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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.
The evaluation of process-oriented cognitive theories through time-ordered observations is crucial for the advancement of cognitive science. The findings presented herein integrate insights from research on eye-movement control and sentence comprehension during reading, addressing challenges in modeling time-ordered data, statistical inference, and interindividual variability. Using kernel density estimation and a pseudo-marginal likelihood for fixation durations and locations, a likelihood implementation of the SWIFT model of eye-movement control during reading (Engbert et al., Psychological Review, 112, 2005, pp. 777–813) is proposed. Within the broader framework of data assimilation, Bayesian parameter inference with adaptive Markov Chain Monte Carlo techniques is facilitated for reliable model fitting. Across the different studies, this framework has shown to enable reliable parameter recovery from simulated data and prediction of experimental summary statistics. Despite its complexity, SWIFT can be fitted within a principled Bayesian workflow, capturing interindividual differences and modeling experimental effects on reading across different geometrical alterations of text. Based on these advancements, the integrated dynamical model SEAM is proposed, which combines eye-movement control, a traditionally psychological research area, and post-lexical language processing in the form of cue-based memory retrieval (Lewis & Vasishth, Cognitive Science, 29, 2005, pp. 375–419), typically the purview of psycholinguistics. This proof-of-concept integration marks a significant step forward in natural language comprehension during reading and suggests that the presented methodology can be useful to develop complex cognitive dynamical models that integrate processes at levels of perception, higher cognition, and (oculo-)motor control. These findings collectively advance process-oriented cognitive modeling and highlight the importance of Bayesian inference, individual differences, and interdisciplinary integration for a holistic understanding of reading processes. Implications for theory and methodology, including proposals for model comparison and hierarchical parameter inference, are briefly discussed.
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.
Amoeboid cell motility takes place in a variety of biomedical processes such as cancer metastasis, embryonic morphogenesis, and wound healing. In contrast to other forms of cell motility, it is mainly driven by substantial cell shape changes. Based on the interplay of explorative membrane protrusions at the front and a slower-acting membrane retraction at the rear, the cell moves in a crawling kind of way. Underlying these protrusions and retractions are multiple physiological processes resulting in changes of the cytoskeleton, a meshwork of different multi-functional proteins. The complexity and versatility of amoeboid cell motility raise the need for novel computational models based on a profound theoretical framework to analyze and simulate the dynamics of the cell shape.
The objective of this thesis is the development of (i) a mathematical framework to describe contour dynamics in time and space, (ii) a computational model to infer expansion and retraction characteristics of individual cell tracks and to produce realistic contour dynamics, (iii) and a complementing Open Science approach to make the above methods fully accessible and easy to use.
In this work, we mainly used single-cell recordings of the model organism Dictyostelium discoideum. Based on stacks of segmented microscopy images, we apply a Bayesian approach to obtain smooth representations of the cell membrane, so-called cell contours. We introduce a one-parameter family of regularized contour flows to track reference points on the contour (virtual markers) in time and space. This way, we define a coordinate system to visualize local geometric and dynamic quantities of individual contour dynamics in so-called kymograph plots. In particular, we introduce the local marker dispersion as a measure to identify membrane protrusions and retractions in a fully automated way.
This mathematical framework is the basis of a novel contour dynamics model, which consists of three biophysiologically motivated components: one stochastic term, accounting for membrane protrusions, and two deterministic terms to control the shape and area of the contour, which account for membrane retractions. Our model provides a fully automated approach to infer protrusion and retraction characteristics from experimental cell tracks while being also capable of simulating realistic and qualitatively different contour dynamics. Furthermore, the model is used to classify two different locomotion types: the amoeboid and a so-called fan-shaped type.
With the complementing Open Science approach, we ensure a high standard regarding the usability of our methods and the reproducibility of our research. In this context, we introduce our software publication named AmoePy, an open-source Python package to segment, analyze, and simulate amoeboid cell motility. Furthermore, we describe measures to improve its usability and extensibility, e.g., by detailed run instructions and an automatically generated source code documentation, and to ensure its functionality and stability, e.g., by automatic software tests, data validation, and a hierarchical package structure.
The mathematical approaches of this work provide substantial improvements regarding the modeling and analysis of amoeboid cell motility. We deem the above methods, due to their generalized nature, to be of greater value for other scientific applications, e.g., varying organisms and experimental setups or the transition from unicellular to multicellular movement. Furthermore, we enable other researchers from different fields, i.e., mathematics, biophysics, and medicine, to apply our mathematical methods. By following Open Science standards, this work is of greater value for the cell migration community and a potential role model for other Open Science contributions.
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.
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.
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.
Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces.
Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness.
In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate.
We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase.
The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy.
The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth’s outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure.
Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset.
The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package.