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Successful methods for a large class of nonlinear convex optimization problems have recently been developed. This class, known as self-scaled optimization problems, has been defined by Nesterov and Todd in 1994. As noticed by Guler in 1996, this class is best described using an algebraic structure known as Euclidean Jordan algebra, which provides an elegant and powerful unifying framework for its study. Euclidean Jordan algebras are now a popular setting for the analysis of algorithms designed for self-scaled optimization problems : dozens of research papers in optimization deal explicitely with them. This thesis proposes an extensive and self-contained description of Euclidean Jordan algebras, and shows how they can be used to design and analyze new algorithms for self-scaled optimization. Our work focuses on the so-called spectral functions on Euclidean Jordan algebras, a natural generalization of spectral functions of symmetric matrices. Based on an original variational analysis technique for Euclidean Jordan algebras, we discuss their most important properties, such as differentiability and convexity. We show how these results can be applied in order to extend several algorithms existing for linear or second-order programming to the general class of self-scaled problems. In particular, our methods allowed us to extend to some nonlinear convex problems the powerful smoothing techniques of Nesterov, and to demonstrate its excellent theoretical and practical efficiency.

In this study two nonlinear analysis methods are proposed for investigation of nonlinear effects of high temperature superconductive(HTSC) thin film planar microwave circuits. The MoM-HB combination method is based on the combination formulation of the moment method(MoM) and the harmonic balance(HB) technique. It consists of linear and nonlinear solvers. The power series method treats the voltages at higher order frequencies as the excitations at the corresponding frequencies, and the higher order current distributions are then obtained by using the moment method again. The power series method is simple and fast for finding the output power at higher order frequencies. The MoM-HB combination method is suitable for strong nonlinearity, and it can be also used to find the fundamental current redistribution, conductor loss, and the scattering parameters variation at the fundamental frequency. These two proposed methods are efficient, accurate, and suitable for distributed-type HTSC nonlinearity. They can be easily incorporated into commercial EM CAD softwares to expand their capabilities. These two nonlinear analysis method are validated by analyzing a HTSC stripline filter and HTSC antenna dipole circuits. HTSC microstrip lines are then investigated for the nonlinear effects of HTSC material on the current density distribution over the cross section and the conductor loss as a function of the applied power. The HTSC microstrip patch filters are then studied to show that the HTSCinterconnecting line could dominate the behaviors of the circuits at high power. The variation of the transmission and reflection coefficients with the applied power and the third output power are calculated. The HTSC microstrip line structure with gilded edges is proposed for improving the power handling capability of HTSC thin film circuit based on a specified limit of harmonic generation and conductor loss. A general analysis approach suitable for any thickness of gilding layer is developed by integrating the multi-port network theory into aforementioned proposed nonlinear analysis methods. The conductor loss and harmonic generation of the gilded HTSC microstrip line are investigated.

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.

In this study two nonlinear analysis methods are proposed for investigation of nonlinear effects of high temperature superconductive(HTSC) thin film planar microwave circuits. The MoM-HB combination method is based on the combination formulation of the moment method(MoM) and the harmonic balance(HB) technique. It consists of linear and nonlinear solvers. The power series method treats the voltages at higher order frequencies as the excitations at the corresponding frequencies, and the higher order current distributions are then obtained by using the moment method again. The power series method is simple and fast for finding the output power at higher order frequencies. The MoM-HB combination method is suitable for strong nonlinearity, and it can be also used to find the fundamental current redistribution, conductor loss, and the scattering parameters variation at the fundamental frequency. These two proposed methods are efficient, accurate, and suitable for distributed-type HTSC nonlinearity. They can be easily incorporated into commercial EM CAD softwares to expand their capabilities. These two nonlinear analysis method are validated by analyzing a HTSC stripline filter and HTSC antenna dipole circuits. HTSC microstrip lines are then investigated for the nonlinear effects of HTSC material on the current density distribution over the cross section and the conductor loss as a function of the applied power. The HTSC microstrip patch filters are then studied to show that the HTSCinterconnecting line could dominate the behaviors of the circuits at high power. The variation of the transmission and reflection coefficients with the applied power and the third output power are calculated. The HTSC microstrip line structure with gilded edges is proposed for improving the power handling capability of HTSC thin film circuit based on a specified limit of harmonic generation and conductor loss. A general analysis approach suitable for any thickness of gilding layer is developed by integrating the multi-port network theory into aforementioned proposed nonlinear analysis methods. The conductor loss and harmonic generation of the gilded HTSC microstrip line are investigated.

We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.

Thesis (Ph.D.); Submitted to New Mexico State Univ., Las Cruces, NM (US); 1 May 2003. / Published through the Information Bridge: DOE Scientific and Technical Information. "JLAB-PHY-03-22" "DOE/ER/40150-2731" David McKee. 05/01/2003. Report is also available in paper and microfiche from NTIS.

Cette thèse traite de la théorie spectrale des opérateurs de Schrödinger discrets H(λ) := - Δ + b sur Zd et plus généralement sur des graphes pondérés infinis. Plus précisément, nous étudions le comportement des fonctions spectrales qui représentent les bornes du spectre de ces opérateurs. Un des principaux résultats est l'obtention d'une condition nécessaire et suffisante sur le potentiel b pour que le bas du spectre soit strictement positif. L'étude du haut du spectre est également considérée.Nous étudions tout d'abord ces questions pour les opérateurs de Schrödinger discrets sur Zd. La régularité de cet espace permet alors d'obtenir des résultats spécifiques dans ce cas particulier. Nous généralisons ensuite nos travaux au cas des graphes infinis pondérés. Les techniques développées dans ce cadre nous permettent également d'étudier le comportement asymptotique du bas du spectre pour les grandes valeurs de λ. / This thesis deals with the spectral theory of discrete Schrödinger operators H(λ) := - Δ + b on Zd and more generally on in#nite weighted graphs. Precisely, we study the behavior of the spectral functions which represent the spectral bounds of these operators. One of the main results is the obtention of a necessary and sufficient condition on the potential b such that the bottom of the spectrum is stricly positive.The study of the top of the spectrum is also treated.We first study these questions for discrete Schrödinger operators on Zd. The regularity of this space provides specific results in this particular case. Then we extend our work to the case of infinite weighted graphs. Moreover, the technics developed in this framework allow us to study the asymptotic behavior of the bottom of the spectrum for large values of λ.

Thesis (Ph.D.); Submitted to Massachusetts Inst. of Tech., Cambridge, MA (US); 1 Sep 2003. / Published through the Information Bridge: DOE Scientific and Technical Information. "JLAB-PHY-03-167" "DOE/ER/40150-2745" Marat Rvachev. 09/01/2003. Report is also available in paper and microfiche from NTIS.

Le plasma de quarks et de gluons (QGP) est un état de la matière observe lors de la collision d'ions lourds dans les accélérateurs tels que le LHC. Il est présent à haute température et/ou à haute densité, les quarks sont alors déconfinés : libres de se mouvoir et interagissant très peu entre eux. A basse température et basse densité, les quarks sont, au contraire, confines dans les hadrons formant la matière hadronique ordinaire. La présence d'une transition entre cette phase hadronique et le QGP a des conséquences importantes que ce soit 'a haute température (expériences RHIC et LHC) ou a haute densité (expérience CBM à FAIR, étude des étoiles compactes). Une première transition de phase est liée à la brisure de la symétrie chirale. Dans la matière hadronique, cette symétrie est brisée spontanément. Elle est restaurée en augmentant la température ou la densité. Au delà de la discussion habituelle sur la transition chirale, nous utiliserons un modèle, le modèle Polyakov Nambu Jona-Lasinio permettant de décrire une deuxième transition : la transition de deconfinement. Ceci permettra de séparer le diagramme Temperature-Densité en trois phases distinctes : la phase hadronique ou les quarks sont confines et o'u la symétrie chirale est brisée, la phase du QGP ou les quarks sont d'confines et o'u la symétrie chirale est restaurée et une phase hypothétique dite quarkyonique à basse température et haute densité ou les quarks sont encore confines mais ou la symétrie chirale est restaurée. On décrira, dans un premier temps les différentes transitions à l'aide des paramètres d'ordre suivant : le condensat de quark pour la transition chirale et la boucle de Polyakov pour le déconfinement. On verra ensuite comment l'évolution des fonctions spectrales des mésons sigma et pi peut nous renseigner sur le diagramme de phase. Le critère de transition chirale sera alors la différence entre les masses de ces mésons, la masse étant prise comme étant le maximum de la fonction spectrale. Le critère de transition de deconfinement sera, quant à lui, l'écart-type de la fonction spectrale. Enfin, nous verrons comment intégrer les mésons vecteurs au modèle, en particulier le méson rho, qui pourra jouer le rôle de sonde du plasma, ses propriétés étant modifiées suivant le milieu dans lequel il est émis / The quark and gluon plasma (QGP) is a state of matter observed in the collision of heavy ions in accelerators such as the LHC. It is formed at high temperature and / or high density, quarks are then deconfined : free to move and interacting very little with each other. At low temperature and low density, the quarks are, however, confined within hadrons forming the ordinary hadronic matter. The presence of the phase transition between hadronic matter and the QGP has observable consequences whatsoever at high temperature (RHIC and LHC experiments) or high density (FAIR experience, study of compact stars). A first phase transition is linked to the chiral symmetry breaking. In hadronic matter, this symmetry is spontaneously broken. It is restored by increasing the temperature or the density. Beyond the usual discussion on the chiral transition, we use a model called Polyakov Nambu Jona-Lasinio for describing a second transition, the deconfinement transition. This allows to separate the temperature-density diagram in three distinct phases : the hadronic phase where quarks are confined and where chiral symmetry is broken, the phase of the QGP where quarks are deconfined and chiral symmetry is restored and a hypothetical phase called quarkyonic at low temperature and high density in which quarks are confined but where chiral symmetry is still restored. We will describe, at first, the various transitions using the following order parameters : the quark condensate for the chiral transition and the Polyakov loop for the deconfinement one. Then we will see how the evolution of the spectral functions of sigma and pi mesons can provide information on the phase diagram. The chiral transition criterion will be the difference between the masses of these mesons, the mass being taken as the maximum of the spectral function. And the criterion for the deconfinement transition will be the standard deviation (also called variance) of the spectral function. Finally, we discuss how the vector mesons fit in the model, especially the meson, which can act as a probe of plasma properties which are modified by the environment from which it is issued