In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according_to_M. Z. Nashed [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in [10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.

<div><div>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</div><div>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.</div></div>

L’étude topologique des variétés algébriques réelles remonte au moins aux travaux de Harnack, Klein, et Hilbert au 19éme siecle; en particulier, la classification des types d’isotopie réalisés par les courbes algébriques réelles d’un degré fixé dans RP2 est un sujet qui a connu un essor considérable jusqu'à aujourd'hui. En revanche, en dehors des études concernants les surfaces de Hirzebruch et les surfaces de degré au plus 3 dans RP3, à peu près rien n’est connu dans le cas de surfaces ambiantes plus générales. Cela est du en particulier au fait que les variétés construites en utilisant le "patchwork" sont des hypersurfaces de variétés toriques. Or, il existe de nombreuses autre surfaces algébriques réelles. Parmi celles-ci se trouvent les surfaces rationnelles réelles, et plus particulièrement les surfaces rèelles minimales. Dans cette thèse, on élargit l’étude des types d’isotopie réalisés par les courbes algébriques réelles aux surfaces réelles minimales de del Pezzo de degré 1 et 2. En outre, on termine la classification des types topologiques réalisés par les courbes algébriques réelles séparantes et non-séparantes de bidegré (5,5) sur la quadrique ellipsoide. / The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in RP2 is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in RP3, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the $mathbb{R}$-minimal surfaces. In this thesis, we extend the study of the topological types realized by real algebraic curves to the real minimal del Pezzo surfaces of degree 1 and 2. Furthermore, we end the classification of separating and non-separating real algebraic curves of bidegree $(5,5)$ in the quadric ellipsoid.

This dissertation thesis deals with the methods for processing image data in X-ray microtomography and digital holographic microscopy. The work aims to achieve significant acceleration of algorithms for tomographic reconstruction and image reconstruction in holographic microscopy by means of optimization and the use of massively parallel GPU. In the field of microtomography, the new GPU (graphic processing unit) accelerated implementations of filtered back projection and back projection filtration of derived data are presented. Another presented algorithm is the orientation normalization technique and evaluation of 3D tomographic data. In the part related to holographic microscopy, the individual steps of the complete image processing procedure are described. This part introduces the new orignal technique of phase unwrapping and correction of image phase damaged by the occurrence of optical vortices in the wrapped image phase. The implementation of the methods for the compensation of the phase deformation and for tracking of cells is then described. In conclusion, there is briefly introduced the Q-PHASE software, which is the complete bundle of all the algorithms necessary for the holographic microscope control, and holographic image processing.

In the present work, we introduce a never before studied soliton equation called the intermediate mixed Manakov (IMM) equation. Through a pole ansatz, we prove that the equation has N-soliton solutions with pole parameters governed by the hyperbolic Calogero-Moser system. We also show that there are spatially periodic N-soliton solutions with poles obeying elliptic Calogero-Moser dynamics. A Lax pair is given in the form of a Riemann-Hilbert problem on a cylinder. A similar Lax pair is shown to imply a novel spin generalization of the intermediate nonlinear Schrödinger equation. Some conservation laws for the IMM are proven. We demonstrate that the IMM can be written as a Hamiltonian system, with one of these conserved quantities as the Hamiltonian. Finally, a physical interpretation is given by showing that the IMM can be rewritten to describe a system of two nonlocally coupled fluids, with nonlinear self-interactions. / Vi presenterar en aldrig tidigare studerad solitonekvation som vi döper till ‘the intermediate mixed Manakov equation’ (ungefär ‘den mellanliggande kopplade Manakovekvationen’. Kortform: IMM). Genom en polansats bevisar vi att ekvationen har N-solitonlösningar där polparametrarna utgör ett hyperboliskt Calogero-Mosersystem. Vi visar också att det finns rumsligt periodiska N-solitonlösningar vars poler följer elliptisk Calogero-Moserdynamik. Ett Laxpar ges i form av ett Riemann-Hilbertproblem på en cylinder. Vi demonstrerar att ett liknande Laxpar leder till en ny spinngeneralisering av den s.k. INLS-ekvationen. Några bevarandelagar för IMM bevisas. Vi visar att IMM-ekvationen kan skrivas som ett Hamiltonskt system, där Hamiltonianen är en av våra tidigare bevarade storheter. Till sist ger vi en fysikalisk tolkning av vår ekvation genom att demonstrera hur den beskriver ett system av ickelokalt interagerande vätskor, med ickelinjära självinteraktioner.

While signatures of attention can be observed in widespread areas within and outside of cortex, the control of attention is thought to be regulated by higher cognitive brain areas, such as the prefrontal cortex. In their recent study on mice Kim et al. could show that successful allocation of attention is characterized by increased spiking of a specific type of inhibitory interneurons, the parvalbumin neurons, and higher oscillatory activity in the gamma band in the local prefrontal network. It was recently demonstrated that encoding of working memory in prefrontal areas is linked to bursts of gamma oscillations, a discontinuous network process characterized by short periods of intense power in the gamma band. The relationship between attention and working memory is unclear, and it is possible that these two cognitive processes share encoding principles. To address this gap, the electrophysiological data collected in the Carlén Lab have been analyzed with advanced spatio-temporal approaches. In particular, we have analyzed bursting gamma activity in medial prefrontal cortex during attentional processing and investigated the similarities to gamma bursting observed during working memory. Gamma-band bursts during attention were reliably detected with several methods. We have characterized several features of the bursts, including the occurrence, duration and amplitude. The neuronal firing rates during and outside of bursts have also been computed. We investigated the correlation between different criteria characterizing the gamma burst and successful vs failed allocation of attention. Control data were generated to discuss the obtained results. The aim of the study was to explore the hypothesis that the medial prefrontal cortex encodes attention trough gamma bursts, which could reveal some similarities and differences in coding of central cognitive processes. No clear difference was found in the characterization between successful and failed allocation of attention. In addition, results were very similar in control set and original data. No underlying mechanism could be identified from this analysis. Therefore, as the bursts occurring in the gamma band in the prefrontal cortex (PFC) were not discriminative with respect to the different tested conditions, they do not seem to encode information related to attention. / Även fast flera olika hjärnområdens aktivitet kan korreleras med uppmärksamhet, anses kontrollen av uppmärksamhet regleras av högre kognitiva hjärnområden, såsom främre hjärnbarken. I en nyligen publicerad artikel studerade Kim et al. hjärnaktiviteten hos möss och kunde visa att en framgångsrik uppmärksamhet kännetecknas av en ökad aktivitet av en specifik typ av inhiberande nervceller, parvalbumin celler, och högre oscillerande aktivitet i gammafrekvens i främre hjärnbarkens lokala nätverk. Det har nyligen visats att kodning av arbetsminne i främre hjärnbarken är kopplat till utbrott av gamma-oscillationer, en diskontinuerlig nätverksprocess som kännetecknas av korta perioder av intensiva oscillationer av det lokala nätverket i gammafrekvens . Relationen mellan uppmärksamhet och arbetsminne är oklar, och det är möjligt att dessa två kognitiva processer delar kodningsprinciper. För att minska detta gap av kunskap har den elektrofysiologiska datan som samlats in i Carlén Lab analyserats med avancerade spatio-temporala tillvägagångssätt. I synnerhet har vi analyserat utbrott i gammaaktivitet i främre hjärnbarken under uppmärksamhet och undersökt likheterna med gamma- utbrott observerade under arbetsminne. Gamma-bandutbrott under uppmärksamhet påvisades på ett tillförlitligt sätt med flera metoder. Vi har karaktäriserat flera funktioner hos utbrotten, inklusive förekomsten, varaktigheten och amplituden. De enskilda cellernas aktivitet undersöktes även under och utanför utprotten av gamma-oscillationer. Vi undersökte sambandet mellan de olika kriterier som karakteriserar gamma-utbrott under framgångsrik mot misslyckad allokering av uppmärksamhet. Kontrolldata genererades för att diskutera de erhållna resultaten. Syftet med studien var att utforska hypotesen att den främre hjärnbarken kodar uppmärksamhet genom gamma-utbrott, vilket kan avslöja vissa likheter och skillnader i kodning av centrala kognitiva processer. Ingen klar skillnad hittades i karaktäriseringen mellan framgångsrik och misslyckad allokering av uppmärksamhet. Dessutom var resultaten mycket likartade i kontrolluppsättningen och den ursprungliga datan. Ingen underliggande mekanism kunde identifieras ur denna analys. Eftersom de utbrott som uppstod i gamma-bandet i främre hjärnbarken inte var unika med hänsyn till de olika testade förhållandena, tycks de därför inte koda information relaterad till uppmärksamhet.

In dieser Arbeit wird eine neue Theorie der quaternionischen Funktionen vorgestellt, welche das Problem der Bestapproximation von Familien prolater und oblater sphäroidalen Funktionen im Hilberträumen behandelt. Die allgemeine Theorie beginnt mit der expliziten Konstruktion von orthogonalen Basen für Räume, definiert auf sphäroidalen Gebieten mit beliebiger Exzentrizität, deren Elemente harmonische, monogene und kontragene Funktionen sind und durch die Form der Gebiete parametrisiert werden. Eine detaillierte Studie dieser grundlegenden Elemente wird in dieser Arbeit durchgeführt. Der Begriff der kontragenen Funktion hängt vom Definitionsbereich ab und ist daher keine lokale Eigenschaft, während die Begriffe der harmonischen und monogenen Funktionen lokal sind. Es werden verschiedene Umwandlungsformeln vorgestellt, die Systeme harmonischer, monogener und kontragener Funktionen auf Sphäroiden unterschiedlicher Exzentrizität in Beziehung setzen. Darüber hinaus wird die Existenz gemeinsamer nichttrivialer kontragener Funktionen für Sphäroide jeglicher Exzentrizität gezeigt. Der zweite wichtige Beitrag dieser Arbeit betrifft eine quaternionische Raumfrequenztheorie für bandbegrenzte quaternionische Funktionen. Es wird eine neue Art von quaternionischen Signalen vorgeschlagen, deren Energiekonzentration im Raum und in den Frequenzbereichen unter der quaternionischen Fourier-Transformation maximal ist. Darüber hinaus werden diese Signale im Kontext der Spektralkonzentration als Eigenfunktionen eines kompakten und selbstadjungierteren quaternionischen Integraloperators untersucht und die grundlegenden Eigenschaften ihrer zugehörigen Eigenwerte werden detailliert beschrieben. Wenn die Konzentrationsgebiete beider Räume kugelförmig sind, kann der Winkelanteil dieser Signale explizit gefunden werden, was zur Lösung von mehreren eindimensionalen radialen Integralgleichungen führt. Wir nutzen die theoretischen Ergebnisse und harmonische Konjugierten um Klassen monogener Funktionen in verschiedenen Räumen zu konstruieren. Zur Charakterisierung der monogenen gewichteten Hardy- und Bergman-Räume in der Einheitskugel werden zwei konstruktive Algorithmen vorgeschlagen. Für eine reelle harmonische Funktion, die zu einem gewichteten Hardy- und Bergman-Raum gehört, werden die harmonischen Konjugiert in den gleichen Räumen gefunden. Die Beschränktheit der zugrundeliegenden harmonischen Konjugationsoperatoren wird in den angegebenen gewichteten Räumen bewiesen. Zusätzlich wird ein quaternionisches Gegenstück zum Satz von Bloch für monogene Funktionen bewiesen. / This work presents a novel Quaternionic Function Theory associated with the best approximation problem in the setting of Hilbert spaces concerning families of prolate and oblate spheroidal functions. The general theory begins with the explicit construction of orthogonal bases for the spaces of harmonic, monogenic, and contragenic functions defined in spheroidal domains of arbitrary eccentricity, whose elements are parametrized by the shape of the corresponding spheroids. A detailed study regarding the elements that constitute these bases is carried out in this thesis. The notion of a contragenic function depends on the domain, and, therefore, it is not a local property in contrast to the concepts of harmonic and monogenic functions. Various conversion formulas that relate systems of harmonic, monogenic, and contragenic functions associated with spheroids of differing eccentricity are presented. Furthermore, the existence of standard nontrivial contragenic functions is shown for spheroids of any eccentricity. The second significant contribution presented in this work pertains to a quaternionic space-frequency theory for band-limited quaternionic functions. A new class of quaternionic signals is proposed, whose energy concentration in the space and the frequency domains are maximal under the quaternion Fourier transform. These signals are studied in the context of spatial-frequency concentration as eigenfunctions of a compact and self-adjoint quaternion integral operator. The fundamental properties of their associated eigenvalues are described in detail. When the concentration domains are spherical in both spaces, the angular part of these signals can be found explicitly, leading to a set of one-dimensional radial integral equations. The theoretical framework described in this work is applied to the construction of classes of monogenic functions in different spaces via harmonic conjugates. Two constructive algorithms are proposed to characterize the monogenic weighted Hardy and Bergman spaces in the Euclidean unit ball. For a real-valued harmonic function belonging to a Hardy and a weighted Bergman space, the harmonic conjugates in the same spaces are found. The boundedness of the underlying harmonic conjugation operators is proven in the given weighted spaces. Additionally, a quaternionic counterpart of Bloch’s Theorem is established for monogenic functions.

Given a full-range simply-invariant shift-invariant subspace <i>M</i> of the vector-valued <i>L²</i> space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function <i>W</i> so that <i>M</i> may be represented as the image of of the Hardy space <i>H²</i> on the disc under multiplication by <i>W</i>. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces <i>(M,M^Ã)</i> which together form a direct-sum decomposition of <i>L²</i>. In the case where the pair <i>(M,M^Ã)</i> are finite-dimensional perturbations of the Hardy space <i>H²</i> and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function <i>W</i>; this realization was parameterized in terms of zero-pole data computed from the pair <i>(M,M^Ã)</i>. Later work by Ball-Raney extended this analysis to the case of nonrational functions <i>W</i> where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs <i>(M,M^Ã)</i> of the <i>L²</i> spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans. / Ph. D.

In dieser Arbeit verwenden wir Flavourinvarianten, um systematisch Lösungen für Probleme des Standardmodells (SM) der Teilchenphysik mit Hilfe verschiedener effektiver Feldtheorien (EFTs) zu untersuchen. In Teil I untersuchen wir die CP-Verletzung im SM und in der SM EFT erweitert mit leichten, sterilen Neutrinos. Wir konstruieren die erzeugende Menge von Flavourinvarianten im νSM, mit der jede Observable als Polynom der Invarianten, sowie die Bedingungen für die CP-Verletzung auf flavourinvariante Weise ausgedrückt werden können. Anschließend weiten wir die Ergebnisse auf die EFT-Wechselwirkungen für verschiedene Szenarien der Neutrinomassen aus. Hier ändert sich die Form der EFT-Flavourinvarianten und ihre Unterdrückung mit der Skala der neuen Physik drastisch mit der untersuchten Art der Neutrinomassen. In Teil II untersuchen wir verschiedene Aspekte der Symmetriebrechung in EFTs von axionartigen Teilchen (ALPs). Wegen ihrer pseudo-Nambu–Goldstone-Natur ist eine wesentliche Eigenschaft der ALPs ihre Shiftsymmetrie (ShS). Wir formulieren flavourinvariante Ordnungsparameter der ShS, die das Powercounting der EFT führender Ordnung bei einer leicht gebrochenen ShS korrekt implementieren lassen. Mit der Hilbertreihe zählen wir die Anzahl der Operatoren, die in der ALP EFT mit und ohne ShS oberhalb und unterhalb der elektroschwachen Skala auftreten, womit wir Operatorbasen konstruieren, die Beziehungen der ShS auf höhere Ordnung verallgemeinern und die CP-verletzenden Flavourinvarianten führender Ordnung konstruieren. Die Axionlösung des starken CP-Problems kann durch neue CP-Verletzung im Ultravioletten durch kleine Instantonen gestört werden. Mit einer SMEFT-Parametrisierung der neuen CP-Verletzung zeigen wir, dass neu konstruierte CP-verletzende SMEFT-Flavourinvarianten explizit in den Instantonberechnungen auftauchen und zur Systematisierung der Berechnungen verwendet werden können, wodurch wir bessere Limits für kleine Instanton- und Flavourszenarien ableiten. / In this thesis, we use flavour invariants to systematically study solutions to problems of the Standard Model (SM) of particle physics with different effective field theories (EFTs). In Part I, we study Charge–Parity (CP) violation in the SM and SM EFT extended with light sterile neutrinos. We construct the generating set of flavour invariants in the νSM allowing us to express any observable as a polynomial of those invariants. In addition, the invariants enable us to express the conditions for CP violation in a flavour-invariant way. We extend the results to the EFT interactions with different scenarios for the neutrino masses. Here, the form of the EFT flavour invariants and their suppression with the scale of new physics changes drastically depending on the nature of the neutrino masses. In Part II, we study different aspects of symmetry breaking in the EFTs of axionlike particles (ALPs). An essential property of ALPs is their shift symmetry (ShS) due to their pseudo-Nambu–Goldstone nature. We formulate flavour-invariant order parameters of ShS, which allow us to properly impose the power counting of the leading order EFT in the presence of a softly broken ShS. Using the Hilbert series, we count the number of operators appearing in the ALP EFT with and without a ShS above and below the electroweak scale. We use this information to construct operator bases, generalise the relations imposing ShS to higher orders and construct the leading order CP-odd flavour invariants. The axion solution to the strong CP problem can be spoiled by new CP violation in the ultraviolet in the presence of small instantons. Parameterising the new CP violation in the SMEFT, we show that newly constructed CP-odd SMEFT flavour invariants, featuring the strong CP angle, explicitly appear in the instanton computations and vice-versa that they can be used to systematise the computations. Using these results, we derive bounds on different small instanton and SMEFT flavour scenarios.

Une écoulement de cavité ouverte tridimensionnel saturé non-linéairement est étudié par une approche spatio-temporelle utilisant des données expérimentales résolues à la fois en temps et en espace. Ces données ont été acquises dans deux plans longitudinaux, respectivement perpendiculaire et parallèle au fond de la cavité, dans le régime incompressible, en air ou en eau. À l'aide de multiples méthodes de décompositions globales en temps et en espace, les ondes et les structures cohérentes constituant la dynamique dans le régime permanent et pouvant être produites par des mécanismes d'instabilités différents sont identifiées et caractérisées.Tout d'abord, on approfondit la compréhension de l'effet des non-linéarités sur les oscillations auto-entretenues de la couche cisaillée impactante et leurs interactions avec l'écoulement intra-cavitaire. En particulier, l'analyse spectrale d'une portion de l'espace des paramètres permet de mettre en évidence un lien entre l'accrochage des modes d'oscillations auto-entretenues, la modulation d'amplitude au niveau du coin impactant et l'intermittence de ces modes. De plus, l'observation des basses fréquences intéragissant fortement avec les oscillations de la couche de mélange démontre l'existence d'une dynamique tridimensionnelle intrinsèque à l'intérieur de la cavité malgré les perturbations causées par la couche cisaillée instable.Les analyses de stabilité linéaire ont montré que des instabilités centrifuges peuvent résulter de la courbure induite par la recirculation. L'étude de la dynamique après saturation révèle de nombreuses structures cohérentes dont les propriétés sont quantifiées et classées en s'appuyant sur la forme des instabilités sous-jacentes: des ondes transverses progressives ou stationnaires. Enfin, certains comportements des structures saturées suggèrent que les mécanismes non-linéaires gouvernant le développement de l'écoulement une fois sorti du régime linéaire pourraient être étudiés dans le cadre des équations d'amplitude. / A space-time study of a three-dimensional nonlinearly saturated open cavity flow is undertaken using time-resolved space-extended experimental data, acquired in both cross-stream and spanwise planes, in incompressible air and water flows. Through use of multiple modal decompositions in time and space, the waves and coherent structures composing the dynamics in the permanent regime are identified and characterised with respect to the instabilities arising in the flow.Effects of nonlinearities are thoroughly investigated in the impinging shear layer, regarding the self-sustained oscillations and their interactions with the inner-flow. In particular, the analysis conducted throughout the parameter space enlightens a global connection between the selection of locked-on modes and the amplitude modulation at the impingement and the mode switching phenomenon. Furthermore, observations of low frequencies interacting drastically with the shear layer flapping motion underline the existence of intrinsic coherent three-dimensional dynamics inside the cavity in spite of the shear layer disturbances.Linear stability analyses have demonstrated that centrifugal instabilities are at play along the main recirculation. The present investigation of the dynamics after onset of the saturation reveals numerous space-time coherent structures, whose properties are quantified and classified with respect to the underlying instabilities: travelling or standing spanwise waves. Finally, some patterns exhibited by the saturated structures suggest that the nonlinear mechanisms governing the mutations of the flow after the linear regime could gain more insight in the frame of amplitude equations.