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[pt] Decompõe-se o espaço das curvas não-degeneradas sobre a n-esfera sujeitas a uma dada matriz de monodromia (munido de uma estrutura de variedade de Hilbert adequada) em uma coleção enumerável de células contráteis parametrizadas pelos itinerários admissíveis para os levantamentos a SOn+1 das referidas curvas através das células obtidas de uma estratificação de SOn+1 estreitamente relacionada com a clássica decomposição de Bruhat de GLn+1. A expressão itinerário admissível significa aqui uma sequência finita de células sujeitas a umas poucas restrições que, ademais, são naturalmente insinuadas pela geometria do problema. O principal interesse dessa nova abordagem é que essa combinatorialização funciona homogeneamente em todas as dimensões n (não obstante óbvias dificuldades computacionais), diferentemente dos métodos ad-hoc, de cunho mais geométrico, até aqui empregados para obter informações topológicas sobre esses e outros espaços de curvas relacionados (que têm sido bem sucedidos apenas em dimensões n baixas). Essa abordagem pode ser considerada como uma primeira tentativa de chegar a um método unificado para a determinação do tipo homotópico de tais espaços, e ajuda a dispensar certos argumentos de análise funcional usualmente empregados na definição da topologia correta para os referidos espaços de curvas. / [en] The space of nondegenerate curves on the n-sphere subject to a fixed monodromy matrix (provided with a suitable Hilbert manifold structure) is decomposed into a countable collection of contractible cells parameterized by the SOn+1-lifted curves admissible itineraries through cells arriving from a stratification of SOn+1 closely related to the classical Bruhat decomposition of GLn+1. The expression admissible itinerary herein stands for a finite sequence of cells subject to a few constraints that are otherwise naturally suggested by the geometry of the problem. The main interest of such a new approach is that this combinatorialization works homogeneously in any dimension n (with obvious computational difficulties), unlike the more geometry-flavoured ad-hoc methods for achieving topological information about these and related spaces of curves (which usually have had a good run only in low dimensions n). This approach can be regarded as a first attempt at a unified method for figuring out the homotopy-type of such spaces, and it helps to override some functional analysis arguments usually deployed in defining the right topology for these spaces of curves.

Induced vibrations due to tire-pavement interaction are one of the main sources of vehicle exterior noise, especially near highways and main roads where traveling speeds are above 50 kph. Its dominant spectral content is approximately within 500-1500 Hz. However, accurate prediction tools within this frequency range are not available. Current methods rely on structural modeling of the complete tire using finite elements and modal expansion approaches that are accurate only at low frequencies. Therefore, alternative physically-based models need to be developed. This work proposes a new approach that incorporates wave behavior along the tire's circumferential direction, while modes are assumed along its transversal direction. The formulation for new infinite plate and cylindrical shell structural models of a tire is presented. These are capable of accounting for orthotropic material properties, different structural parameters between the belt and sidewalls, inflation pressure, and rotation of the tire. In addition, a new contact model between the pavement and the tire is developed presented. The excitation of the tire due to the impact of the tread-pattern blocks in the contact patch region is characterized and coupled to the structure of the tire. Finally, a Boundary Element Method is implemented in order to compute the vibration-induced noise produced by the tire. All the modeling components are combined in a single prediction tool named Wave Pro Tire. Lastly, simulated responses and validation cases are presented in terms of harmonic responses, Frequency Response Functions (FRF), and produced noise. / Doctor of Philosophy / Induced vibrations due to tire-pavement interaction are one of the main sources of vehicle exterior noise, especially near highways and main roads where traveling speeds are above 50 kph. Accurate prediction tools are not currently available. Therefore, new physically based models need to be developed. This work proposes a new approach to model the tire’s structure with a formulation that accounts for multiple physical phenomena. In addition, a model that simulates the contact between the pavement and the tire’s tread is presented. Finally, the vibrations are coupled to the produced noise in a single prediction tool named Wave Pro Tire. This work also includes simulated responses and validation cases.

<p>Computer programs keep finding their ways into new safety-critical applications, while at the same time growing more complex. This calls for new and better methods to verify the correctness of software. We focus on one approach to verifying systems, namely that of <i>model checking</i>. At first, we investigate two categories of problems related to model checking: <i>games</i> and <i>stochastic infinite-state systems</i>. In the end, we join these two lines of research, by studying <i>stochastic infinite-state games</i>.</p><p>Game theory has been used in verification for a long time. We focus on finite-state 2-player parity and limit-average (mean payoff) games. These problems have applications in model checking for the <i>μ</i>-calculus, one of the most expressive logics for programs. We give a simplified proof of memoryless determinacy. The proof applies <i>both</i> to parity and limit-average games. Moreover, we suggest a strategy improvement algorithm for limit-average games. The algorithm is discrete and strongly subexponential.</p><p>We also consider probabilistic infinite-state systems (Markov chains) induced by three types of models. <i>Lossy channel systems (LCS)</i> have been used to model processes that communicate over an unreliable medium. <i>Petri nets</i> model systems with unboundedly many parallel processes. <i>Noisy Turing machines</i> can model computers where the memory may be corrupted in a stochastic manner. We introduce the notion of <i>eagerness</i> and prove that all these systems are eager. We give a scheme to approximate the value of a reward function defined on paths. Eagerness allows us to prove that the scheme terminates. For probabilistic LCS, we also give an algorithm that approximates the limit-average reward. This quantity describes the long-run behavior of the system.</p><p>Finally, we investigate Büchi games on probabilistic LCS. Such games can be used to model a malicious cracker trying to break a network protocol. We give an algorithm to solve these games.</p>

Computer programs keep finding their ways into new safety-critical applications, while at the same time growing more complex. This calls for new and better methods to verify the correctness of software. We focus on one approach to verifying systems, namely that of model checking. At first, we investigate two categories of problems related to model checking: games and stochastic infinite-state systems. In the end, we join these two lines of research, by studying stochastic infinite-state games. Game theory has been used in verification for a long time. We focus on finite-state 2-player parity and limit-average (mean payoff) games. These problems have applications in model checking for the μ-calculus, one of the most expressive logics for programs. We give a simplified proof of memoryless determinacy. The proof applies both to parity and limit-average games. Moreover, we suggest a strategy improvement algorithm for limit-average games. The algorithm is discrete and strongly subexponential. We also consider probabilistic infinite-state systems (Markov chains) induced by three types of models. Lossy channel systems (LCS) have been used to model processes that communicate over an unreliable medium. Petri nets model systems with unboundedly many parallel processes. Noisy Turing machines can model computers where the memory may be corrupted in a stochastic manner. We introduce the notion of eagerness and prove that all these systems are eager. We give a scheme to approximate the value of a reward function defined on paths. Eagerness allows us to prove that the scheme terminates. For probabilistic LCS, we also give an algorithm that approximates the limit-average reward. This quantity describes the long-run behavior of the system. Finally, we investigate Büchi games on probabilistic LCS. Such games can be used to model a malicious cracker trying to break a network protocol. We give an algorithm to solve these games.

We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems. / text

最近，在考慮一個小型開放經濟體系下，貿易條件(Terms of Trade)外生變化對其經 常帳(Current Account) 影響的相關文獻上，以兩期模型(Infinite-horizon Modcl) 處理時，若貿易條件恆常地惡化，則一方面由於實質所得減少造成儲蓄降低，因而使 得經常帳惡化 (此即所謂「財富效果」) ；另一方面則經由實質利率改變影響儲蓄與 投資決策，因此經常帳再度隨之調整 (此稱為「實質利率效果」) 。至於貿易條件惡 化對於經常帳究有改善或惡化的影響則端視各模型處理時假設不同有不同的結論。 在Heckscher-Ohlin 生產技術的假設下，兩要素用在兩部門間生產，會使得貿易條件 透過第三管道影響經常帳。由Stolper-Samuelson 定理可知，貿易條件惡化將會降低 出口財較密集使用要素的報酬，提高進口財較密集使用要素的報酬，在所得重新分配 之後儲蓄決策將有所改變，因而經常帳也跟著受影響。此稱之為 Stolper-Samuelson 效果。 本文擬運用Blanchard 式的跨代模型(Overlapping-generations Mode)考慮一個小型 開放經濟體系在面臨外生貿易條件惡化時，其經常帳與資本累積的動態變化。此模型 假設每個經濟個體(agent) 活有限期，因此長期均衡值不必滿足時間偏好率等於利率 的條件，我們在此考慮投資與儲蓄的動態決策行為。又由於假設Heckscher-Ohlin 生 產技術，本文也將著重於討論Stolper-Samuelson 效果在此模型中的影響。

Performing inference on contemporary samples of homologous DNA sequence data is an important task. By assuming a stochastic model for ancestry, one can make full use of observed data by sampling from the distribution of genealogies conditional upon the sample configuration. A natural such model is Kingman's coalescent, with numerous extensions to account for additional biological phenomena. However, in this model the distribution of interest cannot be written down analytically, and so one solution is to utilize importance sampling. In this context, importance sampling (IS) simulates genealogies from an artificial proposal distribution, and corrects for this by weighting each resulting genealogy. In this thesis I investigate in detail approaches for developing efficient proposal distributions on coalescent histories, with a particular focus on a two-locus model mutating under the infinite-sites assumption and in which the loci are separated by a region of recombination. This model was originally studied by Griffiths (1981), and is a useful simplification for considering the correlated ancestries of two linked loci. I show that my proposal distribution generally outperforms an existing IS method which could be recruited to this model. Given today's sequencing technologies it is not difficult to find volumes of data for which even the most efficient proposal distributions might struggle. I therefore appropriate resampling mechanisms from the theory of sequential Monte Carlo in order to effect substantial improvements in IS applications. In particular, I propose a new resampling scheme and confirm that it ensures a significant gain in the accuracy of likelihood estimates. It outperforms an existing scheme which can actually diminish the quality of an IS simulation unless it is applied to coalescent models with care. Finally, I apply the methods developed here to an example dataset, and discuss a new measure for the way in which two gene trees are correlated.

Soit P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré n et M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$ D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions. / Let P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ a polynomial of degree n and M:=\sup_{|z|=1}|P(z)|$. Without any additional restriction, we know that $|P '(z) | \leq Mn$ for $| z | \leq 1$ (Bernstein's inequality). Now if we assume that the zeros of the polynomial $P$ are outside the circle $| z | = k$, which improvement could be made to the Bernstein inequality? It is already known [{\bf \ref{Mal1}}] that in the case where $k \geq 1$, one has$$ (*) \qquad | P '(z) | \leq \frac{n}{1 + k} M \qquad (| z | \leq 1),$$ what would it be in the case where $k < 1$? What is the analogous inequality for an entire function of exponential type $\tau$? On the other hand, if we assume that $P$ has all its zeros in $| z | \geq k \, \, (k \geq 1),$ which is the estimate of $| P '(z) |$ on the unit circle, in terms of the first four terms of its Maclaurin series expansion. This thesis comprises a contribution to the analytic theory of polynomials in the light of these problems.