (0; 1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. Two equivalent representations of the faces are given, one as (0; 1)-matrices and one as ear decompositions of bipartite graphs. These tools are used to describe properties of the polytope, especially a complete description of the cover relation in the face lattice of the polytope and an exact expression for the diameter. The second paper studies the edge-product space Є(X) for trees on X. This space is generated by the set of edge-weighted finite trees on X, and arises by multiplying the weights of edges on paths in trees. These spaces are closely connected to tree-indexed Markov processes in molecular evolutionary biology. It is known that Є(X) has a natural CW-complex structure, and a combinatorial description of the associated face poset exists which is a poset S(X) of X-forests. In this paper it is shown that the edge-product space is a regular cell complex. One important part in showing that is to conclude that all intervals [Ô, Г], Г Є S(X), have recursive coatom orderings.

El objetivo de esta tesis es desarrollar la interpolación de Birkhoff mediante polinomios lacunarios.En la interpolación algebraica de Birkhoff se determina un polinomio de grado menor que n, para ello se emplean n condiciones que fijan el valor del polinomio o sus derivadas. Los problemas clásicos de interpolación de Lagrange, Taylor, Hermite, Hermite-Sylvester y Abel-Gontcharov son casos particulares de interpolación algebraica de Birkhoff.Un espacio de polinomios lacunarios de dimensión n es el conjunto de los polinomios que pueden generarse por combinación lineal de n potencias distintas de grados, en general, no consecutivos. En particular, cuando tomamos potencias de grados 0,1,.,n-1, se obtiene el espacio de polinomios de grado menor que n, empleado en la interpolación algebraica clásica. En la interpolación algebraica clásica, el número de condiciones determina el espacio de interpolación. En contraste, en la interpolación mediante polinomios lacunarios las condiciones de interpolación determinan únicamente la dimensión del espacio de interpolación y pueden existir una infinidad de espacios sobre los que realizar la interpolación. Esto nos permite construir mejores estrategias de interpolación en ciertos casos, como la interpolación de funciones de gran crecimiento (interpolación de exponenciales y de ramas asintóticas).La aportación de la tesis consiste en la definición de un marco teórico adecuado para la interpolación de Birkhoff mediante polinomios lacunarios y en la extensión al nuevo marco de los principales elementos de la interpolación algebraica de Birkhoff. En concreto, se generaliza la condición de Pólya, se caracteriza la regularidad condicionada, se establecen condiciones suficientes de regularidad ordenada que extienden el teorema de Atkhison-Sharma, se extiende la descomposición normal y se establecen condiciones suficientes de singularidad en los casos indescomponibles.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-02-07Bitstream added on 2014-06-13T20:08:06Z : No. of bitstreams: 1 pereira_ms_me_sjrp.pdf: 2071399 bytes, checksum: 9f8844443f17c4fa7a041cc8bc621d54 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho classificamos problemas de bifurcação Z2-equivariantes de corank 2 até co- dimensão 3 via técnicas da Teoria de Singularidades. A abordagem para classificar tais problemas é baseada no processo de redução à forma normal de Birkhoff para estudar a interação de modos Hopf-Pontos de Equilíbrio. O comportamento geométrico das soluções dos desdobramentos das formas normais obtidas é descrito pelos diagramas de bifurcação e estudamos a estabilidade assintótica desses ramos. / In this work we classify the Z2-equivariant corank 2 bifurcation problems up to codimension 3 via Singularity Theory techniques. The approach to classify such problems is based on the Birkhoff normal form to study Hopf-Steady- State mode interaction. The geometrical behavior of the solutions of the unfolding of the normal forms is described by the bifurcation diagrams and we study the asymptotic stability of such branches.

In this work, we study the set X &#775;(&#966;,f) of points such that the Birkhoff averages do not exist. Following Thompson, our main result here is to show that the topological pressure of X &#775;(&#966;,f) is total. As corollary, we get the some result for the Oseledets Irregular set for Lyapunov exponent in one dimension. For higher dimensions, this question is still open. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Neste trabalho, estudamos o conjunto X &#775;(&#966;,f) de pontos tal que as médias de Birkhoff não existe. Seguindo Thompson, nosso resultado principal aqui é mostrar que a pressão topológica de X &#775;(&#966;,f) é total. Como corolário, damos o mesmo resultado para o conjunto Irregular de Oseledets para os expoentes de Lyapunov em dimensão um. Para dimensões maiores, esta questão está em aberto.