Etude algorithmique du problème algébrique d'estimation de paramètres pour une classe de perturbations

We consider the algebraic parameter estimation problem for a class of standard perturbations. We assume that the measurement z(t) of a solution x(t) of a linear ordinary differential equation -- whose coefficients depend on a set \theta := {\theta_1, ..., \theta_r} of unknown constant parameters -- is affected by a perturbation \gamma(t) whose structure is supposed to be known (e.g., an unknown bias, an unknown ramp), i.e., z(t)=x(t, \theta)+\gamma(t). We investigate the problem of obtaining closed-form expressions for the parameters \theta_i's in terms of iterative indefinite integrals or convolutions of z. The different results are illustrated by explicit examples computed using the NonA package -- developed in Maple -- in which we have implemented our main contributions.Nous considérons le problème algébrique d'estimation de paramètres pour une classe classique de perturbations. Nous supposons que la mesure z(t) d'une solution x(t) d'une équation différentielle linéaire -- dont les coefficients dépendent d'un ensemble \theta := {\theta_1, ..., \theta_r} de paramètres constants inconnus -- est affectée par une perturbation \gamma(t) dont la structure est supposée connue (par exemple, un biais inconnu, une rampe inconnue), c'est-à-dire, nous supposons que z(t)=x(t, \theta)+\gamma(t). Nous étudions alors le problème d'obtenir des formes closes pour les paramètres \theta_i en fonction d'intégrales indéfinies itérées ou de convolutions de z. Les différents résultats sont illustrés par des exemples explicites calculés grâce au package NonA -- développé en Maple -- dans lequel nous avons implanté nos principales contributions

Computational Approaches to Problems in Noncommutative Algebra -- Theory, Applications and Implementations

Noncommutative rings appear in several areas of mathematics. Most prominently, they can be used to model operator equations, such as differential or difference equations. In the Ph.D. studies leading to this thesis, the focus was mainly on two areas: Factorization in certain noncommutative domains and matrix normal forms over noncommutative principal ideal domains. Regarding the area of factorization, we initialize in this thesis a classification of noncommutative domains with respect to the factorization properties of their elements. Such a classification is well established in the area of commutative integral domains. Specifically, we define conditions to identify so-called finite factorization domains, and discover that the ubiquitous G-algebras are finite factorization domains. We furthermore realize a practical factorization algorithm applicable to G-algebras, with minor assumptions on the underlying field. Since the generality of our algorithm comes with the price of performance, we also study how it can be optimized for specific domains. Moreover, all of these factorization algorithms are implemented. However, it turns out that factorization is difficult for many types of noncommutative rings. This observation leads to the adjunct examination of noncommutative rings in the context of cryptography. In particular, we develop a Diffie-Hellman-like key exchange protocol based on certain noncommutative rings. Regarding the matrix normal forms, we present a polynomial-time algorithm of Las Vegas type to compute the Jacobson normal form of matrices over specific domains. We will study the flexibility, as well as the limitations of our proposal. Another core contribution of this thesis consists of various implementations to assist future researchers working with noncommutative algebras. Detailed reports on all these programs and software-libraries are provided. We furthermore develop a benchmarking tool called SDEval, tailored to the needs of the computer algebra community. A description of this tool is also included in this thesis

Unraveling hadron structure with generalized parton distributions

The generalized parton distributions, introduced nearly a decade ago, have emerged as a universal tool to describe hadrons in terms of quark and gluonic degrees of freedom. They combine the features of form factors, parton densities and distribution amplitudes--the functions used for a long time in studies of hadronic structure. Generalized parton distributions are analogous to the phase-space Wigner quasi-probability function of non-relativistic quantum mechanics which encodes full information on a quantum-mechanical system. We give an extensive review of main achievements in the development of this formalism. We discuss physical interpretation and basic properties of generalized parton distributions, their modeling and QCD evolution in the leading and next-to-leading orders. We describe how these functions enter a wide class of exclusive reactions, such as electro- and photo-production of photons, lepton pairs, or mesons. The theory of these processes requires and implies full control over diverse corrections and thus we outline the progress in handling higher-order and higher-twist effects. We catalogue corresponding results and present diverse techniques for their derivations. Subsequently, we address observables that are sensitive to different characteristics of the nucleon structure in terms of generalized parton distributions. The ultimate goal of the GPD approach is to provide a three-dimensional spatial picture of the nucleon, direct measurement of the quark orbital angular momentum, and various inter- and multi-parton correlations.Comment: 370 pages, 62 figures; Dedicated to Anatoly V. Efremov on occasion of his 70th anniversar

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

The nonperturbative functional renormalization group and its applications

The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson's RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity.Comment: v2) Review article, 93 pages + bibliography, 35 figure

Quantum Field Theoretical Green Functions and Electronic Structure

Die vorliegende Arbeit stellt eine rein theoretische Untersuchung dar, deren Hauptziel darin besteht, die Theorie der elektronischen Struktur vom Standpunkt der Quantenfeld- und Vielteilchentheorie aus zu entwickeln. Dabei habe ich versucht, alle Konzepte und Definitionen im quantenfeldtheoretischen Rahmen einzuf\"{u}hren.Die Einteilchenn\"{a}herung hingegen wird nur als Spezialfall behandelt, spielt jedoch keine konzeptionelle Rolle. Ein weiteres Ziel dieser Arbeit besteht darin, Analogien zwischen der ``Electronic Structure Theory'' und anderen Gebieten der Physik zu erhellen. Das konkrete Thema dieser Dissertation ist die Anwendung von Greenschen Funktionen auf die Quantentheorie der Materialeigenschaften des Festk\"{o}rpers, insbesondere der elektronischen Struktur. Die Arbeit ist in zwei Teile gegliedert: (i) eine Zusammenfassung und Diskussion des allgemeinen vielteilchen- und quantenfeldtheoretischen Rahmens und (ii) die Theorie der elektronischen Green-Funktionen.he present thesis is a purely theoretical study, the main goal of which is the theoretical development of Electronic Structure Theory from the quantum field theoretical and many-body theory point of view. I have tried to introduce all concepts and definitions in the quantum field theoretical framework. The independent electron approximation is always treated as a special case of the general many-body theory, but does not play any conceptual role. Another goal of this thesis has been the elucidation of various analogies between Electronic Structure Theory and other branches of physics. The concrete topic of this thesis is the application of Green function methods to the quantum theory of materials properties in the solid state, in particular to Electronic Structure Theory. The thesis is divided into two parts, (i) a review and discussion of some aspects of the general Many-Body Theory and non-relativistic Quantum Field Theory background and (ii) electronic Green functions methods