Cette thèse est constituée de deux volets principaux, le premier volet est consacré à l'étude des opérateurs de Hankel de symboles antiméromorphes, plus précisément, on s'intéresse à la possibilité d'obtenir des opérateurs de Hankel bornés (resp. compacts, dans les classes de Schatten) dont les symboles ne sont pas nécessairement des polynômes.Nous allons donner dans un premier temps, des conditions nécessaires et suffisantes pour que l'opérateur H_{f} définit sur une couronne dans le plan complexe, soit borné (resp. compact, dans la p-ième classe de Schatten) et nous allons traiter des exemples sur les quels nous montrons que les opérateurs de Hankel H_{f} et H_{Uf} sont bornés simultanément (resp. compacts, dans les classes de Schatten) si et seulement si f est un polynôme de Laurent et les conditions établies portent sur son L-degré.Le deuxième volet traite les propriétés spectrales en commun des opérateurs A et B vérifiant l'équation A²=ABA et B²=BAB. Nous allons généraliser les résultats de Christopher Schmoeger sur l'égalité des différents spectres de ces opérateurs, ensuite nous allons élargir le champ d'étude de ces opérateurs dans la direction de la théorie spectrale locale (Propriété de l'extension unique, décomposabilité...). / This thesis consists of two main parts, the first part is devoted to the study of Hankel operators of antiméromorphes symbols, more precisely, we are interested in the possibility of obtaining Hankel operators bounded (resp. compact, in Schatten classes) which the symbols are not necessarily polynomials.We will give in first step, the necessary and sufficient conditions for the operator H_ {f} defined on a ring in the complex plane is bounded (resp. compact in the p-th Schatten class) and we treat examples on which we show that the Hankel operators H_ {f} and H_ {Uf} are simultaneously bounded (resp. compact, in the Schatten classes) if and only if f is a Laurent polynomial and conditions set relate to its L-degree.The second part deals with common spectral properties of operators A and B satisfying the equation A ² = ABA and B ² = BAB. We will generalize the results of Christopher Schmoeger on equality different spectra of these operators, then we will expand the field of study of these operators in the direction of the local spectral theory (SVEP, Decomposability).

Nous nous intéressons aux opérateurs de Hankel $H_{bar{f}}$ de symbole anti holomorphe $bar{f}$ et regardons l'espace de Dixmier $mathcal{D}^{p}$ associé ($pgeq1$), c'est à dire l'ensemble des $f$ tel que $|H_{bar{f}}|^{p}$ soit dans l'idéal de Macaev $mathcal{S}^{+}_{1}$. Notre approche est de voir l'espace de Dixmier comme une certaine limite des classes de Schatten. Quand $f in mathcal{D}^{p}$, nous étudions $Tr_{omega}(|$H_{bar{f}}$|^{p})$ la trace de Dixmier de $|H_{bar{f}}|^{p}$. Nous redémontrons certains résultats classiques quand $f$ est holomorphe sur le disque alors que nous donnons de nouveaux résultats quand $f$ est entière. Nous utilisons notre méthode pour étudier l'espace de Dixmier du petit opérateur de Hankel, des opérateurs de Toeplitz $T_{varphi}$ ($varphi$ définie sur le disque ou sur le plan complexe tout entier) ainsi que pour l'opérateur de composition. / We study Hankel operators $H_{bar{f}}$ with anti holomorphic symbol $bar{f}$ and we are interested to the Dixmier space $mathcal{D}^{p}$ ($pgeq1$), the set of functions $f$ such that $|H_{bar{f}}|^{p} in mathcal{S}^{+}_{1}$ the Macaev ideal. We look Dixmier space as a limit of Schatten class. When $f in mathcal{D}^{p}$, we study $Tr_{omega}(|$H_{bar{f}}$|^{p})$ the Dixmier trace of $|H_{bar{f}}|^{p}$. We have different results when $f$ is an entire or a holomorphic function of the unit disk in the complex plan. We study also the Dixmier space of the little Hankel operator, Toeplitz operator and composition operator.

Indiana University-Purdue University Indianapolis (IUPUI) / In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.

L'étude du contrôle optique de composants hyperfréquences consiste en la compréhension et la simulation des phénomènes physiques mis en jeu, à savoir, l'injection optique et la propagation d'une onde hyperfréquence au sein de la charge photo-induite. L'objectif est d'obtenir une simulation complète. Ainsi la partie optique est résolue par un calcul semi-analytique tridimensionnel de l'injection optique pour obtenir la concentration de porteurs excédentaires dans le substrat. Ce calcul semi-analytique permet de plus la réalisation d'une étude de sensibilité par rapport aux paramètres physiques et géométriques du substrat et du faisceau laser. Dans un deuxième temps, cette charge photo-induite, déduite de la concentration, est modélisée sous un logiciel électromagnétique pour obtenir le comportement fréquentiel du composant contrôlé optiquement. Toutes ces simulations sont enfin comparées aux mesures pratiques réalisées sur des composants simples en technologie coplanaire.

In this habilitation thesis a factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip is established. These operators are considered with matrix-valued symbols and are thought of acting on the vector-valued analogues of the Hardy and Lebesgue spaces. A factorization theory for pure Toeplitz operators and singular integral operators without flip is known since decades and provides necessary and sufficient conditions for Fredholmness and formulas for the defect numbers. In particular, the invertibility of such operators is equivalent to the existence of a certain type of Wiener-Hopf factorization. In this thesis an analogous theory for the afore-mentioned more general classes of operators is developed. It turns out that a completely different kind of factorization is needed. This kind of factorization is studied extensively, and a corresponding Fredholm theory is established. A connection with the Hunt-Muckenhoupt-Wheeden condition is made, and several examples and applications are given as well. / In dieser Habilitationsschrift wird eine Faktorisierungstheorie für Toeplitz plus Hankel-Operatoren und singuläre Integraloperatoren mit Flip aufgestellt. Diese Operatoren werden mit matrixwertigem Symbol betrachtet und sind auf den vektorwertigen Analoga der Hardy- und Lebesgue-Räumen definiert. Eine Faktorisierungstheorie für reine Toeplitz bzw. singuläre Integraloperatoren ohne Flip ist seit Jahrzehnten bekannt. Sie liefert notwendige und hinreichende Bedingungen für die Fredholmeigenschaft und Formeln für die Defektzahlen. Insbesondere ist die Invertierbarkeit derartiger Operatoren äquivalent zur Existenz einer bestimmten Art der Wiener-Hopf-Faktorisierung. In dieser Habilitationsschrift wird eine entsprechende Theorie für die erwähnten, allgemeineren Klassen von Operatoren aufgestellt. Es stellt sich heraus, dass eine völlig andere Art der Faktorisierung benötigt wird. Diese Art der Faktorisierung wird eingehend studiert und eine entsprechende Fredholmtheorie wird entwickelt. Ein Zusammenhang mit der Hunt-Muckenhoupt-Wheeden Bedingung wird hergestellt. Mehrere Beispiele und Anwendungen werden ebenfalls angegeben.

Technology trends toward lower power, higher speed and higher density devices have pushed the package performance to its limits. The high frequency effects e.g., crosstalk and signal distortion, may cause high bit error rates or malfunctioning of the circuit. Therefore, the successful release of a new product requires constant attention to the high frequency effects through the whole design process. Full-wave electromagnetic tools must be used for this purpose. Unfortunately, currently available full-wave tools require excessive computational resources to simulate large-scale interconnect structures.A prototype version of the Full-Wave Layered-Interconnect Simulator (UA-FWLIS), which employs the Method of Moments (MoM) technique, was developed in response to this design need. Instead of using standard numerical integration techniques, the MoM reaction elements were analytically evaluated, thereby greatly improving the computational efficiency of the simulator. However, the expansion and testing functions that are employed in the prototype simulator involve filamentary functions across the wire. Thus, many problems cannot be handled correctly. Therefore, a fundamental extension is made in this dissertation to incorporate rectangular-based, finite-width expansion and testing functions into the simulator. The critical mathematical issues and theoretical issues that were met during the extension are straightened out. The breakthroughs that were accomplished in this dissertation lay the foundation for future extensions. A new bend-cell expansion function is also introduced, thus allowing the simulator to handle bends on the interconnects with fewer unknowns. In addition, the Incomplete Lipschitz-Hankel integrals, which are used in the analytical solution, are studied. Two new series expansions were also developed to improve the computational efficiency and accuracy.

Medicinoje duomenys dažniausiai yra analizuojami statistiniais metodais. Tačiau šiame darbe elektrokardiogramos (EKG) parametrus tirsime naudodamiesi algebriniais metodais, nes į kiekvieno žmogaus organizmą žvelgiant kaip į sudėtingą ir unikalią kompleksinę sistemą, statistinio tyrimo nepakanka. Egzistuoja specialus algoritmas, leidžiantis funkciją , aprašytą Teiloro eilute arba jos reikšmėmis, išreikšti baigtine eksponenčių suma. Šio darbo tikslas – baigtine eksponenčių suma aproksimuoti EKG parametrų funkcijos dalis, kurios turi baigtinį kompleksiškumą, bei tirti parametrų kompleksiškumo pokyčius ilgame EKG įraše. / Medical data are often analysed using statistical methods. However the aim of this work is to analyse the complexity of some electrocardiogram (ECG) parameters using algebraic mathematical methods, because if we consider a human organism a complex and individual system, the former methods are not adequate. Mainly, we analyse two parameters of ECG: the complexity of R wave amplitude and duration between R waves. The method of Hankel matrices, in which functions of the finite h-rank are written as linear combinations of exponential functions, is used.

Šiame darbe tiriamos naujo metodo, skirto funkcijų aproksimavimui baigtine eksponenčių suma galimybės, taikant šį metodą konkrečios diferencialinių lygčių sistemos, aprašančios mechaninius virpesius, sprendiniams. Viena iš galimų darbe pristatomos mechaninių virpesių sistemos taikymo sričių – jūros bangų arba vėjo sukeltus virpesius panaudoti kaip atsinaujinantį energijos šaltinį. Tokių mechanizmų veikimo principai prieš pradedant kurti realų veikiantį modelį analizuojami taikant matematinį modeliavimą. Sudėtingos lygčių sistemos sprendiniai, priklausomai nuo sprendimo metodo, gaunami laipsninių eilučių pavidale arba kaip taškų aibė, bet nei viena iš šių formų nėra patogi sprendinio kokybiniam tyrimui. Tačiau turint sprendinio išraišką eksponentinių funkcijų su kompleksiniais koeficientais suma, žinomi ir šį sprendinį sudarančių harmonikų dažniai – svarbi konkretaus virpesių sistemos režimo charakteristika. Atliekant skaitinius eksperimentus nustatyta, jog nusistovėjusį sistemos sprendinį galima įvertinti baigtine eksponenčių suma. Aproksimavimo paklaidos priklauso nuo žingsnio, aproksimuojamos funkcijos ir skaičiavimo paklaidos. / The aim of this work was to explore the possibilities of a new method, which gives an ability to approximate functions by a finite sum of exponential functions. This method was applied to the solutions of the concrete differential equations that describe the system of mechanical oscillations. One of the possible application areas of the system of oscillations presented in the paper is to use oscillations caused by the wind or water waves as a source of renewable energy. The action principles of such mechanisms are investigated using mathematical simulation before the real working model. The solutions of the sophisticated system of differential equations are obtained either in the form of power series or a set of points, depending of the solving method chosen. However, none of these forms is convenient for exploring properties of the solution. Therefore, we have a problem to approximate the solutions with linear formations of exponential functions. It is possible then to express the solutions as the linear formations of harmonics. It is demonstrated that a steady solution of the system can be expressed as a finite sum of exponential functions. Approximation errors vary depending on the distance between the points used, the function, which is being approximated, and the computation errors.

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