In this thesis we consider problems Ax b arising from the discretization of ill-posed problems, where the right-hand side b is polluted by (unknown) noise. It was shown in [29] that under some natural assumptions, using the Golub-Kahan iterative bidiagonalization the noise level in the data can be estimated at a negligible cost. Such information can be further used in solving ill-posed problems. Here we suggest criteria for detecting the noise revealing iteration in the Golub-Kahan iterative bidiagonalization. We discuss the presence of noise of different colors. We study how the loss of orthogonality affects the noise revealing property of the bidiagonalization.

<p>Multi-linear operators play an important role in analysis due to their multiple connections with and applications to other mathematical areas such as ergodic theory, elliptic regularity, and other problems in partial differential equations.</p> <p>Within the area of multi-linear operators, powerful methods were developed originating from the problem of the almost everywhere convergence of Fourier series. Indeed, in their work, Carleson and Fefferman lay the foundation of time-frequency analysis. By further refining their methods, M. Lacey and C. Thiele proved the boundedness of the classical bilinear Hilbert transform for a suitable range of Hölder indices.</p> <p>In this thesis, we consider the general boundedness properties of the bilinear Hilbert transform and the sub-bilinear maximal function along a suitable family of curves.</p> <p>In the first part of our work, we present a short proof of the maximal boundedness range for the sub-bilinear maximal function along non-flat curves, giving a unified treatment of both the singular and the maximal operators.</p> <p>In the second part, we discuss the boundedness of these operators along hybrid curves. This work aims to present a unified perspective that treats the case obtained by joining the zero-curvature features of the operators along flat curves with the non-zero curvature features along non-flat curves.</p>

In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough.

In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough.:1 Introduction 2 On angular measures 3 Types of convex quadrilaterals 4 On conic sections

<p>The concept of the transfer Hamiltonian formalism has been reconsidered and generalized to include the non-orthogonality between the electron states in an interacting region, e.g. quantum dot (QD), and the states in the conduction bands in the attached contacts. The electron correlations in the QD are described by means of a diagram technique for Hubbard operator Green functions for non-equilibrium states. </p><p>It is shown that the non-orthogonality between the electrons states in the contacts and the QD is reflected in the anti-commutation relations for the field operators of the subsystems. The derived forumla for the current contains corrections from the overlap of the same order as the widely used conventional tunneling coefficients. </p><p>It is also shown that kinematic interactions between the QD states and the electrons in the contacts, renormalizes the QD energies in a spin-dependent fashion. The structure of the renormalization provides an opportunity to include a spin splitting of the QD levels by polarizing the conduction bands in the contacts and/or imposing different hybridizations between the states in the contacts and the QD for the two spin channels. This leads to a substantial amplification of the spin polarization in the current, suggesting applications in magnetic sensors and spin-filters.</p>

The concept of the transfer Hamiltonian formalism has been reconsidered and generalized to include the non-orthogonality between the electron states in an interacting region, e.g. quantum dot (QD), and the states in the conduction bands in the attached contacts. The electron correlations in the QD are described by means of a diagram technique for Hubbard operator Green functions for non-equilibrium states. It is shown that the non-orthogonality between the electrons states in the contacts and the QD is reflected in the anti-commutation relations for the field operators of the subsystems. The derived forumla for the current contains corrections from the overlap of the same order as the widely used conventional tunneling coefficients. It is also shown that kinematic interactions between the QD states and the electrons in the contacts, renormalizes the QD energies in a spin-dependent fashion. The structure of the renormalization provides an opportunity to include a spin splitting of the QD levels by polarizing the conduction bands in the contacts and/or imposing different hybridizations between the states in the contacts and the QD for the two spin channels. This leads to a substantial amplification of the spin polarization in the current, suggesting applications in magnetic sensors and spin-filters.

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

Ce travail de thèse porte sur différentes caractérisations des modèles multivariés de stables-Tweedie multiples dans le cadre des familles exponentielles naturelles sous la propriété de "steepness". Ces modèles parus en 2014 dans la littérature ont été d’abord introduits et décrits sous une forme restreinte des stables-Tweedie normaux avant les extensions aux cas multiples. Ils sont composés d’un mélange d’une loi unidimensionnelle stable-Tweedie de variable réelle positive fixée, et des lois stables-Tweedie de variables réelles indépendantes conditionnées par la première fixée, de même variance égale à la valeur de la variable fixée. Les modèles stables-Tweedie normaux correspondants sont ceux du mélange d’une loi unidimensionnelle stable-Tweedie positive fixé et les autres toutes gaussiennes indépendantes. A travers des cas particuliers tels que normal, Poisson, gamma, inverse gaussienne, les modèles stables-Tweedie multiples sont très fréquents dans les études de statistique et probabilités appliquées. D’abord, nous avons caractérisé les modèles stables-Tweedie normaux à travers leurs fonctions variances ou matrices de covariance exprimées en fonction de leurs vecteurs moyens. La nature des polynômes associés à ces modèles est déduite selon les valeurs de la puissance variance à l’aide des propriétés de quasi orthogonalité, des systèmes de Lévy-Sheffer, et des relations de récurrence polynomiale. Ensuite, ces premiers résultats nous ont permis de caractériser à l’aide de la fonction variance la plus grande classe des stables-Tweedie multiples. Ce qui a conduit à une nouvelle classification laquelle rend la famille beaucoup plus compréhensible. Enfin, une extension de caractérisation des stables-Tweedie normaux par fonction variance généralisée ou déterminant de la fonction variance a été établie via leur propriété d’indéfinie divisibilité et en passant par les équations de Monge-Ampère correspondantes. Exprimées sous la forme de produit des composantes du vecteur moyen aux puissances multiples, la caractérisationde tous les modèles multivariés stables-Tweedie multiples par fonction variance généralisée reste un problème ouvert. / In the framework of natural exponential families, this thesis proposes differents characterizations of multivariate multiple stables-Tweedie under "steepness" property. These models appeared in 2014 in the literature were first introduced and described in a restricted form of the normal stables-Tweedie models before extensions to multiple cases. They are composed by a fixed univariate stable-Tweedie variable having a positive domain, and the remaining random variables given the fixed one are reals independent stables-Tweedie variables, possibly different, with the same dispersion parameter equal to the fixed component. The corresponding normal stables-Tweedie models have a fixed univariate stable-Tweedie and all the others are reals Gaussian variables. Through special cases such that normal, Poisson, gamma, inverse Gaussian, multiple stables-Tweedie models are very common in applied probability and statistical studies. We first characterized the normal stable-Tweedie through their variances function or covariance matrices expressed in terms of their means vector. According to the power variance parameter values, the nature of polynomials associated with these models is deduced with the properties of the quasi orthogonal, Levy-Sheffer systems, and polynomial recurrence relations. Then, these results allowed us to characterize by function variance the largest class of multiple stables-Tweedie. Which led to a new classification, which makes more understandable the family. Finally, a extension characterization of normal stable-Tweedie by generalized variance function or determinant of variance function have been established via their infinite divisibility property and through the corresponding Monge-Ampere equations. Expressed as product of the components of the mean vector with multiple powers parameters reals, the characterization of all multivariate multiple stable- Tweedie models by generalized variance function remains an open problem.

This dissertation is dedicated to the topic of multidimensional liquid phase separations. This separation techniques are developed for analysis of complex samples containing thermally labile, low volatile or high molecular weight components that can´t be analysed by two-dimensional (2D) gas chromatography. Concepts of peak capacity and orthogonality are explained and various methods of their determination are stated in theoretical part of dissertation. High performance column liquid chromatography (HPLC) and high performance capillary electrophoresis (HPCE) are suggested as the most suitable methods for automated multidimensional liquid phase separations on-line coupled to mass spectrometry. Configuration of simplified miniaturized liquid chromatograph is described in experimental part of this thesis. Original concept of the system has been extended by simple mobile phase gradient generation technique. Correct function was demonstrated on repeatable separation of alkylphenones, peptides, nitroaromatics, and nitroesters. This system has been utilized as a base for a couple of simple two-dimensional separation platforms for HILIC-MALDI-MS analysis of glycans, for separation of peptides based on off-line coupling of isoelectric focusing and capillary liquid chromatography, and finally for on-line IEC×RPLC, RPLC×RPLC, and HILIC×RPLC two-dimensional liquid chromatography. Correct operation of submitted platforms has been proved.

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common