Le groupe de difféomorphismes hamiltoniens à support compact d’une variété symplectique admet une distance naturelle bi-invariante, d’après les travaux de Viterbo, Schwarz, Oh, Frauenfelder et Schlenk, construite à partir des invariants spectraux en homologie de Floer Hamiltonienne. Cette distance, appelée la norme spectrale, s’est révélée être un outil fort utile en topologie symplectique. Par contre, son diamètre reste inconnu en général. En fait, pour les variétés symplectiques fermées, il n’existe même pas de critère pour déterminer si la norme spectrale a un diamètre fini ou infini. Il a été conjecturé que, pour les variétés symplectiquement asphériques, le diamètre de la norme spectrale est infini. Dans cette thèse, nous démontrons que pour tout domaine de Liouville, la norme spectrale a un diamètre infini si et seulement si la cohomologie symplectique du domaine de Liouville en question est non nulle. Ceci généralise un résultat de Monzner-Vichery-Zapolsky et admet plusieurs applications dans le cadre des variétés symplectiques fermées. En particulier, nous démontrons que le produit de deux variétés symplectiquement asphériques a un diamètre spectral infini. Plus généralement, nous démontrons que toute variété symplectiquement asphérique contenant un domaine de Liouville incompressible de codimension zéro avec cohomologie symplectique non nulle doit avoir un diamètre spectral infini. / The group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold is endowed with a natural bi-invariant distance, due to Viterbo, Schwarz, Oh, Frauenfelder and Schlenk, coming from spectral invariants in Hamiltonian Floer homology. This distance, called the spectral norm, has found numerous applications in symplectic topology. However, its diameter is still unknown in general. In fact, for closed symplectic manifolds there is no unifying criterion for the diameter to be finite or infinite. It has been conjectured that for closed symplectically aspherical manifolds, the spectral norm has infinite diameter. In this thesis, we prove that for any Liouville domain the spectral norm has infinite diameter if and only if its symplectic cohomology does not vanish. This generalizes a result of Monzner-Vichery-Zapolsky and has applications in the setting of closed symplectic manifolds. For instance, we show that the product of two closed symplectically aspherical manifold has an infinite spectral diameter . More generally, we prove that any symplectically aspherical manifold which contains an incompressible Liouville domain of codimension zero with non-vanishing symplectic cohomology must have infinite spectral diameter.

Práce se zabývá teorií nelineární spektroskopie a projevy kvantové koherentní dynamiky v nelineární spektroskopii. Poskytuje stručný přehled spektroskopických metod se zaměřením na metodu pump-probe. Dále rozví- jíme teorii nelineární odezvy, přičemž vycházíme z obecného N-wave mixing experimentu, a dospíváme ke tvaru odezvové funkce třetího řádu vyjádřené pomocí Liouvillových drah. Pro vybrané modelové systémy sledujeme koher- entní efekty, které se projevují v 2D a pump-probe spektrech, a provádíme jejich porovnání. D·raz je kladen na objasnění jev· relaxace a excitonové koherence mezi dvěma excitovanými stavy molekulárního dimeru. 1

Neste trabalho buscamos caracterizar o espectro de uma classe de operadores bloco--Jacobi limitados definidos em $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ representa uma faixa de largura $L\\ge 2$ no semi--plano $\\mathbb{Z}_+^2$) e sujeitos a perturbações esparsas (no sentido que as distâncias entre as ``barreiras\'\' crescem geometricamente à medida que estas se afastam da origem) distribuídas aleatoriamente. Tais operadores são construídos a partir da soma de Kronecker de matrizes de Jacobi $J$, cada qual atuando em uma direção do espaço. Demonstramos, por meio da bloco--diagonalização do operador, que %o estudo de suas principais propriedades espectrais dependem da %se limita à caracterização da ``medida de mistura\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ a medida espectral associada à matriz de Jacobi $J^j=J+2\\cos(2\\pi j/L)I $. Para tanto, buscamos primeiramente caracterizar cada uma das medidas $\\mu_j$, explorando e aperfeiçoando algumas técnicas bastante conhecidas no estudo de operadores esparsos unidimensionais. Demonstramos, por exemplo, que a seqüência de ângulos de Prüfer (variáveis que, juntamente com os raios de Prüfer, parametrizam as soluções da equação de autovalores) é uniformemente distribuída no intervalo $[0,\\pi)$, o %que %resultado que nos permite determinar o comportamento assintótico médio das soluções da equação de autovalores. Tal resultado, aliado às técnicas desenvolvidas por Marchetti \\textit{et. al.} em \\cite{MarWre} e a uma adaptação dos critérios de Last e Simon \\cite{LS} para operadores esparsos, nos permitem demonstrar a existência de uma transição aguda (pontual) entre os espectros singular--contínuo e puramente pontual. Empregamos em seguida os resultados de Jitomirskaya e Last presentes em \\cite{JitLast} e obtemos a dimensão Hausdorff exata associada à medida $\\mu_j$, dada por $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4- (\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[-2,2]$), recuperando um resultado análogo obtido por Zlato\\v s em \\cite{Zla}. Por fim, adaptamos tais resultados à situação da medida de mistura associada à matriz bloco--Jacobi, obtendo $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, como sua dimensão Hausdorff exata. Estudamos modelos idênticos com esparsidades sub e super-geométricas, obtendo na primeira situação um espectro puramente pontual (de dimensão Hausdorff nula) e na segunda um espectro puramente singular--contínuo (de dimensão Hausdorff 1). Finalmente, verificamos a existência de transição entre os espectros puramente pontual e singular--contínuo em um modelo com esparsidade super-geométrica cuja dimensão Hausdorff associada à medida espectral é nula. / In this work we attempt to caracterize the spectrum of a class of limited block--Jacobi operators defined in $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ represents a strip of width $L\\ge 2$ on the semi--plane $\\mathbb{Z}_+^2$) subject to a sparse perturbation (which means that the distance between the ``barries\'\' grow geometrically with their distance to the origin) randomly distributed. Such operators are defined as Kronecker sums of unidimensional Jacobi matrices $J$, each one acting in different directions of the space. We prove, by means of a block--diagonalization of the operator, that %the study of its most relevant spectral properties depend on %is related to the caracterization of the ``mixture measure\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ the spectral measure of the Jacobi matrix $J^j=J+2\\cos(2\\pi j/L)I$. For this, we must characterize at first each one of the measures $\\mu_j$, exploiting and improving some well known techniques developed in the study of unidimensional sparse operators. We prove, for instance, that the sequence of Prüfer angles (variables which parametrize the solutions of the eigenvalue equation) are uniform distributed on the interval $[0,\\pi)$, a result which gives us condition to determine the average asymptotic behavior of the solutions of the eigenvalue equation. Such result, in association with the techniques developed by Marchetti \\textit{et. al.} in \\cite{MarWre} and with an adaptation of Last--Simon \\cite{LS} criteria for sparse operator, permit us to prove the existence of a sharp transition between singular continuous and pure point spectra. Following on, we use the results from Jitomirskaya--Last of \\cite{JitLast} and obtain the exact Hausdorff dimension of the measure $\\mu_j$, given by $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4-(\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[- 2,2]$), recovering an analogous result due to Zlato\\v s in \\cite{Zla}. At last, we adapt these results to the mixture measure of the block--Jacobi matrix, obtaining $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, as its exact Hausdorff dimension. We study as well identical models with sub and super geometric sparsities conditions, obtaining a pure point spectrum (with null Hausdorff dimension) in the first case, and a purely singular continuous spectrum (such that its Hausdorff dimension is 1) in the second. Finally, we prove the existence of a transition between pure point and singular continuous spectra in a model with sub--geometric sparsity whose Hausdorff dimension related to the spectral measure is null.

C'est l'histoire d'un mathématicien qui est allé à la recherche des tores perdus<br />dans la jungle des systèmes complètement intégrables. Il a trouvé des feuilles<br />particulières et des tores pour construire une petite cabane qui donne une vue<br />topologique sur la jungle.

Cette thèse aborde des sujets d'approximation diophantienne et de transcendance liés aux fonctions exponentielles. Il est tout d'abord établit des liens entre les coefficients d'approximants de Hermite-Padé, ceux de polynômes d'interpolation de Hermite et certains cofacteurs d'un déterminant de Vandermonde généralisé. Nous utilisons ensuite la notion de hauteur d'une matrice (que nous majorons grâce aux liens précédemment fournis) afin de donner une nouvelle démonstration de la transcendance de $e$. Ces résultats nous permettent finalement d'obtenir de nouveaux énoncés d'approximation diophantienne tels que la minoration de la distance de l'exponentielle d'un nombre algébrique (de hauteur absolue logarithmique de Weil bornée) à un autre nombre algébrique (lui aussi de hauteur absolue logarithmique de Weil bornée) en fonction de ces mêmes bornes. Il est ensuite donné, pour différentes valeurs de nombres rationnels $a$, quelques estimations remarquables telles que le minimum, sur l'ensemble des entiers non nuls $b$ et $c$, de la distance $|e^(b)-a^(c)|$.

<p>This thesis describes water proton and deuterium relaxation processes, as seen by Nuclear Magnetic Resonance (NMR) spectroscopy, using Brownian Dynamics (BD) or Molecular Dynamics (MD) simulations. The MD simulations reveal new detailed information about the dynamics and order of water molecules outside of a lipid bilayer. This is very important information in order to fully understand deuterium NMR measurements in lipid bilayer systems, which require an advanced analysis, because of the complicated water motion (such as tumbling and self-diffusion). The BD simulation methods are combined with the powerful Stochastic Liouville Equation (SLE) in its Langevin form (SLEL) to give new insight into both ¹H₂O and ²H₂O relaxation. The new simulation techniques which combine BD and SLEL can give important new information in cases where other methods do not apply. The deuterium relaxation is described in the context of a water/lipid interface and is in a very elegant way combined with the simulation of diffusion on curved surfaces developed by our research group. ¹H₂O spin-lattice relaxation is described for paramagneticsystems. With this we mean systems with paramagnetic transition metal ions or complexes, that are dissolved into a water solvent. The theoretical description of such systems are quite well investigated but such systems are not yet fully understood. An important consequence of the Paramagnetic Relaxation Enhancement (PRE) calculations when using the SLEL approach combined with BD simulations is that we obtain the electron correlation functions, which describe the relaxation of the paramagnetic electron spins. This means for example that it is also straight forward to generate Electron Spin Resonance (ESR) lineshapes.</p>

This thesis describes water proton and deuterium relaxation processes, as seen by Nuclear Magnetic Resonance (NMR) spectroscopy, using Brownian Dynamics (BD) or Molecular Dynamics (MD) simulations. The MD simulations reveal new detailed information about the dynamics and order of water molecules outside of a lipid bilayer. This is very important information in order to fully understand deuterium NMR measurements in lipid bilayer systems, which require an advanced analysis, because of the complicated water motion (such as tumbling and self-diffusion). The BD simulation methods are combined with the powerful Stochastic Liouville Equation (SLE) in its Langevin form (SLEL) to give new insight into both 1H2O and 2H2O relaxation. The new simulation techniques which combine BD and SLEL can give important new information in cases where other methods do not apply. The deuterium relaxation is described in the context of a water/lipid interface and is in a very elegant way combined with the simulation of diffusion on curved surfaces developed by our research group. 1H2O spin-lattice relaxation is described for paramagneticsystems. With this we mean systems with paramagnetic transition metal ions or complexes, that are dissolved into a water solvent. The theoretical description of such systems are quite well investigated but such systems are not yet fully understood. An important consequence of the Paramagnetic Relaxation Enhancement (PRE) calculations when using the SLEL approach combined with BD simulations is that we obtain the electron correlation functions, which describe the relaxation of the paramagnetic electron spins. This means for example that it is also straight forward to generate Electron Spin Resonance (ESR) lineshapes.

Relaxation models are developed using numerical solutions of the Stochastic Liouville Equation of motion. Simplified descriptions such as the stochastic master equation is described in the context of fluorescence depolarisation experiments. Redfield theory is used in order to describe NMR relaxation in bicontinuous phases. The stochastic fluctuations in the relaxation models are accounted for using Brownian Dynamics simulation technique. A novel approach to quantitatively analyse fluorescence depolarisation experiments and to determine intramolecular distances is presented. A new Brownian Dynamics simulation technique is developed in order to characterize translational diffusion along the water lipid interface of bicontinuous cubic phases.

No description available.

MASCARENHAS, Sebastião Pontes. A irracionalidade e transcendência dos números. 2017. 77 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-07-28T13:20:00Z No. of bitstreams: 1 2017_dis_spmascarenhas.pdf: 904841 bytes, checksum: a2a7ea7aa1e426a76fd5027662179d42 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Bom dia, Motivos da rejeição: Na capa o nome do curso deve ser Programa de Pós-Graduação em Matemática em Rede Nacional. Faltou a ficha catalográfica. Atenciosamente, Rocilda on 2017-07-28T13:58:59Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-07-28T19:21:17Z No. of bitstreams: 1 2017_dis_spmascarenhas.pdf: 469506 bytes, checksum: b542e915b1564ae42c5160fee08ccedc (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-07-31T11:41:11Z (GMT) No. of bitstreams: 1 2017_dis_spmascarenhas.pdf: 469506 bytes, checksum: b542e915b1564ae42c5160fee08ccedc (MD5) / Made available in DSpace on 2017-07-31T11:41:11Z (GMT). No. of bitstreams: 1 2017_dis_spmascarenhas.pdf: 469506 bytes, checksum: b542e915b1564ae42c5160fee08ccedc (MD5) Previous issue date: 2017 / This present work is an explanation orientated for the check of the irracionality of some real numbers, for the construction of some transcendents numbers (in especial, the Liouville´s numbers) and for the transcendency of , and anothers numbers. The understanding of the presents demonstrations in this work involves some basics knowledge in theory of numbers (divisibility, highest divisor common, number prime, etc), theory of conjunct (enumerate), Differential and Integral Calculation in a real variable, a few of functions of two variables e some facts about convergence of sequences and series. As a consequence, will be seen the solution of the old problem of the quadrature of a circle, that is, a possibility ou not of the construction with ruler and compass of a square, whose area be equal to area of a circle radius gived. / O presente trabalho é uma exposição voltada à verificação da irracionalidade de certos números reais, à construção de certos números transcendentes (em especial, os números de Liouville) e à transcendência de , e outros números. O entendimento das demonstrações presentes nesse trabalho envolve alguns conhecimentos básicos em teoria dos números (divisibilidade, máximo divisor comum, números primos, etc), teoria dos conjuntos (enumerabilidade), Cálculo Diferencial e Integral em uma variável real, um pouco de funções de duas variáveis e alguns fatos sobre convergência de sequências e séries. Como consequência, veremos a solução do antigo problema da quadratura de um círculo, isto é, a possibilidade ou não da construção com régua e compasso de um quadrado, cuja área equivale-se à área de um círculo de raio dado.