On étudie dans cette thèse l'équation de Szegö sur la droite réelle ainsi que ses perturbations. Cette équation a été introduite il y a quelques années par Gérard et Grellier comme modèle mathématique d'une équation non linéaire totalement non dispersive.L'équation de Szegöapparait naturellement dans l'étude de l'équation de Schrödinger non linéaire (NLS) danscertaines situations sur-critiques où l'on constate un manque de dispersion, par exemplelorsque l'on considère NLS sur le groupe de Heisenberg. Par conséquent, une des motivationsde cette thèse est d'établir des résultats concernant l'équation de Szegö qui pourrontéventuellement être utilisés dans le contexte de l'équation de Schrödinger non linéaire.Le premier résultat de cette thèse est la classification des solitons de l'équation de Szegö.On montre que ce sont tous des fonctions rationnelles ayant un unique pôle qui est simple.De plus, on prouve que les solitons sont orbitalement stables.La propriété la plus remarquable de l'équation de Szegö est le fait qu'elle est complètement intégrable, ce qui permet notamment d'établir une formule explicite de sa solution.Comme applications de cette formule, on obtient les trois résultats suivants. (A) On montreque les solutions fonctions rationnelles génériques se décomposent en une somme de solitonset d'un reste qui est petit lorsque le temps tend vers l'infini. (B) On met en évidence unexemple de solution non générique dont les grandes normes de Sobolev tendent vers l'infiniavec le temps. (C) On détermine des coordonnées action-angle généralisées lorsque l'on restreintl'équation de Szegö à une sous-variété de dimension finie. En particulier, on en déduitqu'une grande partie des trajectoires de cette équation sont des spirales autour de cylindrestoroïdaux.Comme l'équation de Szegö est complètement intégrable, il est ensuite naturel d'étudierses perturbations et d'établir de nouvelles propriétés pour celles-ci à partir des résultatsconnus pour l'équation de Szegö. Une des perturbations de l'équation de Szegö est une équation desondes non linéaire (NLW) de donnée bien préparée.On prouve que si la donnée initiale de NLW est petite et à support dans l'ensemble desfréquences positives, la solution de NLW est alors approximée pour un temps long par lasolution de l'équation de Szegö. Autrement dit, on démontre ainsi que l'équation de Szegöest la première approximation de NLW. On construit ensuite une solution de NLW dont lesgrandes normes de Sobolev augmentent (relativement à la norme de la donnée initiale).Sur le tore T, Gérard et Grellier ont démontré un résultat analogue d'approximation deNLW. On améliore ce résultat en trouvant une approximation plus fine, de deuxième ordre.Dans une dernière partie, on s'intéresse à l'équation de Szegö perturbée par un potentielmultiplicatif petit. On étudie l'interaction de ce potentiel avec les solitons. Plus précisément,on montre que, si la donnée initiale est celle d'un soliton pour l'équation non perturbée, lasolution de l'équation perturbée garde la forme d'un soliton sur un long temps. De plus, ondéduit la dynamique effective, i.e. les équations différentielles satisfaites par les paramètresdu soliton.

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejérinterpolation problem on the polydisc$\D^n. $ in the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. We discuss an alternative approach to the Carathéodory-Fejérinterpolation problem, in the special case of $n=2$, adapting a theorem of Korányi and Pukánzsky. As a consequence, a class of polynomials are isolated for which a complete solution to the Carathéodory-Fejér interpolation problem is easily obtained. Many of our results remain valid for any $n\in \mathbb N$, however the computations are somewhat cumbersome. Recall the well known inequality due to Varopoulos, namely, $\lim{n\to \infty}C_2(n)\leq 2 K^\C_G$, where $K^\C_G$ is the complex Grothendieck constant and \[C_2(n)=sup\{\|p(\boldsymbolT)\|:\|p\|_{\D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1\}.\] Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples$\boldsymbolT:=(T_1,\ldots,T_n)$ of contractions. We show that \[\lim_{n\to \infty} C_2 (n)\leq \frac{3\sqrt{3}}{4} K^\C_G\] obtaining a slight improvement in the inequality of Varopoulos. We also discuss several finite and infinite dimensional operator space structures on $\ell^1(n) $, $n>1. $

This work is concerned with the applicability of several approximation methods (finite section method, Galerkin and collocation methods with maximum defect splines for uniform and non uniform meshes) to operators belonging to the closed subalgebra generated by operators of multiplication bz piecewise continuous functions and convolution operators also with piecewise continuous generating function.

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.:1 Introduction 2 Balanced truncation for LTV systems 3 Solving differential Lyapunov equations 4 Solving the reduced-order system 5 Numerical experiments 6 Conclusion

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For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

Dans cette thèse nous nous intéressons aux opérateurs de composition sur les espaces de Hardy et Dirichlet et aux opérateurs de Hankel sur les espaces des fonction polyanalytiques. On s’'intéresse à l’'opérateur de composition sur les espaces de Dirichlet : $mathcal{D}_alpha=\left{f \in Hol(D): |f|_alpha^{2}=| f(0)| ^{2}+int_{D}| f'(z)| ^{2}dA_alpha(z)<infty \right}.$ La fonction de comptage généralisée de Nevanlinna associée à l'espace de Dirichlet $\mathcal{D}_\alpha$ est donnée par:$$ N_{\varphi,\alpha}(z):=\sum_{z=\varphi(w),{w\in\D}}(1-|w| )^\alpha,\qquad z\in\D.$$Nous étudions dans la première partie de ce travail la relation entre la fonction de comptage généralisée de Nevanlinna associée à $\varphi$ et la norme de ses ses puissances sur les espaces de Dirichlet. Nous aussi des examples d’'opérateurs de composition de Hilbert-Schmidt sur les espaces de Dirichlet. Nous étudions aussi l’'appartenance de $C_\varphi$ à la classe de Schatten en termes de la taille de l’ensemble de niveau et la norme de $\varphi^n$. Dans la deuxième partie nous considérons l’'espace de Fock-Bargmann des fonctions polyanalytiques, $f in F^n(mathbb{C})$. Nous montrons que si $f (z) = z^k\overline{z}^l$ avec $k, l \in \mathbb{N},$, alors l’'opérateur de Hankel $ H_{f}$ est borné sur $F^n(\mathbb{C})$ si et seulement si $\sup_{m,j}\|H_{f}e_{j, m}\|_{F^n(\mathbb{C})} < +\infty$.On montre aussi que si $f$ une fonction entière sur $\mathbb{C}$, alors l’'opérateur de Hankel $ H_{\bar f}$ est borné sur $F_n(C)$ si et seulement si f est un polynôme de degré au plus 1, et l’'opérateur de Hankel $ H_{\bar f}$ est compact sur $F_n(C)$ si et seulement si f est un polynôme constant. / In this thesis we focus on the composition operators on Hardy and Dirichlet spaces and Hankel operators on spaces of polyanalytiques functions. We are interested in the composition operator on the Dirichlet spaces: $$ mathcal{D}_alpha=left{ f in Hol(D): |f|_alpha^{2}=| f(0)|^{2}+int_{D}| f'(z)| ^{2}dA_alpha(z)<infty \right}. $$ The generalized Nevanlinna counting function associated to $ mathcal{D}_alpha $, is given by: $ N_{varphi,alpha}(z)=sum_{z=phi(w),{winD}}(1-|w| )^alpha,qquad zinDsetminus{phi(0)} .$ We study in the first part of this work the relationship between the generalized Nevanlinna counting function associated with $varphi$ and the norms of its iterated in the Dirichlet spaces. We give examples of Hilbert-Schmidt composition operators on the Dirichlet spaces. We study the composition operators on the Dirichlet spaces belong to Schatten class and the link with the size of contact points of its symbol with the unit circle. In the second part we consider the Bargmann-Fock space of polyanalytic functions, $f in F^n(mathbb{C})$. We prove that if $f (z) = z^koverline{z}^l$ with $k, l in mathbb{N},$ then the Hankel operator $ H_{f}$ is bounded on $F^n(mathbb{C})$ if and only if $sup_{m,j}|H_{f}e_{j, m}|_{F^n(mathbb{C})} < +infty$. We also establish that if $f $ an entire function on $mathbb{C}$, then the Hankel operator $ H_{bar f}$ is bounded on $F^n(mathbb{C})$ if and only if $f$ is a polynomial of degree at most $1,$ and the Hankel operator $ H_{bar f}$ is compact on $F^n(mathbb{C})$ if and only if $f$ is a constant polynomial.

For noiseless sampled data, we describe the close connections between Prony--like methods, namely the classical Prony method, the matrix pencil method and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.

Contact mechanical study of layered structures is useful to various fields of engineering, such as - mechanical engineering, civil engineering, materials engineering and biomechanics. Thin hard film coating on a compliant substrate used in cutting tool industry is an example of a layered structure. The protective coating saves the substrate from fracture and wear. However, due to film material brittleness, fracture in the films is of concern. We have developed an analytical model for a film-substrate bilayer system under normal contact loading, which helps us to obtain the stress distribution in the film and fracture behaviour. Our contact model is based on Hankel’s Transform technique, where we assume a Hertzian pressure boundary condition. At each depth of penetration of the indenter in the film-substrate system, we estimate effective modulus of the system based on Gao’s approach. We have validated our analysis by surface strain measurements and photoelastic stress study in the film on a substrate. Experimental observations from literatures show the dependence of different fracture modes in a thin hard film with columnar structure on film thickness and substrate plasticity. We perform fracture analysis, a parametric study of the fracture modes in the film under contact loading. When the film thickness is small and the substrate is relatively hard (e.g. tool steel), the film and the substrate deform conformally under contact loading and the columns of TiN slide against each other into the substrate. On the other hand, when the film is thicker and the substrate is soft (e.g. mild steel or aluminium), the strain mismatch between the film and substrate acts as an added traction at the interface and drives cracks, such as radial tensile stress driven bending cracks that start from the interface at the center of indentation; maximum shear stress driven inclined shear crack that starts inside the film and propagate at an angle to the indentation axis and tensile stress driven edge crack that starts from the free surface outside the contact. We can draw a fracture map based on these calculations which provides a guide to select film thickness depending on the substrate hardness, so that the benign mode of damage, i.e., columnar shear occurs in the film. Apart from generating the fracture map, we can obtain rationale for different fracture phenomenon in the film by studying the indentation stress field. Principal tensile stresses, responsible for driving edge cracks from the free surface outside the contact, become compressive as one approaches the substrate if the substrate is compliant. The cracks therefore do not penetrate deep into the film rather curve away from the axis of indentation. At the transition zone from one mode of damage to other in the fracture map, different modes of fracture may co-exist. The whole column may not shear, rather the shear can start from somewhere in the middle of the film, where the shear stress is maximum and it can end without reaching the interface. The indentation energy is then dissipated in other forms of damage. The contact analysis is further applied to TiN /AlTiN multilayered films having similar elastic properties. Experimental observations suggest that with decreasing layer thickness the fracture resistance of the multilayers increase and some plastic yielding occurs at the top layers of the film. However no substantial change in strain capacity (Hardness/ Young’s Modulus) of the film is observed. Hence we attribute the increase of fracture resistance of multilayers to film plasticity and mimic it by reducing the modulus of the film. The analysis validates the propensity of edge cracking and transgranular cracking as they decrease with increasing number of layers in a multilayer. We next extend our bilayer analysis to a more general trilayer problem where the moduli of the layers vary by several orders. The test system here is a mica-glue-glass system which is used in surface force apparatus experiments. Gao’s trilayer analysis is used to fit the experimental data obtained from surface force apparatus experiments, where a glass sphere indents the trilayer. The parallel spring model used in Gao’s approximation is found to be inadequate to rationalize the experimental data. We have modified Gao’s formulations by reducing the problem to a bilayer problem where the layers are the first layer (in contact) and an equivalent layer which has properties determined by a rule of mixture of the properties of all the layers excluding the top layer set out as a set of springs in series. The modified formulations give a better fit to the experimental data and it is validated from nanoindentation experiments on the same system. The formulation is used to obtain the compression of the glue, which contributes significantly to the deformation of the trilayer system in the SFA experiments. Thus, the analysis can be used to deconvolute the influence of glue in the actual mechanical response of the system in an SFA experiment, which has so far been neglected.

Recognition of complex events in consumer uploaded Internet videos, captured under realworld settings, has emerged as a challenging area of research across both computer vision and multimedia community. In this dissertation, we present a systematic decomposition of complex events into hierarchical components and make an in-depth analysis of how existing research are being used to cater to various levels of this hierarchy and identify three key stages where we make novel contributions, keeping complex events in focus. These are listed as follows: (a) Extraction of novel semi-global features – firstly, we introduce a Lie-algebra based representation of dominant camera motion present while capturing videos and show how this can be used as a complementary feature for video analysis. Secondly, we propose compact clip level descriptors of a video based on covariance of appearance and motion features which we further use in a sparse coding framework to recognize realistic actions and gestures. (b) Construction of intermediate representations – We propose an efficient probabilistic representation from low-level features computed from videos, based on Maximum Likelihood Estimates which demonstrates state of the art performance in large scale visual concept detection, and finally, (c) Modeling temporal interactions between intermediate concepts – Using block Hankel matrices and harmonic analysis of slowly evolving Linear Dynamical Systems, we propose two new discriminative feature spaces for complex event recognition and demonstrate significantly improved recognition rates over previously proposed approaches.