323 research outputs found

    A conformal approach to the existence and asymptotic properties of solutions to the Einstein field equations

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    The study of the initial value problem in General Relativity by means of conformal methods was initiated by H. Friedrich in 1986. In this seminal work, the standard conformal Einstein field equations are used to prove the non-linear stability of the de Sitter spacetime. These equations constitute the main technical tool of this thesis. In the first part of this thesis, a technique based on a more general formulation of these equations, the extended conformal Einstein field equations, and a conformal Gaussian gauge is used to establish the non-linear stability of de Sitter-like spacetimes. The gauge freedom associated to the field equations is fixed using the properties of the conformal geodesics. The conformal Gaussian gauge system allows recasting the evolution equations as a symmetric hyperbolic system, which enables the use of standard Cauchy stability results. The same strategy is used to study the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime. The key observation is that this region can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces can be expressed in terms of a conformal Gaussian gauge. A perturbative argument allows then to prove existence and stability results close to the conformal boundary, excluding the asymptotic points where the Cosmological horizon intersects the conformal boundary. In the second part of this thesis, the asymptotic properties of the Maxwell-scalar field system on a flat spacetime are studied by means of the framework of the cylinder at spatial infinity. The analysis is aimed to understand the effects of the non-linearities of this system on the regularity of solutions and polyhomogeneous expansions at the critical sets. The main result is that the non-linear interaction causes both fields to be more singular at the conformal boundary than when the fields are non-interacting

    Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

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    Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results

    Elements, Government, and Licensing: Developments in phonology

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    Elements, Government, and Licensing brings together new theoretical and empirical developments in phonology. It covers three principal domains of phonological representation: melody and segmental structure; tone, prosody and prosodic structure; and phonological relations, empty categories, and vowel-zero alternations. Theoretical topics covered include the formalisation of Element Theory, the hotly debated topic of structural recursion in phonology, and the empirical status of government. In addition, a wealth of new analyses and empirical evidence sheds new light on empty categories in phonology, the analysis of certain consonantal sequences, phonological and non-phonological alternation, the elemental composition of segments, and many more. Taking up long-standing empirical and theoretical issues informed by the Government Phonology and Element Theory, this book provides theoretical advances while also bringing to light new empirical evidence and analysis challenging previous generalisations. The insights offered here will be equally exciting for phonologists working on related issues inside and outside the Principles & Parameters programme, such as researchers working in Optimality Theory or classical rule-based phonology

    On astrophysical solutions in the constructive gravity program and cosmological tests for weakly birefringent spacetime

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    Via gravitational closure [Dü+18]; [Wol22]; [Due20]; [Wie18] could show, how gravitational theories based on the matter content of spacetime can be systematically constructed. While this successfully reproduces general relativity for metric spacetimes, finding a solution for the simplest generalization of Maxwell electrodynamics with a vacuum birefringence allowing, area-metric structure has in general not been possible so far. For highly symmetric FLRW spacetimes a metric, as well as an area-metric solution could be derived [Due20]; [Fis17]. Based on this result, the constructive gravity program will be applied for spherically symmetric, stationary metric spacetimes. Furthermore, an according ansatz is worked out for area-metric geometries, and it is discussed which difficulties arise in finding a corresponding solution. Furthermore, the Etherington-duality is violated in the case of weakly area-metric gravitation [Sch+17]; [Ale20b]; [SW17], and this violation will be investigated with weak gravitational lensing experiments. The observable is the surface brightness, which is, however, heavily influenced by astrophysical processes like physical interaction of galaxies with tidal fields. Beyond that, it is studied how galaxies also get bent due to tidal interactions and how strong this effect is compared to its analog in gravitational lensing

    Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions

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    We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. This is the first work in which we study the dual codes in the framework of the two generic constructions; in particular, we propose a Gram-Schmidt (complexity of O(n3)\mathcal{O}(n^3)) process to compute them explicitly. The originality of this contribution is in the study of the existence or not of defining sets D′D', which can be used as ingredients to construct the dual code C′\mathcal{C}' for a given code C\mathcal{C} in the context of the second generic construction. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of C\mathcal{C} to belong in the dual. This achievement was done in general and when the involved functions are weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals

    Holographic duals of evaporating black holes

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    We describe the dynamical evaporation of a black hole as the classical evolution in time of a black hole in an Anti-de Sitter braneworld. A bulk black hole whose horizon intersects the brane yields the classical bulk dual of a black hole coupled to quantum conformal fields. The evaporation of this black hole happens when the bulk horizon slides off the brane, making the horizon on the brane shrink. We use a large-D effective theory of the bulk Einstein equations to solve the time evolution of these systems. With this method, we study the dual evaporation of a variety of black holes interacting with colder radiation baths. We also obtain the dual of the collapse of holographic radiation to form a black hole on the brane. Finally, we discuss the evolution of the Page curve of the radiation in our evaporation setups, with entanglement islands appearing and then shrinking during the decreasing part of the curve.Comment: 27 pages, 13 figures. v2: 31 pages, 16 figures. Improved discussions, refs adde

    SS-arithmetic (co)homology and pp-adic automorphic forms

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    We study the SS-arithmetic (co)homology of reductive groups over number fields with coefficients in (duals of) certain locally algebraic and locally analytic representations for finite sets of primes SS. We use our results to construct eigenvarieties associated to parabolic subgroups at places in SS and certain classes of supercuspidal and algebraic representations of their Levi factors. We show that these agree with eigenvarieties constructed using overconvergent homology and that for definite unitary groups they are closely related to the Bernstein eigenvarieties constructed by Breuil-Ding.Comment: 85 pages. Comments are welcom

    Magnetic quivers – a new perspective on supersymmetric gauge theories

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    We focus on supersymmetric gauge theories with eight superchrages in spacetime dimensions d = 3, 4, 5, 6. These theories have very rich vacuum structures so our focus will be on their moduli spaces of vacua. For d =, 4, 5, 6, we look at the Higgs branch moduli space. The usual story is that the Higgs branch is a classical object that can be easily computed from its Lagrangian. However, non-perturbative contributions can enhance the Higgs branch and a classical description no longer works. In 6d N = (1, 0) and 5d N = (1, 0), these contributions originate from tensionless BPS-strings and massless gauge instantons respectively as we tune gauge coupling(s) to infinity. For 4d N = 2 theories, many gauge theories, and in particular superconformal field theories (SCFTs), do not even have a Lagrangian description. We offer a unifying solution to these problems in the form of magnetic quivers. These are 3d N = 4 gauge theories whose Coulomb branch is the same as the Higgs branch of the higher dimensional theories. Using brane systems of D_d−D_(d+2)−NS5, with the possible inclusion of Od orientifold planes, we show how the magnetic quivers of these theories can be extracted. Then, a) using the monopole formula we study the moduli space as an algebraic variety by computing its Hilbert series and b) using the new concept of Quiver subtraction we extract the phase diagram (Hasse diagram) of these moduli spaces. Examples we explore include 5d SQCD theories at UV fixed point, 4d rank one SCFTs, class S theories, S-fold theories etc. For the second outcome of the thesis, we focus on new features of gauge theories with orthosymplectic gauge groups such as discrete subgroups and non-simply laced edges, leading to a general classification of such theories. For the final outcome, we study gauge theories with a mixture of unitary and special unitary gauge groups which lead to a slew of new gauge theories related by 3d mirror symmetry.Open Acces

    The emergence of continuum gravitational physics from group field theory models of quantum gravity

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    QG Theorien streben danach, die grundlegenden Theorien von GR und QFT in der theoretischen Physik mit einer kohärenten Weltbeschreibung in Verbindung zu bringen. Dieses dynamische Forschungsgebiet ist eine Kreuzung verschiedener physikalischer Disziplinen, die von der phänomenologischen bis hin zur abstrakten mathematischen Physik reichen. In dieser Arbeit widmen wir uns der Entstehung der Kontinuumsgravitationsphysik aus dem QG-Hintergrund unabhängigen Theorie der GFT. Insbesondere untersuchen wir seine Beziehung zu anderen QG-Modellen wie SF im Rahmen der Modellbildung für 4d Lorentzsche Quantengeometrien, die Berührung zwischen Quantenverschränkung (als Preview von Entstehung) und Quantengeometrie durch die Ausnutzung von Spin-Netzwerk-Zuständen, die Quantengeometrien charakterisieren, und schließlich die kulminierende Phase der Extraktion einer effektiven Beschreibung der kosmologischen Fassung der Theorie in der Sprache einer Feldtheorie, die sich auf einem gekrümmten Hintergrund ausbreitet. Diese Forschungsschwerpunkte dieser Dissertation lassen sich wie folgt präzise zusammenfassen: Zunächst stellen wir die Konstruktion eines neuen SF-Modells für 4d Lorentz’sche Quantengravitation vor, das auf der Beschreibung einer vereinfachten Quantengeometrie beruht, die sich auf edge vector-Variablen stützt. Auf der repräsentationstheoretischen Seite werden Quantenzustände der Geometrie aus irreduziblen Darstellungen der Translationsgruppe auf dem Minkowski-Raum oder Funktionen auf der Translationsgruppe selbst gebildet. Wir zeigen auch, wie das neue Modell mit dem Lorentzschen Barrett-Crane BC-Spinschaum-Modell für einen Sektor seiner Quantenkonfigurationen zusammenhängt. Das neue Modell besitzt offensichtlich alle relevanten Freiheitsgrade, um die vereinfachte Geometrie auf Quantenebene zu beschreiben, und stellt somit einen vielversprechenden Vorschlag für die Lorentzsche Quantengravitation dar. Es kann daher auch als Ergänzung (oder notwendiger Bezugspunkt) für bekannte Spin-Schaum-Modelle angesehen werden, die auf einer eingeschränkten BF-Quantisierung und einer Formulierung der Quantengeometrie in Form von Kantenvektoren basieren. Anschließend untersuchen wir die Verschränkung/geometrische Charakterisierung von generischen überlagerten Quantengeometrien. Der Vorschlag, dass die Raumzeit und ihre geometrischen Eigenschaften aus rein nicht-geometrischen Freiheitsgraden entstehen, die dann wiederum eng mit Verschränkungsmaßen verknüpft sind, hat im Bereich der Quanteninformation QIT viel Aufmerksamkeit erregt. Wir stellen eine unkomplizierte Umsetzung dieser Techniken in QG-Modellen vor, bei denen wir uns auf eine bestimmte Gruppe von QG-Zuständen konzentrieren. Konkret zeigen wir, wie die Untersuchung der Verschränkungseigenschaften einer Überlagerung von QG-Zuständen, genauer gesagt von Spin-Netzwerk-Graph-Zuständen mit unterschiedlichen kombinatorischen Strukturen, auf natürliche Weise zu einer Verallgemeinerung der üblichen von-Neumann-Entropie führt, die für Spin-Netzwerk-Zustände in LQG-Berechnungen erhalten wird. Dies wird in der Tat erreicht, wenn wir verschiedene entropische Begriffe und Maße aus der Quanteninformationstheorie entlehnen, wobei der untersuchte Fall der Überlagerung von Zuständen, die von-Neumann-Entropie verschränkter Regionen bereits auf der kinematischen Ebene der Theorie zur so genannten Interaktionsentropie in QIT führt. Darüber hinaus wird ein Vergleich zwischen dem zweiten Quantisierungsformalismus dieses Schemas, der auf der Überlagerung von Zuständen beruht, und dem der LQG-Ergebnisse vorgestellt. Schließlich besteht die verfügbare Technologie zur Erzeugung der Dynamik in der Verwendung des relationalen Rahmens, wenn kein Begriff des metrischen Hintergrunds oder alternativ dazu die Diffeomorphismusinvarianz in der Theorie vorhanden ist. Ein GFTModell, in dem dies umgesetzt wurde, ist verfügbar, und vor allem ist es gelungen, Kontinuumsphysik in einem kosmologischen Kontext zu extrahieren, der auf GFT-Kondensaten beruht, genauer gesagt in einem homogenen FLRW-Universum mit eingeschlossenen Störungen. Ausgehend von diesen Ergebnissen leiten wir die explizite Lösung der effektiven Dynamik von GFT-Kondensaten her, die auch skalare Störungen berücksichtigt. Dieser erste Schritt ermöglichte es uns, den Materiegehalt weiter zu untersuchen und seine Dynamik in Form einer Feldtheorie auf einem gekrümmten Hintergrund zu formulieren. Dies wiederum führte zu zusätzlichen emergenten Eigenschaften, die die Feldtheorie im Vergleich zur klassischen Theorie besitzt, was sich auch auf der Ebene der Störung widerspiegelt. Im letzteren Fall haben wir eine modifizierte Dispersionsrelation für das gestörte Skalarfeld erhalten.Quantum Gravity (QG) theories pursue the goal of reconciling the pillar theories of General Relativity (GR) and Quantum Field Theory (QFT) in theoretical physics and a coherent description of the physical world surrounding us. This prosperous field of research is a crossroad of various disciplines in physics, ranging from the phenomenological to the most abstract mathematical ones. In this thesis, we devote our focus to the emergence of continuum gravitational physics from the QG background independent approach of Group Field Theory (GFT). In particular, we explore its relation to other QG models such as Spin Foam (SF) in the context of model-building of 4d Lorentzian quantum geometries, the interface between quantum entanglement (considered as the preview of emergence) and quantum geometry through the exploitation of spin network states characterizing quantum geometries, and finally the culminating stage of extracting an effective description of the cosmological version of the theory in the language of a field theory propagating on a curved background. More precisely, the three research focal points of this thesis are summarized as follows: First, we present the construction of a new SF model for 4d Lorentzian quantum gravity based on the description of quantum simplicial geometry relying on edge vector variables. On the representation theoretic side, quantum states of geometry are built from irreducible representations of the translation group on Minkowski space or functions on the translation group itself. We also show how the new model connects to the Lorentzian Barrett-Crane Barrett-Crane (BC) spin foam model, for a sector of its quantum configurations. The new model manifestly possesses all the relevant degrees of freedom to describe simplicial geometry at the quantum level and thus constitutes a promising proposal for Lorentzian quantum gravity. Hence, it may be seen also as a completion (or a necessary reference point) for known spin foam models based on constrained BF quantization and a formulation of quantum geometry in terms of quantum edge vectors. We then move on to inspecting the entanglement/geometric characterization of generic superposed quantum geometries. The proposal that spacetime and its geometric properties are emergent entities from purely non-geometric degrees of freedom that are subsequently closely related to entanglement measures has attracted a lot of attention in the sector of quantum information Quantum Information Theory (QIT). We present a straightforward implementation of these techniques in QG models where we focus on a particular set of QG states. More concretely, we show how studying the entanglement properties of a superposition of QG states, precisely spin network graph states endowed with different combinatorial structures, naturally leads to a generalization of the usual von Neumann entropy obtained for spin network states in Loop Quantum Gravity (LQG) calculations. This is indeed achieved once we borrow different entropic notions and measures from quantum information theory, wherein the studied case of the superposition of states, the von Neumann entropy of entangled regions gives rise to the so-called interaction entropy in QIT already at the kinematical level of the theory. Lastly, in the absence of any notion of metric background or alternatively, in the presence of diffeomorphism invariance in the theory, the available technology to generate the dynamics is to employ the relational framework. A GFT model where this has been implemented is available and importantly it succeeded in extracting continuum physics in a cosmological context relying on GFT condensates, and more precisely that of a homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) universe with perturbations included. Starting from such results, we derive the explicit solution to the GFT condensate effective dynamics including the treatment of scalar perturbations. This first step allowed us to investigate further the matter content, and formulate its dynamics in the form of field theory on a curved background. This in turn produced additional emergent properties the field theory possesses in comparison with the classical one, which was further mirrored at the level of the perturbation. Where it the latter case, we attained a modified dispersion relation for the perturbed scalar field

    Central and Eastern European Literary Theory and the West

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    The twentieth century saw intensive intellectual exchange between Eastern and Central Europe and the West. Yet political and linguistic obstacles meant that many important trends in East and Central European thought and knowledge hardly registered in Western Europe and the US. This book uncovers the hidden westward movements of Eastern European literary theory and its influence on Western scholarship
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