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- Zahlerwerb (1)
- Zellmotilität (1)
- Zeta-function (1)
- Zig-zag order (1)
- Zufallsvariable (1)
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- Zählprozesse (1)
- a posteriori stopping rule (1)
- absorbing boundary (1)
- absorbing set (1)
- absorption (1)
- accelerated life time model (1)
- accelerated small (1)
- accuracy (1)
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- adaptive (1)
- adaptivity (1)
- adipose tissue (1)
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- aerosol (1)
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- aerosols (1)
- affine (1)
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- algebra (1)
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- algebras (1)
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- asymptotic (1)
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- asymptotics of solutions (1)
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- attenuated Radon transform (1)
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- b-value (1)
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- bar with variable cross-section (1)
- basic ideas ('Grundvorstellungen') (1)
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- behavior (1)
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- coercivity (1)
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- compressible Euler equations (1)
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- conditioned (1)
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- conditions (1)
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- congenital adrenal hyperplasia (1)
- congruence (1)
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- convergence rate (1)
- convolution (1)
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- corona virus (1)
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- cortisol (1)
- coupled solution (1)
- covering (1)
- critical and subcritical Dawson-Watanabe process (1)
- criticality theorem (1)
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- critically ill (1)
- crohn's disease (1)
- crossed product (1)
- curvature varifold (1)
- cusp (1)
- cusped bar (1)
- das Cauchyproblem (1)
- das Goursatproblem (1)
- das charakteristische Cauchyproblem (1)
- data-driven (1)
- dbar-Neumann problem (1)
- de Sitter model ; Fundamental solutions ; Decay estimates (1)
- decay of eigenfunctions (1)
- decomposition (1)
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- degenerate elliptic equations (1)
- degenerate elliptic systems (1)
- delaney-dress tiling theory (1)
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- disagreement percolation (1)
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- discrete Schrodinger (1)
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- distribution (1)
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- doppelsemigroup (1)
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- edge Sobolev spaces (1)
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- edge quantizations (1)
- edge spaces (1)
- edge symbol (1)
- edge- and corner-degenerate symbols (1)
- eigenfunction (1)
- eigenvalue asymptotics (1)
- eigenvalue decay (1)
- eigenvalue estimate (1)
- eigenvalues (1)
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- elliptic complex (1)
- elliptic differential operators of firstorder (1)
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- elliptic problem (1)
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- elliptic quasicomplexes (1)
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- ellipticity in the edge calculus (1)
- ellipticity of cone operators (1)
- ellipticity of corners operators (1)
- ellipticity with interface conditions (1)
- ellipticity with parameter (1)
- ellipticity with respect to interior and edge symbols (1)
- elliptische Gleichungen (1)
- elliptische Quasi-Komplexe (1)
- embedded Markov chain (1)
- embeddings (1)
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- endomorphism semigroup (1)
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- enlargement of filtration (1)
- ensembles (1)
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- equation (1)
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- equivalence (1)
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- error diagram (1)
- erste Variation (1)
- essential position in terms (1)
- estimation of regression (1)
- eta forms (1)
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- expansion (1)
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- exponential function (1)
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- exterior tensor product (1)
- extinction probability (1)
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- fat-free mass (1)
- feedback (1)
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- fibration (1)
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- fibroblasts (1)
- field-aligned currents (1)
- filtering (1)
- finite transformation semigroup (1)
- finiteness theorem (1)
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- illposed problem (1)
- indecomposable varifold (1)
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- invariance (1)
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- manifold (1)
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- minor planets, asteroids: individual: (162173) Ryugu (1)
- mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung (1)
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- relative ranks (1)
- relative η-invariant (1)
- removable set (1)
- removable sets (1)
- representations of groups as automorphism groups of (1)
- reproduction rate (1)
- resampling (1)
- rescaled lattice (1)
- residue (1)
- resolvents (1)
- resonances (1)
- restricted isometry property (1)
- retrieval (1)
- reziproke Invarianten (1)
- rho-invariants (1)
- right limits (1)
- robust filtering (1)
- robustes Filtern (1)
- rock mechanics (1)
- rooted trees (1)
- rough metrics (1)
- saccade detection (1)
- saccades (1)
- scalable (1)
- scaled lattice (1)
- scattering amplitude (1)
- scattering theory (1)
- schlecht gestellt (1)
- seismic hazard (1)
- seismic source inversion (1)
- seismicity (1)
- seismische Quellinversion (1)
- self-assembly (1)
- semi-Lagrangian method (1)
- semi-classical difference operator (1)
- semi-classical limit (1)
- semi-classical spectral estimates (1)
- semiclassical Agmon estimate (1)
- semiclassical spectral asymptotics (1)
- semiclassics (1)
- semiconductors (1)
- semigroup (1)
- semigroup representations (1)
- semigroup theory (1)
- semigroups on infinite chain (1)
- semipermeable barriers (1)
- semiprocess (1)
- sequences of microsaccades (1)
- sequential learning (1)
- series representation (1)
- shallow-water equations (1)
- shock wave (1)
- short-range prediction (1)
- signal detection (1)
- simulation (1)
- simulations (1)
- singular Sturm-Liouville (1)
- singular drifts (1)
- singular foliation (1)
- singular integral equations (1)
- singular point (1)
- singular points (1)
- singuläre Mannigfaltigkeiten (1)
- skew diffusion (1)
- skew field of fraction (1)
- small ball probabilities (1)
- small ball probabilities; (1)
- small noise asymptotics (1)
- smooth drift dependence (1)
- smoother (1)
- socialisation (1)
- soft matter (1)
- space-time Gibbs field (1)
- spacecraft operations (1)
- spacetimes with timelike boundary (1)
- sparsity (1)
- specific entropy (1)
- spectral boundary value problems (1)
- spectral cut-off (1)
- spectral independence (1)
- spectral kernel function (1)
- spectral regularization (1)
- spectral resolution (1)
- spectral theory (1)
- spin Hall effect (1)
- spirallike function (1)
- spread correction (1)
- stability (1)
- stable variety (1)
- stark Hughes-frei (1)
- starker Halbverband von Halbgruppen (1)
- stars: early-type (1)
- stars: individual: Vega (1)
- stars: oscillations (1)
- stars: rotation (1)
- starspots (1)
- state estimation (1)
- state space change point detection (1)
- statistical (1)
- statistical inference (1)
- statistical inverse problem (1)
- statistical machine learning (1)
- statistical methods (1)
- statistical model Mis-specification (1)
- statistics (1)
- statistische Inferenz (1)
- statistisches maschinelles Lernen (1)
- step process (1)
- step-up (1)
- stiff ODE (1)
- stochastic Burgers equations (1)
- stochastic Marcus (canonical) differential equation (1)
- stochastic bridges (1)
- stochastic completeness (1)
- stochastic interacting particles (1)
- stochastic mechanics (1)
- stochastic models (1)
- stochastic process (1)
- stochastic reaction diffusion equation with heavy-tailed Levy noise (1)
- stochastic systems (1)
- stochastics (1)
- stochastische Anordnung (1)
- stochastische Differentialgleichungen (1)
- stochastische Mechanik (1)
- stochastische Zellulare Automaten (1)
- stochastisches interagierendes System (1)
- stress variability (1)
- strong Feller property (1)
- strongly Hughes-free (1)
- strongly pseudoconvex domains (1)
- strongly tempered stable Levy measure (1)
- structure formation (1)
- structured cantilever (1)
- structured numbers (1)
- strukturierte Zahlen (1)
- sub-grid scale (1)
- subRiemannian geometry (1)
- subspaces (1)
- supergeometry (1)
- superposition (1)
- superposition of n-ary operations and n-ary (1)
- superposition of operations (1)
- surrogate loss (1)
- survival analysis (1)
- symbols (1)
- symmetry group (1)
- symplectic (canonical) transformations (1)
- symplectic manifold (1)
- symplectic methods (1)
- symplectic reduction (1)
- system Lame (1)
- systems of partial differential equations (1)
- systems pharmacology (1)
- target dimensions (1)
- targeted antineoplastic drugs (1)
- teacher training mathematics (1)
- teaching (1)
- temporal discretization (1)
- terms (1)
- terms and (1)
- terrigener Staub (1)
- terrigenous dust (1)
- test ability (1)
- tetration (1)
- the Dirichlet problem (1)
- the Goursat problem (1)
- the characteristic Cauchy problem (1)
- the first boundary value problem (1)
- the linearised Einstein equation (1)
- theorem (1)
- theory (1)
- therapeutic (1)
- thermospheric wind (1)
- thymoproteasome (1)
- thymus (1)
- tiling theory (1)
- time (1)
- time reversal (1)
- time series (1)
- time series with heavy tails (1)
- time symmetry (1)
- time-fractional derivative (1)
- tomogrphy (1)
- topic modeling (1)
- torsion forms (1)
- trace formula (1)
- tracer tomography (1)
- transceiver (TRX) (1)
- transdimensional inversion (1)
- transformation (1)
- transformation semigroups (1)
- transformations on infinite set (1)
- transition paths (1)
- trapped surfaces (1)
- travel time tomography (1)
- triply periodic minimal surface (1)
- truncated SVD (1)
- two-dimensional topology (1)
- two-level interacting processes (1)
- tyrosine kinase inhibitors (1)
- ulcerative colitis (1)
- uncertainty (1)
- unendlich teilbare Punktprozesse (1)
- unendliche Teilbarkeit (1)
- uniform compact attractor (1)
- universal Hopf algebra of renormalization (1)
- unknown variance (1)
- unsteady flow (1)
- unzerlegbare Varifaltigkeit (1)
- upper atmosphere model (1)
- vagal sympathetic activity (1)
- value problems (1)
- variable Ricci curvature (1)
- variable projection method (1)
- variational calculus (1)
- variational iteration method (1)
- variational principle (1)
- varifold (1)
- verification (1)
- vibration (1)
- video study (1)
- viral fitness (1)
- wahrscheinlichkeitserzeugende Funktion (1)
- wave structure (1)
- wavelet analysis (1)
- weak dependence (1)
- weakly almost periodic (1)
- weiche Materie (1)
- weight-based formulations (1)
- weighted (1)
- weighted Hölder spaces (1)
- weighted Sobolev space (1)
- weighted Sobolev spaces (1)
- weighted Sobolev spaces with discrete saymptotics (1)
- weighted edge and corner spaces (1)
- weighted graphs (1)
- weighted spaces with asymptotics (1)
- well-posedness (1)
- zero-noise limit (1)
- zufällige Summe (1)
- η-invariant (1)
- когомологии (1)
- комплекс де Рама (1)
- проблема Неймана (1)
- теория Ходжа (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (2166) (remove)
State space models enjoy wide popularity in mathematical and statistical modelling across disciplines and research fields. Frequent solutions to problems of estimation and forecasting of a latent signal such as the celebrated Kalman filter hereby rely on a set of strong assumptions such as linearity of system dynamics and Gaussianity of noise terms.
We investigate fallacy in mis-specification of the noise terms, that is signal noise
and observation noise, regarding heavy tailedness in that the true dynamic frequently produces observation outliers or abrupt jumps of the signal state due to realizations of these heavy tails not considered by the model. We propose a formalisation of observation noise mis-specification in terms of Huber’s ε-contamination as well as a computationally cheap solution via generalised Bayesian posteriors with a diffusion Stein divergence loss resulting in the diffusion score matching Kalman filter - a modified algorithm akin in complexity to the regular Kalman filter. For this new filter interpretations of novel terms, stability and an ensemble variant are discussed. Regarding signal noise mis-specification, we propose a formalisation in the frame work of change point detection and join ideas from the popular CUSUM algo-
rithm with ideas from Bayesian online change point detection to combine frequent reliability constraints and online inference resulting in a Gaussian mixture model variant of multiple Kalman filters. We hereby exploit open-end sequential probability ratio tests on the evidence of Kalman filters on observation sub-sequences for aggregated inference under notions of plausibility.
Both proposed methods are combined to investigate the double mis-specification problem and discussed regarding their capabilities in reliable and well-tuned uncertainty quantification. Each section provides an introduction to required terminology and tools as well as simulation experiments on the popular target tracking task and the non-linear, chaotic Lorenz-63 system to showcase practical performance of theoretical considerations.
Hardy inequalities on graphs
(2024)
The dissertation deals with a central inequality of non-linear potential theory, the Hardy inequality. It states that the non-linear energy functional can be estimated from below by a pth power of a weighted p-norm, p>1. The energy functional consists of a divergence part and an arbitrary potential part. Locally summable infinite graphs were chosen as the underlying space. Previous publications on Hardy inequalities on graphs have mainly considered the special case p=2, or locally finite graphs without a potential part.
Two fundamental questions now arise quite naturally: For which graphs is there a Hardy inequality at all? And, if it exists, is there a way to obtain an optimal weight? Answers to these questions are given in Theorem 10.1 and Theorem 12.1. Theorem 10.1 gives a number of characterizations; among others, there is a Hardy inequality on a graph if and only if there is a Green's function. Theorem 12.1 gives an explicit formula to compute optimal Hardy weights for locally finite graphs under some additional technical assumptions. Examples show that Green's functions are good candidates to be used in the formula.
Emphasis is also placed on illustrating the theory with examples. The focus is on natural numbers, Euclidean lattices, trees and star graphs. Finally, a non-linear version of the Heisenberg uncertainty principle and a Rellich inequality are derived from the Hardy inequality.
Das Eigene und das Fremde
(2023)
Die vorliegende Arbeit stellt eine Untersuchung des Fremdverstehens von Lehrkräften im Mathematikunterricht dar. Mit ‚Fremdverstehen‘ soll dabei – in Anlehnung an den Soziologen Alfred Schütz – der Prozess bezeichnet werden, in welchem eine Lehrkraft versucht, das Verhalten einer Schülerin oder eines Schülers zu verstehen, indem sie dieses Verhalten auf ein Erleben zurückführt, das ihm zugrunde gelegen haben könnte. Als ein wesentliches Merkmal des Prozesses stellt Schütz in seiner Theorie des Fremdverstehens heraus, dass das Fremdverstehen eines Menschen immer auch auf seinen eigenen Erlebnissen basiert. Aus diesem Grund wird in der Arbeit ein methodischer Zweischritt vorgenommen: Es werden zunächst die mathematikbezogenen Erlebnisse zweier Lehrkräfte nachgezeichnet, bevor dann ihr Fremdverstehen in konkreten Situationen im Mathematikunterricht rekonstruiert wird. In der ersten Teiluntersuchung (= der Rekonstruktion eigener Erlebnisse der untersuchten Lehrkräfte) erfolgt die Datenerhebung mit Hilfe biographisch-narrativer Interviews, in denen die untersuchten Lehrkräfte angeregt werden, ihre mathematikbezogene Lebensgeschichte zu erzählen. Die Analyse dieser Interviews wird im Sinne der rekonstruktiven Fallanalyse vorgenommen. Insgesamt führt die erste Teiluntersuchung zu textlichen Darstellungen der rekonstruierten mathematikbezogenen Lebensgeschichte der untersuchten Mathematiklehrkräfte. In der zweiten Teiluntersuchung (= der Rekonstruktion des Fremdverstehens der untersuchten Lehrkräfte) werden dann narrative Interviews geführt, in denen die untersuchten Lehrkräfte von ihrem Fremdverstehen in konkreten Situationen im Mathematikunterricht erzählen. Die Analyse dieser Interviews erfolgt mit Hilfe eines dreischrittigen Analyseverfahrens, welches die Autorin eigens zum Zweck der Rekonstruktion von Fremdverstehen entwickelte. Am Ende dieser zweiten Teiluntersuchung werden sowohl das rekonstruierte Fremdverstehen der Lehrkräfte in verschiedenen Unterrichtssituationen dargestellt als auch Strukturen, die sich in ihrem Fremdverstehen abzeichnen. Mit Hilfe einer theoretischen Verallgemeinerung werden schließlich – auf Basis der Ergebnisse der zweiten Teiluntersuchung – Aussagen über fünf Merkmale des Fremdverstehens von Lehrkräften im Mathematikunterricht im Allgemeinen gewonnen. Mit diesen Aussagen vermag die Arbeit eine erste Beschreibung davon hervorzubringen, wie sich das Phänomen des Fremdverstehens von Lehrkräften im Mathematikunterricht ausgestalten kann.
Zahlen in den Fingern
(2023)
Die Debatte über den Einsatz von digitalen Werkzeugen in der mathematischen Frühförderung ist hoch aktuell. Lernspiele werden konstruiert, mit dem Ziel, mathematisches, informelles Wissen aufzubauen und so einen besseren Schulstart zu ermöglichen. Doch allein die digitale und spielerische Aufarbeitung führt nicht zwingend zu einem Lernerfolg. Daher ist es umso wichtiger, die konkrete Implementation der theoretischen Konstrukte und Interaktionsmöglichkeiten mit den Werkzeugen zu analysieren und passend aufzubereiten.
In dieser Masterarbeit wird dazu exemplarisch ein mathematisches Lernspiel namens „Fingu“ für den Einsatz im vorschulischen Bereich theoretisch und empirisch im Rahmen der Artifact-Centric Activity Theory (ACAT) untersucht. Dazu werden zunächst die theoretischen Hintergründe zum Zahlensinn, Zahlbegriffserwerb, Teil-Ganze-Verständnis, der Anzahlwahrnehmung und -bestimmung, den Anzahlvergleichen und der Anzahldarstellung mithilfe von Fingern gemäß der Embodied Cognition sowie der Verwendung von digitalen Werkzeugen und Multi-Touch-Geräten umfassend beschrieben. Anschließend wird die App Fingu erklärt und dann theoretisch entlang des ACAT-Review-Guides analysiert. Zuletzt wird die selbstständig durchgeführte Studie mit zehn Vorschulkindern erläutert und darauf aufbauend Verbesserungs- und Entwicklungsmöglichkeiten der App auf wissenschaftlicher Grundlage beigetragen. Für Fingu lässt sich abschließend festhalten, dass viele Prozesse wie die (Quasi-)Simultanerfassung oder das Zählen gefördert werden können, für andere wie das Teil-Ganze-Verständnis aber noch Anpassungen und/oder die Begleitung durch Erwachsene nötig ist.
Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary
(2023)
Non-local boundary conditions – for example the Atiyah–Patodi–Singer (APS) conditions – for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, Bär and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While Bär and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions.
In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed.
We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems.
Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions – the analogue of the Atiyah–Patodi–Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.
This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincaré-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions.
The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincaré-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincaré-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts.
The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.
Point processes are a common methodology to model sets of events. From earthquakes to social media posts, from the arrival times of neuronal spikes to the timing of crimes, from stock prices to disease spreading -- these phenomena can be reduced to the occurrences of events concentrated in points. Often, these events happen one after the other defining a time--series.
Models of point processes can be used to deepen our understanding of such events and for classification and prediction. Such models include an underlying random process that generates the events. This work uses Bayesian methodology to infer the underlying generative process from observed data. Our contribution is twofold -- we develop new models and new inference methods for these processes.
We propose a model that extends the family of point processes where the occurrence of an event depends on the previous events. This family is known as Hawkes processes. Whereas in most existing models of such processes, past events are assumed to have only an excitatory effect on future events, we focus on the newly developed nonlinear Hawkes process, where past events could have excitatory and inhibitory effects. After defining the model, we present its inference method and apply it to data from different fields, among others, to neuronal activity.
The second model described in the thesis concerns a specific instance of point processes --- the decision process underlying human gaze control. This process results in a series of fixated locations in an image. We developed a new model to describe this process, motivated by the known Exploration--Exploitation dilemma. Alongside the model, we present a Bayesian inference algorithm to infer the model parameters.
Remaining in the realm of human scene viewing, we identify the lack of best practices for Bayesian inference in this field. We survey four popular algorithms and compare their performances for parameter inference in two scan path models.
The novel models and inference algorithms presented in this dissertation enrich the understanding of point process data and allow us to uncover meaningful insights.
Übungsbuch zur Stochastik
(2023)
Dieses Buch stellt Übungen zu den Grundbegriffen und Grundsätzen der Stochastik und ihre Lösungen zur Verfügung. So wie man Tonleitern in der Musik trainiert, so berechnet man Übungsaufgaben in der Mathematik. In diesem Sinne soll dieses Übungsbuch vor allem als Vorlage dienen für das eigenständige, eigenverantwortliche Lernen und Üben.
Die Schönheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Phänomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Glücksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.
Das Mathematik-Teilprojekt SPIES-M zielt auf eine stärkere Professionsorientierung und die Verknüpfung von Fachwissenschaft und Fachdidaktik in der universitären Lehrkräftebildung. Zu allen großen Inhaltsgebieten der Mathematik wurden neue Lehrveranstaltungen konzipiert und in den Studienordnungen sämtlicher Lehrämter Mathematik an der Universität Potsdam implementiert. Für die Konzeption wurden theoriebasiert Gestaltungsprinzipien herausgearbeitet, die sowohl für das Design als auch für die Evaluation und Weiterentwicklung der Lehrveranstaltungen nach dem Design-Research-Ansatz genutzt werden können. Die Umsetzung der Gestaltungsprinzipien wird am Beispiel der Fundamentalen Idee der Proportionalität verdeutlicht und dabei aufgezeigt, wie Studierende dazu befähigt werden können, fachdidaktisches Wissen aus fachmathematischen Inhalten zu generieren. Die Entwicklung des Professionswissens der Studierenden wird mithilfe unterschiedlicher Instrumente untersucht, um Rückschlüsse auf die Wirksamkeit der neu konzipierten Lehrveranstaltungen zu ziehen. Für die Untersuchungen im Mixed-Methods-Design werden neben Beobachtungen in Lehrveranstaltungen eigens konzipierte Wissenstests, Gruppeninterviews, Unterrichtsentwürfe aus Praxisphasen und Lerntagebücher genutzt. Die Studierendenperspektive wird durch Befragungen zur wahrgenommenen (Berufs-)Relevanz der Lehrveranstaltungen erhoben. Weiteres wesentliches Element der Begleitforschung ist die kollegiale Supervision durch sogenannte „Spies“ (Spione), die die Veranstaltungen kriteriengeleitet beobachten und anschließend gemeinsam mit den Dozierenden reflektieren. Die bisherigen Ergebnisse werden hier präsentiert und hinsichtlich ihrer Implikationen diskutiert. Die im Projekt entwickelten Gestaltungsprinzipien als Werkzeug für Design und Evaluation sowie das Spies-Konzept der kollegialen Supervision werden für die Qualitätsentwicklung von Lehrveranstaltungen zum Transfer vorgeschlagen.