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We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians (-△)ˢ of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of s-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for s ∈ (1; √3 + 3/2) in any dimension n ≥ 2. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion transformations, it follows that a variety of bounded and unbounded domains do not satisfy positivity preserving properties either and we give some examples.
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. We investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets.
Adaptive, synchronous, and mobile online education: developing the ASYMPTOTE learning environment
(2022)
The COVID-19-induced distance education was perceived as highly challenging by teachers and students. A cross-national comparison of five European countries identified several challenges occurred during the distance learning period. On this basis, the article aims to develop a theoretical framework and design requirements for distance and online learning tools. As one example for online learning in mathematics education, the ASYMPTOTE system is introduced. It will be freely available by May 2022. ASYMPTOTE is aimed at the adaptive and synchronous delivery of online education by taking a mobile learning approach. Its core is the so-called digital classroom, which not only allows students to interact with each other or with the teacher but also enables teachers to monitor their students’ work progress in real time. With respect to the theoretical framework, this article analyses to what extent the ASYMPTOTE system meets the requirements of online learning. Overall, the digital classroom can be seen as a promising tool for teachers to carry out appropriate formative assessment and—partly—to maintain personal and content-related interaction at a distance. Moreover, we highlight the availability of this tool. Due to its mobile learning approach, almost all students will be able to participate in lessons conducted with ASYMPTOTE.
This dissertation is concerned with the task of map-based self-localization, using images of the ground recorded with a downward-facing camera. In this context, map-based (self-)localization is the task of determining the position and orientation of a query image that is to be localized. The map used for this purpose consists of a set of reference images with known positions and orientations in a common coordinate system. For localization, the considered methods determine correspondences between features of the query image and those of the reference images.
In comparison with localization approaches that use images of the surrounding environment, we expect that using images of the ground has the advantage that, unlike the surrounding, the visual appearance of the ground is often long-term stable. Also, by using active lighting of the ground, localization becomes independent of external lighting conditions.
This dissertation includes content of several published contributions, which present research on the development and testing of methods for feature-based localization of ground images. Our first contribution examines methods for the extraction of image features that have not been designed to be used on ground images. This survey shows that, with appropriate parametrization, several of these methods are well suited for the task.
Based on this insight, we develop and examine methods for various subtasks of map-based localization in the following contributions. We examine global localization, where all reference images have to be considered, as well as local localization, where an approximation of the query image position is already known, which allows for disregarding reference images with a large distance to this position.
In our second contribution, we present the first systematic comparison of state-of-the-art methods for ground texture based localization. Furthermore, we present a method, which is characterized by its usage of our novel feature matching technique. This technique is called identity matching, as it matches only those features with identical descriptors, in contrast to the state-of-the-art that also matches features with similar descriptors. We show that our method is well suited for global and local localization, as it has favorable scaling with the number of reference images considered during the localization process. In another contribution, we develop a variant of our localization method that is significantly faster to compute, as it applies a sampling approach to determine the image positions at which local features are extracted, instead of using classical feature detectors.
Two further contributions are concerned with global localization. The first one introduces a prediction model for the global localization performance, based on an evaluation of the local localization performance. This allows us to quickly evaluate any considered parameter settings of global localization methods. The second contribution introduces a learning-based method that computes compact descriptors of ground images. This descriptor can be used to retrieve the overlapping reference images of a query image from a large set of reference images with little computational effort.
The most recent contribution included in this dissertation presents a new ground image database, which was recorded with a dedicated platform using a downward-facing camera. In addition to the data, we also explain our guidelines for the construction of the platform. In comparison with existing databases, our database contains more images and presents a larger variety of ground textures. Furthermore, this database enables us to perform the first systematic evaluation of how localization performance is affected by the time interval between the point in time at which the reference images are recorded and the point in time at which the query image is recorded. We find out that for outdoor areas all ground texture based localization methods have reliability issues, if the time interval between the recording of the query and reference images is large, and also if there are different weather conditions. These findings point to remaining challenges in ground texture base localization that should be addressed in future work.
Matroids are combinatorial objects that generalize linear independence. A matroid can be represented geometrically by its Bergman fan and we compare the symmetries of these two objects. Sometimes, the Bergman fan has additional automorphisms, which are related to Cremona transformations in projective space. Their existence depends on a combinatorial property of the matroid, as has been shown by Shaw and Werner, and we study the consequences for the structure of such matroids. This allows us to gain a better understanding of the so-called Cremona group of a matroid and we apply our results to root system matroids.
Das adaptive Immunsystem schützt den Menschen vor extra- wie auch intrakorporal auftretenden Pathogenen und Krebszellen. Die Funktionalität dieses Prozesses geht hierbei auf die Interaktion und Kooperation einer Vielzahl verschiedener Zelltypen des Körpers zurück und ist vorwiegend innerhalb der Lymphknoten lokalisiert. Ist auch nur ein Bestandteil dieses sensiblen Prozesses gestört, kann dies zu einem teilweisen oder vollständigen Verlust der immunologischen Fitness des Menschen führen. Daher war es das Ziel dieser Arbeit, solche Aberrationen des humanen Lymphknotengewebes umfassend digital-pathologisch zu detektieren und zu definieren.
Hierfür wurde zunächst eine digitale Gewebedatenbank etabliert. Diese basiert auf dem im Rahmen dieser Arbeit implementierten Content-Management-System Digital Tissue Management Suite. Weiterhin wurde die Software Feature analysis in tissue histomorphometry entwickelt, welche die Analyse von zweidimensionalen whole slide images ermöglicht. Hierbei werden Methoden aus dem Bereich Computer Vision und Graphentheorie eingesetzt, um morphologische und distributionale Eigenschaften der Zelltypen des Lymphknotens zu charakterisieren. Darüber hinaus enthält diese Software Plug-ins zur Visualisierung und statistischen Analyse der Daten.
Aufbauend auf der eigens implementierten, digitalen Infrastruktur, in Kombination mit der Software Imaris wurden zweidimensional und dreidimensional gescannte, reaktive und neoplastische Gewebeproben digital phänotypisiert. Hierbei konnten neue mechanische Barrieren zur Kompartimentalisierung der Keimzentren aufgeklärt werden. Weiterhin konnte der Erhalt des quantitativen Verhältnisses einzelner Zellpopulationen innerhalb der Keimzentren beschrieben werden. Ausgehend von den reaktiven Phänotypen des Lymphknotens, wurden pathophysiologische Aberrationen in verschiedenen lymphatischen Neoplasien untersucht. Hierbei konnte gezeigt werden, dass speziell die strukturelle Destruktion häufig mit einer morphologischen Veränderung der fibroblastischen Retikulumzellen einhergeht.
Neben strukturellen Veränderungen sind auch zytologische Veränderungen der Tumormikroumgebung zu verzeichnen. Eine besondere Rolle spielen hierbei sogenannte Tumor-assoziierte Makrophagen. Im Rahmen dieser Arbeit konnte gezeigt werden, dass speziell Makrophagen in der Tumormikroumgebung des diffus großzelligen B-Zell-Lymphoms und der chronisch lymphatischen Leukämie spezifische pathophysiologische Veränderungen aufzeigen. Auch konnte gezeigt werden, dass genetische Änderungen neoplastischer B-Zellen mit einer generellen Reduktion der CD20-Antigendichte einhergehen.
Zusammenfassend ermöglichten die Ergebnisse die Generierung eines umfassenden digital-pathologischen Profils des klassischen Hodgkin-Lymphoms. Hierbei konnten morphologische Veränderungen neoplastischer, CD30-positiver Hodgkin-Reed-Sternberg-Zellen validiert und beschrieben werden. Auch konnten pathologische Veränderungen des Konnektoms und der Tumormikroumgebung dieser Zellen parametrisiert und quantifiziert werden. Abschließend wurde unter Anwendung eines Random forest-Klassifikators die diagnostische Potenz digital-pathologischer Profile evaluiert und validiert.
FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D
(2022)
We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.
The development of epilepsy (epileptogenesis) involves a complex interplay of neuronal and immune processes. Here, we present a first-of-its-kind mathematical model to better understand the relationships among these processes. Our model describes the interaction between neuroinflammation, blood-brain barrier disruption, neuronal loss, circuit remodeling, and seizures. Formulated as a system of nonlinear differential equations, the model reproduces the available data from three animal models. The model successfully describes characteristic features of epileptogenesis such as its paradoxically long timescales (up to decades) despite short and transient injuries or the existence of qualitatively different outcomes for varying injury intensity. In line with the concept of degeneracy, our simulations reveal multiple routes toward epilepsy with neuronal loss as a sufficient but non-necessary component. Finally, we show that our model allows for in silico predictions of therapeutic strategies, revealing injury-specific therapeutic targets and optimal time windows for intervention.
AttendAffectNet-emotion prediction of movie viewers using multimodal fusion with self-attention
(2021)
In this paper, we tackle the problem of predicting the affective responses of movie viewers, based on the content of the movies. Current studies on this topic focus on video representation learning and fusion techniques to combine the extracted features for predicting affect. Yet, these typically, while ignoring the correlation between multiple modality inputs, ignore the correlation between temporal inputs (i.e., sequential features). To explore these correlations, a neural network architecture—namely AttendAffectNet (AAN)—uses the self-attention mechanism for predicting the emotions of movie viewers from different input modalities. Particularly, visual, audio, and text features are considered for predicting emotions (and expressed in terms of valence and arousal). We analyze three variants of our proposed AAN: Feature AAN, Temporal AAN, and Mixed AAN. The Feature AAN applies the self-attention mechanism in an innovative way on the features extracted from the different modalities (including video, audio, and movie subtitles) of a whole movie to, thereby, capture the relationships between them. The Temporal AAN takes the time domain of the movies and the sequential dependency of affective responses into account. In the Temporal AAN, self-attention is applied on the concatenated (multimodal) feature vectors representing different subsequent movie segments. In the Mixed AAN, we combine the strong points of the Feature AAN and the Temporal AAN, by applying self-attention first on vectors of features obtained from different modalities in each movie segment and then on the feature representations of all subsequent (temporal) movie segments. We extensively trained and validated our proposed AAN on both the MediaEval 2016 dataset for the Emotional Impact of Movies Task and the extended COGNIMUSE dataset. Our experiments demonstrate that audio features play a more influential role than those extracted from video and movie subtitles when predicting the emotions of movie viewers on these datasets. The models that use all visual, audio, and text features simultaneously as their inputs performed better than those using features extracted from each modality separately. In addition, the Feature AAN outperformed other AAN variants on the above-mentioned datasets, highlighting the importance of taking different features as context to one another when fusing them. The Feature AAN also performed better than the baseline models when predicting the valence dimension.