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On Entanglement Entropy for magnetic Hamiltonians
Get PDFThis doctoral thesis concerns the entanglement entropy of ideal Fermi gases subject to some simple magnetic Hamiltonians in the asymptotic for large domains. To mathematically formalize this, we first introduce a Hamiltonian operator H acting on a dense subset of L2(ℝd) for d≤3. It describes the kinetic energy of an electron and all external forces acting on our ideal Fermi gas. This Hamiltonian fully describes the behaviour of our idealised Fermions, as they do not have pairwise interactions. We mostly work with the Landau Hamiltonian in two or three spatial dimensions, which stems from a constant magnetic field of strength B>0. In two dimensions, it is given by HB≔(-i∂1-x2)2+(-i∂2+x1)2, while in three dimensions, it is given by HB≔(-i∂1-x2)2+(-i∂2+x1)2-∂32 . In both cases xj are the euclidean spatial coordinates and ∂j are the partial differentials with respect to these coordinates. Furthermore, we introduce a domain Λ⊂ℝd, which is an open set such that the boundary ∂ Λ satisfies some regularity assumptions, which vary throughout this thesis. Let μ>0 be the so-called chemical potential. We work with the ground state projector, which agrees with the thermal equilibrium at zero temperature. It is written as 1_(≤μ)(H) and is the spectral projector associated to the self-adjoint operator H and the part of the spectrum which lies inside the interval (-∞,μ]. The final piece of notation we need is the spatial projector 1_Λ, which is the multiplication operator with the characteristic function of the domain Λ. For some test functions h:[0,1]→R vanishing at 0, we study the asymptotic expansion as L→∞ of the expression tr h(1_(LΛ) 1_(≤μ)(H)1_(LΛ)) , where tr refers to the usual Hilbert space trace of the trace class operator inside. When we insert the entropy function h(t)≔-t ln(t)-(1-t)ln(1-t), we arrive at the bipartite (or local) entanglement entropy of the system. This is of some physical interest and the motivation for this study. Our first main result is the following theorem. Suppose that Λ⊂ℝ3 is a piecewise Lipschitz region and let μ>B>0. Let ν≔⌈(μ/B-1)/2⌉ and let h:[0,1]→R be a continuous function, which is β-Hölder continuous at 0 and 1 for some 1≥β>0, and assume that h(0)=h(1)=0. Then, we have the asymptotic expansion trh(1_(LΛ)1≤μ(HB)1_(LΛ))=L2ln(L)νB/π I(h) ∫_(∂Λ) dH2(v) |n(v)⋅e3|+o(L2ln(L)) . In particular, as the γ-Rényi entropy function h_γ is β-Hölder continuous for any βγ,1), the γ-Rényi entanglement entropy, S_γ(LΛ), of the ground state at Fermi energy μ localized to LΛ, satisfies the asymptotic expansion S_γ(LΛ)=L2ln(L)νB (1+ γ)/(24γπ) ∫_(∂Λ) dH2(v) |n(v)⋅e3|+o(L2ln(L)) as L→∞. Essentially, this theorem shows an enhanced area law for HB, the Landau Hamiltonian in three dimensions. The noteworthy point here is that, in contrast to the free Hamiltonian H = -Δ, the dependence on the domain is not rotationally invariant, but it depends strongly on the direction of the magnetic field, which is oriented in e3 direction. The proof of this theorem takes about a quarter of this thesis. The first step is to work with polynomial test functions h and find appropriate arguments to reduce the leading order asymptotic to the known asymptotic of the one-dimensional free case (with the Laplace operator on R as the Hamiltonian) and the two-dimensional Landau operator HB. Then, we need some p-Schatten quasi-norm estimates to go from polynomials to merely Hölder continuous test functions h. The remaining part of this thesis is dedicated to the study of the two-dimensional case with the Hamiltonian HB. For fixed magnetic field strength and chemical potential, this has essentially already been fully understood before the work on this thesis began. This thesis focuses on the joint limit B→0 and L→∞. Here, depending on which one approaches its limit faster, we get different, interesting behaviours. We prove the following conjecture in a variety of circumstances. For any Hölder-continuous function f with Hölder exponent strictly bigger than 0, which satisfies f(0)=f(1)=0, any bounded Lipschitz domain Λ, and any μ>0 we have the asymptotic expansion tr f (1_(LΛ)1_(HB≤μ)1_(LΛ))= |∂Λ| 2√μ /π I(f)Lln(μL)+o(μLln(μL)) if BL<μ , |∂Λ| 2√μ /π I(f)Lln(μ/B)+o(μLln(μ/B)) if BL≥μ , as L→∞ and B→0, where we defined the functional f↦I(f)≔1/(4π2) ∫01f(t) /(t(1-t)) dt . The proofs of this conjecture under varied assumptions make up the bulk of the remaining three quarters of this thesis. For this study, due to scaling invariance, we assume μ≔2. After we tackle the case f(t)=t(1-t), where we only need to assume that Λ has C2-boundary and can handle the full range of B,L, we continue with the study of polynomial test functions f. Here, we prove the conjecture in the case BL2.5≪1 using that the projection 1≤μ(HB) converges pointwise to 1≤μ(-Δ), as B tends to zero. Furthermore, we prove it in the case B2L≫1, assuming that ∂ Λ is C2-smooth. Lastly, we prove it under the assumption BL/ln(L)≫1, assuming that Λ is a polygon. This part of the proof relies on a very good understanding of the sine kernel asymptotics of the Hermite kernel, which are discussed along the way. It also relies on the previously established results for a fixed constant magnetic field. After that, we try to get from the polynomial test functions f to the Hölder continuous functions. In the case BL≫1, we recover the same results as for polynomial test functions. This thesis does not study the case BL≪1, as the author has not been able to find a satisfactory approach to this case. The author still considers the case BL≫1 more interesting, as it results in a new formula, while the case BL≪1 yields the same result as B=0, which can thus be regarded as a slight perturbation of the free case H=-Δ. For the purpose of proving this, it is, however, not quite that simple. To get the required bounds, we estimate some p-Schatten quasi-norms. As the methods the author had been aware of prior to working on this thesis were not applicable, the author utilized a method, which relies on “guessing” an orthonormal basis, which is sufficiently close to the eigenbasis of an operator. We discuss this in a lot of detail and provide two other motivational examples, where this method can be applied. This also leads to a (as far as the author is aware) new such quasi-norm estimate for the Hermite kernel associated to the ground state of the harmonic oscillator. The doctoral thesis closes with a brief outlook of potential future research which might be inspired by this work
Animierte Lehrvideos zum Einstieg in die Themenbereiche der Rechtpsychologie
Get PDFUm Studierenden den Einstieg in die anwendungsbezogenen Bestandteile der Rechtspsychologie zu erleichtern, wurden im Rahmes des Projekts drei kurze animierte Videos erstellt. Ziel war es, komplexe Inhalte verständlich und praxisnah aufzubereiten und den asynchronen Lernprozess abwechslungsreicher zu gestalten. Die Materialien sind wiederholt einsetzbar und bieten somit eine nachhaltige Grundlage für die zukünftige Lehre
Slapstick: Ein Kompendium Teil II
Get PDFDer zweite Teil des Slapstick-Kompendiums führt zunächst in einer Einleitung noch einmal aus, warum das Kompendium an den Kategorien Körper- und Kulturtechnik, Situation und Milieu ausgerichtet ist. Sodann folgen zwanzig weitere – reich bebilderte – Artikel teils bewährter, teils neuer Autorinnen und Autoren, in denen diese Kategorien eine ganz unterschiedliche Rolle spielen. In alphabetischer Reihenfolge: Aufzug, Baustelle, Boxkampf, Cross-Dressing, Fast Food, Lokomotive, Malträtieren, Rauchen, Retten, Rutschen, Schienen, Spazierstock, Springen, Straßenbahn, Stürzen, Tablett, Trittbrettfahren, Trunkenheit, Verstecken, Werfen, Würfeln. Wie man sieht, liegt dem Konzept kein systematischer Anspruch zugrunde. Der nächste – dritte – Teil des Kompendiums wird Lücken schließen, aber nichts komplettieren können. Obwohl so viele Filme aus dieser Zeit verloren sind, ist es – wie die angehängte Filmographie zeigt – ein außerordentlich reicher Fundus, aus dem die Beiträge schöpfen. Die Filme sind fast ausnahmslos im Internet verfügbar, sodass die Leserinnen und Leser die Beobachtungen in den Artikeln jederzeit überprüfen können
Stressbewältigung im Arbeitskontext: Eine Mixed-Methods-Untersuchung zu Ressourcen und psychosozialem Wohlbefinden von Mitarbeitenden mit neurodiversen Entwicklungsprofilen
Get PDFThis dissertation investigates the challenges faced by adults with neurodevelopmental disorders (NDDs), such as autism and attention-deficit / hyperactivity disorder (ADHD), in the workplace. While there is increasing recognition of their specific needs, the role of stress, coping strategies, and workplace accommodations in improving work outcomes for individuals with NDDs remains underexplored. The primary objectives of this thesis are to explore the coping strategies used by adults with ADHD and autism at work, investigate the relationship between these strategies, stress, and burnout, and contribute to the development of inclusive workplace practices. To achieve these goals, this dissertation employs a variety of methodological approaches. Study 1 uses qualitative methods, specifically thematic analysis of in-depth interviews, to explore coping strategies employed by autistic adults at work. Study 2 applies a quantitative design with two measurement points to investigate the relationship between coping strategies and stress/ burnout in autistic employees. Study 3 combines diary methodology and cross-sectional quantitative data to analyze daily stressors, coping behaviors, and burnout risk among working students with and without ADHD. The findings reveal that employees with NDDs engage in both problem-oriented and emotion-focused coping strategies, influenced by workplace characteristics such as autonomy and support. Participants reported stress-related challenges but also exhibited problem-oriented coping mechanisms, particularly in environments with flexible structures. However, emotion-focused strategies also emerged, contributing to higher stress and burnout risk. Additionally, employees with ADHD and autistic employees were found to be highly vulnerable to stress, leading to increased risk of burnout and reduced quality of life. This dissertation emphasizes the need for a deeper understanding of the coping strategies used by employees with ADHD and autistic employees and their impact on stress management and workplace outcomes. The findings highlight the importance of tailored interventions and workplace accommodations that promote inclusivity and well-being. Bridging clinical psychology and industrial-organizational psychology, this research contributes to understanding the intersection of personal resources, stress, and work outcomes for individuals with NDDs
Balancing Autonomy and Guidance in Online Multimedia Learning
Get PDFIn different learning environments, students can be guided by instructions, sequencing and feedback, which can lead to them experiencing various levels of autonomy. This experience especially occurs in online learning environments. Therefore, it is important to determine the extent to which students should be granted autonomy as well as the amount of guidance they should be provided in order to ensure the fulfilment of their learning goals. To create a nourishing learning environment that aids students in fulfilling their learning goals, it is necessary to foster students’ perceptions of autonomy, competence and relatedness, which can be enhanced by supporting students’ autonomy, offering structure through guidance and involving them actively. This dissertation tackles the question of the balance between student autonomy and guidance by investigating learning environments (i.e., a Moodle online courses and an R Shiny application) that offer students structural guidance (i.e., a sequential course structure that guides students through a Moodle course and a learning material page instructing students to read a text, select pictures and answer reflection questions) and endow them with higher and lower levels of autonomy, specifically in terms of students’ behaviour (i.e., students’ mouse click log data from online Moodle courses regarding the starting and ending of multiple choice quizzes and picture selections in an R Shiny Application) and learning outcomes. In the first article contributing to this dissertation, student behaviour and preferences are studied within the scope of two Moodle online courses, targeting the question of whether students prefer unrestricted access to learning materials or structured guidance through the subsequent unlocking of learning materials. This first article unveiled a discrepancy in students’ proclaimed preference for higher levels of autonomy through unrestricted access to learning materials and their behaviour within the Moodle online courses, in which they mostly followed the provided structured guidance. The second article contributing to this dissertation, analyses a Moodle course in which students are guided by a sequential lecture unit structure unlocking subsequent lecture units through the completion of quizzes, for which formative outcome feedback (i.e., a score), which may aid them in the decision when to progress between lecture units. The results indicate that most students, while asserting their need for autonomy, tend to follow structured guidance in online multimedia learning environments. Particularly, higher-accuracy students invest more time in learning activities and adapt their learning behaviour in accordance with the formative outcome feedback. Lower-accuracy students (lower-accuracy students achieve lower scores in an exam defined by median split) tend to fall short in distributing their time and adapting their learning behaviour to the feedback; these students may require additional digital support. In the third article contributing to this dissertation, student behaviour and learning outcomes are observed in an unguided R Shiny application in which students perform all learning activities autonomously. In this article it was discovered, that picture types may influence interaction patterns (i.e., the selection of pictures) and instructional pictures may guide students, specifically those achieving higher learning outcomes (higher learning outcomes, or higher-accuracy students refer to students achieving higher scores in a knowledge test or exam defined by median split). It is noted that autonomy and guidance need not exclude each other. Instead, structured guidance can be offered as a variant for students to choose alongside autonomy. Digital support should, in line with the current research, focus on fostering student autonomy while also providing students with guidance through the imposition of structure to fulfil students’ needs in online learning
Prädiktoren des Studienerfolgs von internationalen Studierenden in Deutschland
No full textInternationale Studierende sind eine bedeutsame und zugleich wenig erforschte Gruppe im deutschen Hochschulsystem. Die vorliegende Dissertation untersuchte zentrale Einflussfaktoren des Studienerfolgs dieser Zielgruppe und leistete einen wesentlichen Beitrag, um diese Forschungslücke zu schließen. Aufbauend auf dem SeSaBa-Modell widmete sich die Arbeit der sozialen und akademischen Integration, zwei zentrale Faktoren des Studienerfolgs, die in bisherigen Modellen des Studienabbruchs als entscheidend hervorgehoben wurden, und überprüfte deren Übertragbarkeit auf die besondere Gruppe internationaler Studierender. Die Datengrundlage bildet die Panelstudie „International Student Survey“, die im Rahmen des SeSaBa-Projekts durchgeführt wurde. Befragt wurden N = 4.751 internationale Bachelor- und Masterstudierende aus 134 Ländern an deutschen Hochschulen über sechs Messzeitpunkte hinweg. Im Rahmen dieser Dissertation wurden in vier empirischen Studien unterschiedliche Faktoren des Studienerfolgs analysiert: Lernerprofile und Motivationsregulation als akademische Integrationsmaße (Studie 1 und 4) sowie das Hochschulzugehörigkeitsgefühl als soziales Integrationsmaß (Studie 2 und 3). Die Ergebnisse zeigten, dass soziale und akademische Integration maßgeblich die Studienzufriedenheit, die Studienabbruchsintentionen und die akademischen Leistungen beeinflussen. Diese Dissertation bietet nicht nur theoretische Beiträge zur Bildungspsychologie und Hochschulforschung, sondern sie liefert auch praxisrelevante Erkenntnisse. Sie unterstreicht die Bedeutung, gezielte Maßnahmen zur Förderung der sozialen und akademischen Integration internationaler Studierender zu entwickeln, um deren Studienerfolg zu steigern. In Anbetracht der hohen Bleibequoten internationaler Studierender nach dem Studienabschluss könnten diese Maßnahmen somit auch langfristig zur Bewältigung des Fachkräftemangels in Deutschland beitragen. Zukünftige Forschung sollte auf diesen Erkenntnissen aufbauen und zusätzlich die kulturelle Integration sowie weitere individuelle und institutionelle Faktoren berücksichtigen, um ein noch umfassenderes Verständnis des Studienerfolgs internationaler Studierender zu ermöglichen
Relevance of Sentiment Indicators for Expectations and Uncertainty in Foreign Exchange and Energy Markets
Get PDFThe foreign exchange (FX) and energy markets have undergone significant changes due to increased trading activity and global financial shifts. Foreign exchange markets, in particular, are seeing increasing daily trading volumes. Oil markets are influenced by geopolitical and production events, which impact expectations, uncertainties, and overall market dynamics. This cumulative dissertation examines how media sentiment influences the FX and oil markets, focusing on expectation formation and uncertainty. Building on traditional models such as the rational expectations hypothesis, the study emphasizes the role of media coverage as a public signal that shapes expectations and uncertainties. The dissertation consists of six empirical studies. It finds that expectations about future exchange rates and oil prices are affected by the intensity and tonality of media coverage. In addition, the study illustrates the effects on uncertainties, including forecast errors and disagreements among professionals. Furthermore, media coverage is more significant for information rigidities in the FX market compared to order flow and is valuable for understanding the weak link between FX and fundamentals through scapegoat effects. The implications of the dissertation show that media coverage and tonality, as sources of public information, capture a substantial part of the market, providing an important channel through which information is processed and potentially influencing expectation formations and uncertainties in the FX and oil markets
Intervenierendes Erinnern – Kritische Aufarbeitung des NSU-Komplexes in Theater, Film und Hörspiel
Get PDFJahrelang blieben die Anschläge und Morde des NSU-Komplexes unaufgeklärt, bis sich 2011 das ‚Kerntrio‘ des NSU-Komplexes enttarnte. Bis dahin wurden die Morde sowohl in der Ermittlungsarbeit als auch in der öffentlichen Berichterstattung an der Schnittstelle von antimigrantischen und antimuslimischen Diskursen als Taten migrantischer Menschen verhandelt. Auch die anschließende Aufarbeitung war von rassistischen Zuschreibungen und einer mangelnden Einbeziehung der Betroffenenperspektive geprägt. Bereits kurz nach Bekanntwerden des NSU-Komplexes entstanden zahlreiche Theaterstücke, Filme, Romane und Hörspiele, die diese Leerstellen und blinden Flecken der Aufarbeitung thematisierten und kritische Reflexionsräume jenseits tradierter Gedenkformate schufen. Inwieweit Literatur, Theater, Hörspiel und Film zur kritischen Reflexion des NSU-Komplexes beitragen können und wie dies aus literatur- und medienwissenschaftlicher Perspektive untersucht werden kann, ist Gegenstand dieser Studie, die Ansätze der Memory Studies mit intermedialen und interdisziplinären Ansätzen der Erzähltheorie verbindet
Warum wurde in Afrika anders Krieg geführt als in Europa?
Get PDFInnerhalb der Gesamtreihe Europa transnational, deren thematischer Schwerpunkt auf der transnationalen Geschichte Europas im 19. und 20. Jahrhundert liegt, erscheint die Unterreihe En passant. Mit kleinen Essaybänden erprobt sie neue Formen geschichtswissenschaftlichen Schreibens und greift kurzfristig in aktuelle historische Debatten ein. Den Auftakt bildet der von Dieter Langewiesche verfasste Essay „Warum wurde in Afrika anders Krieg geführt als in Europa?“, der sich den Entstehungsbedingungen der europäischen Kolonialkriege des 19. Jahrhunderts zuwendet
Digitalisierung im Handwerk: Herausforderungen für die Lehre in überbetrieblichen Berufsbildungsstätten: Eine explorative Studie im Kontext der Neuordnung des Berufs Anlagenmechaniker/in für Sanitär-, Heizungs- und Klimatechnik
Get PDFAusgehend vom Phänomen der Digitalisierung der Arbeitswelt untersucht diese explorative Studie die Nutzung digitaler Bildungstechnologien im Rahmen der beruflichen Erstausbildung im dualen System. Der Fokus wird dazu auf drei überbetriebliche Berufsbildungsstätten des Handwerks gelegt. In einem qualitativen Ansatz wird untersucht, welche digitalen Bildungsmedien zu welchen didaktischen Zielsetzungen eingesetzt werden und welche Faktoren die Nutzung dieser Medien beeinflussen. Auf Grundlage eines sozio-kulturellen Ökologiemodells wird ein Leitfaden entwickelt, mit welchem Problemzentrierte Interviews mit sechs Ausbildern geführt werden. Die Auswertung mittels qualitativer Inhaltsanalyse zeigt, dass die untersuchten Bildungsstätten ein relativ einheitliches Muster hinsichtlich der Ausschöpfung didaktischer Funktionen aufweisen, das sich auf das Zusammenspiel von Strukturen, Kulturpraktiken und individuellen Handlungsdispositionen zurückführen lässt. Als Einflussfaktoren lassen sich insbesondere Aspekte beruflicher Handlungsfähigkeit und Konfigurationen medienpädagogischer Kompetenzen, die strukturellen Rahmenbedingungen des dualen Systems und organisational verankerte betriebliche Lernformen ausmachen. Es werden Schlussfolgerungen für die medienpädagogische Kompetenzentwicklung des Bildungspersonals gezogen, die jedoch durch weitere Forschung zu fundieren wären