Considérons une variété régulière analytique M sur le corps réel ou complexe, un faisceau d'idéaux J défini sur M, un diviseur à croisement normaux simples E et une distribution singulière involutive Θ tangent à E.L'objectif principal de ce travail est d'obtenir une résolution des singularités du faisceau d'idéaux J qui préserve certaines ``bonnes" propriétés de la distribution singulière Θ. Plus précisément, la propriété de R-monomialité : l'existence d'intégrales premières monomiales. Ce problème est naturel dans le contexte où on doit étudier l'interaction d'une variété et d'un feuilletage et, donc, est aussi reliée au problème de la monomilisation des applications et de résolution ``quasi-lisse" des familles d'idéaux.- Le premier résultat donne une résolution globale si le faisceau d'idéaux J est invariant par la distribution singulière;- Le deuxième résultat donne une résolution globale si la distribution singulière Θ est de dimension 1 ;- Le troisième résultat donne une uniformisation locale si la distribution singulière Θ est de dimension 2.On présente aussi deux utilisations des résultats précédents. La première application concerne la résolution des singularités en famille analytique, soit pour une famille d'idéaux, soit pour une famille de champs de vecteurs. Pour la deuxième, on applique les résultats à un problème de système dynamique, motivé par une question de Mattei. / Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and Θ an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some ``good" properties of the singular distribution Θ. More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the ``interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of ``quasi-smooth" resolution of families of ideal sheaves.- The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution Θ;- The second result is a global resolution if the the singular distribution Θ has leaf dimension 1;- The third result is a local uniformization if the the singular distribution Θ has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.

Nous présentons les outils fondamentaux du formalisme de la mécanique quantique non relativiste basée sur un principe dincertitude généralisé, impliquant lexistence dune longueur élémentaire. En considérant deux systèmes simples, à savoir le potentiel delta de Dirac à 1 dimension et le potentiel de Coulomb à 3 dimensions, nous illustrons comment on peut résoudre léquation de Schrödinger et extraire le spectre dénergie, analytiquement ou perturbativement, dans ce formalisme. Nous appliquons ce formalisme au potentiel singulier -α/r²(α > 0) à 3 dimensions, qui nécessite une régularisation aux petites distances en mécanique quantique ordinaire. Nous étudions la solution de léquation de Schrödinger dans lespace des impulsions. Nous montrons que la longueur élémentaire régularise le potentiel naturellement. Le spectre dénergie est calculé comme dans le cas des potentiels réguliers, sans introduction dun paramètre arbitraire, et le système possède un état fondamental avec une énergie finie. Nous généralisons notre étude en étudiant léquation de Schrödinger déformée pour le potentiel −α/r² à N dimensions, pour toutes les valeurs du nombre quantique du moment orbital l. La solution analytique est une fonction de Heun qui se réduit à une fonction hypergéométrique dans certains cas particuliers. Nous appliquons nos résultats à 2 dimensions spatiales au problème dun dipôle dans le champ dune corde cosmique. Nous étudions en détail lexistence des états liés du système pour différentes valeurs de la constante de couplage, qui d´epend de langle (θ) entre la corde cosmique et le dipôle. Nous montrons en particulier que la corde cosmique ne peut pas lier le dipôle si θ ≤ π/4. Nous éxaminons également le nombre des états liés du potentiel −α/x² à 1 dimension dans ce nouveau formalisme de la mécanique quantique. Les résultats sont en accord qualitatif avec ceux de la mécanique quantique ordinaire. Nous concluons que dans une théorie quantique non relativiste incluant une longueur élémentaire, celle-ci représenterait une dimension intrinsèque du système étudié. Le formalisme de cette nouvelle version de la mécanique quantique serait utile pour résoudre des problèmes caractérisés par des anomalies dues à des singularités aux petites distances./We discuss the fundamental tools of the formalism of nonrelativistic quantum mechanics based on a generalized uncertainty principle, implying the existence of a minimal length. We consider two simple systems, namely the one-dimensional Dirac delta potential and the three-dimensional Coulomb potential to illustrate how the Schrödinger equation and the eigenvalue problem in the presence of the minimal length can be solved exactly or perturbatively. We apply this formalism to the singular potential −α/r² (α > 0), whose short distance behavior must be regularized in ordinary quantum mechanics. We solve analytically the three-dimensional Schrödinger equation in momentum space. We show that the presence of a minimal length in the formalism regularizes the potential in a natural way. The energy spectrum is calculated as in the case of regular potentials, without introducing any arbitrary parameters, and the system possesses a finite energy in the ground state. We generalize our study by solving analytically the deformed Schrödinger equation for the potential −α/r² in N-dimensions, and for all values of orbital momentum quantum number l. The solution is a Heun function which reduces to a hypergeometric function in some special cases. We apply our results in two spatial dimensions to the problem of a dipole in a cosmic string background. We study in detail the existence of bound states of the system for all values of the coupling constant, depending on the angle (θ between the cosmic string and the dipole. We show in particular that the cosmic string cannot bind the dipole if θ ≤ π/4. We investigate also the number of bound states for the one-dimensional −α/x² potential in this new formalism of quantum mechanics. The results are in qualitative agreement with those of ordinary quantum mechanics. We conclude that the minimal length in a non relativistic quantum theory may represent an intrinsic dimension of the system under study. The formalism of this deformed version of quantum mechanics would be useful to solve problems characterized by anomalies dues to singularities at small distances.

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ ψ(A), σ ^(A), σ ^(A)), where σ ψ is the interior symbol and σ ^(A)(y, η), (y, η) 2 T*Y 0, the (operator-valued) edge symbol of ‘first generation’, cf. [15]. The novelty here is the edge symbol σ^ of ‘second generation’, parametrised by (z, Ϛ) 2 T*Z 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.

With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero. The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions. Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations. While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly. One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution. One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

This study seeks to gain an understanding about the scribal tendencies observed in the Fourth Gospel in Codex Alexandrinus using the method of isolating and classifying singular readings similar to what was first proposed by Colwell, and later modified by Royse and others. In addition to singular readings made before corrections, this study considers singular readings in relation to punctuation markers and line breaks. First, a brief introduction to Codex Alexandrinus is given. Second, the method used to undertake this study is set forth and explained. Third, each singular reading in the Fourth Gospel in Codex Alexandrinus is listed under each respective group in which it is classified, the text of the exemplar is reconstructed if possible and each singular reading is discussed. Finally, the resulting data are analyzed and conclusions are given regarding the tendencies of the scribe responsible for the Fourth Gospel in Codex Alexandrinus. In general, this thesis demonstrates that the strongest tendency of the scribe was that of omission; the scribe was reluctant to add or to harmonize. Harmonizations, when they do occur, are never corrected. Both punctuation and line breaks often afforded the opportunity for the scribe to become distracted and to commit error.

Führungsgetriebe die durch Servomotoren angetrieben werden, benötigen für definierte Stellungen des Abtriebsglieds eine programmierte Funktion (elektronische Kurvenscheibe). Diese leitet sich aus dem möglicherweise fehlerbehafteten kinematischen Modell des Getriebes ab (inverse Kinematik). Zur Verbesserung der Genauigkeit der Führungsbewegung wird ein Verfahren zur Justierung der Übertragungsfunktion auf Basis des Newton-Verfahrens unter Nutzung der Singulärwertzerlegung vorgestellt. Dabei werden die realen Getriebeabmessungen anhand einer Messung berechnet und werden anschließend korrigiert zur Anpassung der Übertragungsfunktion verwendet.

This dissertation addresses the control problem for the general class of control non-affine, non-standard singularly perturbed continuous-time systems. The problem of control for nonlinear multiple time scale systems is addressed here for the first time in a systematic manner. Toward this end, this dissertation develops the theory of feedback passivation for non-affine systems. This is done by generalizing the Kalman-Yakubovich-Popov lemma for non-affine systems. This generalization is used to identify conditions under which non-affine systems can be rendered passive. Asymptotic stabilization for non-affine systems is guaranteed by using these conditions along with well-known passivity-based control methods. Unlike previous non-affine control approaches, the constructive static compensation technique derived here does not make any assumptions regarding the control influence on the nonlinear dynamical model. Along with these control laws, this dissertation presents novel hierarchical control design procedures to address the two major difficulties in control of multiple time scale systems: lack of an explicit small parameter that models the time scale separation and the complexity of constructing the slow manifold. These research issues are addressed by using insights from geometric singular perturbation theory and control laws are designed without making any assumptions regarding the construction of the slow manifold. The control schemes synthesized accomplish asymptotic slow state tracking for multiple time scale systems and simultaneous slow and fast state trajectory tracking for two time scale systems. The control laws are independent of the scalar perturbation parameter and an upper bound for it is determined such that closed-loop system stability is guaranteed. Performance of these methods is validated in simulation for several problems from science and engineering including the continuously stirred tank reactor, magnetic levitation, six degrees-of-freedom F-18/A Hornet model, non-minimum phase helicopter and conventional take-off and landing aircraft models. Results show that the proposed technique applies both to standard and non-standard forms of singularly perturbed systems and provides asymptotic tracking irrespective of the reference trajectory. This dissertation also shows that some benchmark non-minimum phase aerospace control problems can be posed as slow state tracking for multiple time scale systems and techniques developed here provide an alternate method for exact output tracking.

No description available.

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.