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Tasks are a key resource in the process of teaching and learning mathematics, which is why task design continues to be one of the main research issues in mathematics education. Different settings can influence the principles underlying the formulation of tasks, and so does the outdoor context. Specifically, a math trail can be a privileged context, known to promote positive attitudes and additional engagement for the learning of mathematics, confronting students with a sequence of real-life tasks, related to a particular mathematical theme. Recently, mobile devices and apps, i.e., MathCityMap, have been recognized as an important resource to facilitate the extension of the classroom to the outdoors. The study reported in this paper intends to identify the principles of design for mobile theme-based math trails (TBT) that result in rich learning experiences in early algebraic thinking. A designed-based research is used, through a qualitative approach, to develop and refine design principles for TBT about Sequences and Patterns. The iterative approach is described by cycles with the intervention of the researchers, pre-service and in-service teachers and students of the targeted school levels. The results are discussed taking into account previous research and data collected along the cycles, conducing to the development of general design principles for TBT tasks.
Die Emergenz digitaler Netzwerke ist auf die ständige Entwicklung und Transformation neuer Informationstechnologien zurückzuführen.
Dieser Strukturwandel führt zu äußerst komplexen Systemen in vielen verschiedenen Lebensbereichen.
Es besteht daher verstärkt die Notwendigkeit, die zugrunde liegenden wesentlichen Eigenschaften von realen Netzwerken zu untersuchen und zu verstehen.
In diesem Zusammenhang wird die Netzwerkanalyse als Mittel für die Untersuchung von Netzwerken herangezogen und stellt beobachtete Strukturen mithilfe mathematischer Modelle dar.
Hierbei, werden in der Regel parametrisierbare Zufallsgraphen verwendet, um eine systematische experimentelle Evaluation von Algorithmen und Datenstrukturen zu ermöglichen.
Angesichts der zunehmenden Menge an Informationen, sind viele Aspekte der Netzwerkanalyse datengesteuert und zur Interpretation auf effiziente Algorithmen angewiesen.
Algorithmische Lösungen müssen daher sowohl die strukturellen Eigenschaften der Eingabe als auch die Besonderheiten der zugrunde liegenden Maschinen, die sie ausführen, sorgfältig berücksichtigen.
Die Generierung und Analyse massiver Netzwerke ist dementsprechend eine anspruchsvolle Aufgabe für sich.
Die vorliegende Arbeit bietet daher algorithmische Lösungen für die Generierung und Analyse massiver Graphen.
Zu diesem Zweck entwickeln wir Algorithmen für das Generieren von Graphen mit vorgegebenen Knotengraden, die Berechnung von Zusammenhangskomponenten massiver Graphen und zertifizierende Grapherkennung für Instanzen, die die Größe des Hauptspeichers überschreiten.
Unsere Algorithmen und Implementierungen sind praktisch effizient für verschiedene Maschinenmodelle und bieten sequentielle, Shared-Memory parallele und/oder I/O-effiziente Lösungen.
In the human brain, the incoming light to the retina is transformed into meaningful representations that allow us to interact with the world. In a similar vein, the RGB pixel values are transformed by a deep neural network (DNN) into meaningful representations relevant to solving a computer vision task it was trained for. Therefore, in my research, I aim to reveal insights into the visual representations in the human visual cortex and DNNs solving vision tasks.
In the previous decade, DNNs have emerged as the state-of-the-art models for predicting neural responses in the human and monkey visual cortex. Research has shown that training on a task related to a brain region’s function leads to better predictivity than a randomly initialized network. Based on this observation, we proposed that we can use DNNs trained on different computer vision tasks to identify functional mapping of the human visual cortex.
To validate our proposed idea, we first investigate a brain region occipital place area (OPA) using DNNs trained on scene parsing task and scene classification task. From the previous investigations about OPA’s functions, we knew that it encodes navigational affordances that require spatial information about the scene. Therefore, we hypothesized that OPA’s representation should be closer to a scene parsing model than a scene classification model as the scene parsing task explicitly requires spatial information about the scene. Our results showed that scene parsing models had representation closer to OPA than scene classification models thus validating our approach.
We then selected multiple DNNs performing a wide range of computer vision tasks ranging from low-level tasks such as edge detection, 3D tasks such as surface normals, and semantic tasks such as semantic segmentation. We compared the representations of these DNNs with all the regions in the visual cortex, thus revealing the functional representations of different regions of the visual cortex. Our results highly converged with previous investigations of these brain regions validating the feasibility of the proposed approach in finding functional representations of the human brain. Our results also provided new insights into underinvestigated brain regions that can serve as starting hypotheses and promote further investigation into those brain regions.
We applied the same approach to find representational insights about the DNNs. A DNN usually consists of multiple layers with each layer performing a computation leading to the final layer that performs prediction for a given task. Training on different tasks could lead to very different representations. Therefore, we first investigate at which stage does the representation in DNNs trained on different tasks starts to differ. We further investigate if the DNNs trained on similar tasks lead to similar representations and on dissimilar tasks lead to more dissimilar representations. We selected the same set of DNNs used in the previous work that were trained on the Taskonomy dataset on a diverse range of 2D, 3D and semantic tasks. Then, given a DNN trained on a particular task, we compared the representation of multiple layers to corresponding layers in other DNNs. From this analysis, we aimed to reveal where in the network architecture task-specific representation is prominent. We found that task specificity increases as we go deeper into the DNN architecture and similar tasks start to cluster in groups. We found that the grouping we found using representational similarity was highly correlated with grouping based on transfer learning thus creating an interesting application of the approach to model selection in transfer learning.
During previous works, several new measures were introduced to compare DNN representations. So, we identified the commonalities in different measures and unified different measures into a single framework referred to as duality diagram similarity. This work opens up new possibilities for similarity measures to understand DNN representations. While demonstrating a much higher correlation with transfer learning than previous state-of-the-art measures we extend it to understanding layer-wise representations of models trained on the Imagenet and Places dataset using different tasks and demonstrate its applicability to layer selection for transfer learning.
In all the previous works, we used the task-specific DNN representations to understand the representations in the human visual cortex and other DNNs. We were able to interpret our findings in terms of computer vision tasks such as edge detection, semantic segmentation, depth estimation, etc. however we were not able to map the representations to human interpretable concepts. Therefore in our most recent work, we developed a new method that associates individual artificial neurons with human interpretable concepts.
Overall, the works in this thesis revealed new insights into the representation of the visual cortex and DNNs...
People can describe spatial scenes with language and, vice versa, create images based on linguistic descriptions. However, current systems do not even come close to matching the complexity of humans when it comes to reconstructing a scene from a given text. Even the ever-advancing development of better and better Transformer-based models has not been able to achieve this so far. This task, the automatic generation of a 3D scene based on an input text, is called text-to-3D scene generation. The key challenge, and focus of this dissertation, now relate to the following topics:
(a) Analyses of how well current language models understand spatial information, how static embeddings compare, and whether they can be improved by anaphora resolution.
(b) Automated resource generation for context expansion and grounding that can help in the creation of realistic scenes.
(c) Creation of a VR-based text-to-3D scene system that can be used as an annotation and active-learning environment, but can also be easily extended in a modular way with additional features to solve more contexts in the future.
(d) Analyze existing practices and tools for digital and virtual teaching, learning, and collaboration, as well as the conditions and strategies in the context of VR.
In the first part of this work, we could show that static word embeddings do not benefit significantly from pronoun substitution. We explain this result by the loss of contextual information, the reduction in the relative occurrence of rare words, and the absence of pronouns to be substituted. But we were able to we have shown that both static and contextualizing language models appear to encode object knowledge, but require a sophisticated apparatus to retrieve it. The models themselves in combination with the measures differ greatly in terms of the amount of knowledge they allow to extract.
Classifier-based variants perform significantly better than the unsupervised methods from bias research, but this is also due to overfitting. The resources generated for this evaluation are later also an important component of point three.
In the second part, we present AffordanceUPT, a modularization of UPT trained on the HICO-DET dataset, which we have extended with Gibsonien/telic annotations. We then show that AffordanceUPT can effectively make the Gibsonian/telic distinction and that the model learns other correlations in the data to make such distinctions (e.g., the presence of hands in the image) that have important implications for grounding images to language.
The third part first presents a VR project to support spatial annotation respectively IsoSpace. The direct spatial visualization and the immediate interaction with the 3D objects should make the labeling more intuitive and thus easier. The project will later be incorporated as part of the Semantic Scene Builder (SeSB). The project itself in turn relies on the Text2SceneVR presented here for generating spatial hypertext, which in turn is based on the VAnnotatoR. Finally, we introduce Semantic Scene Builder (SeSB), a VR-based text-to-3D scene framework using Semantic Annotation Framework (SemAF) as a scheme for annotating semantic relations. It integrates a wide range of tools and resources by utilizing SemAF and UIMA as a unified data structure to generate 3D scenes from textual descriptions and also supports annotations. When evaluating SeSB against another state-of-the-art tool, it was found that our approach not only performed better, but also allowed us to model a wider variety of scenes. The final part reviews existing practices and tools for digital and virtual teaching, learning, and collaboration, as well as the conditions and strategies needed to make the most of technological opportunities in the future.
For an abeloid variety A over a complete algebraically closed field extension K of Qp, we construct a p-adic Corlette–Simpson correspondence, namely an equivalence between finite-dimensional continuous K-linear representations of the Tate module and a certain subcategory of the Higgs bundles on A. To do so, our central object of study is the category of vector bundles for the v-topology on the diamond associated to A. We prove that any pro-finite-étale v-vector bundle can be built from pro-finite-étale v-line bundles and unipotent v-bundles. To describe the latter, we extend the theory of universal vector extensions to the v-topology and use this to generalise a result of Brion by relating unipotent v-bundles on abeloids to representations of vector groups.
We present a symmetry result to solutions of equations involving the fractional Laplacian in a domain with at least two perpendicular symmetries. We show that if the solution is continuous, bounded, and odd in one direction such that it has a fixed sign on one side, then it will be symmetric in the perpendicular direction. Moreover, the solution will be monotonic in the part where it is of fixed sign. In addition, we present also a class of examples in which our result can be applied.
This thesis is concerned with the study of symmetry breaking phenomena for several different semilinear partial differential equations. Roughly speaking, this encompasses equations whose symmetries are not necessarily inherited by their solutions, which is particularly interesting for ground state solutions.
Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset X of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets X. We introduce the X-circuits of a finite subset A⊂Rn , which generalize the simplicial circuits of the affine-linear matroid induced by A to a constrained setting. The X-circuits serve as the main tool in our analysis and exhibit particularly rich combinatorial properties for polyhedral X, in which case the set of X-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of X-circuits transparently reveals when an X-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler X-nonnegative signomials. We develop a duality theory for X-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when X is polyhedral. In conjunction with a notion of reduced X-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization.
Motivated by Gröbner basis theory for finite point configurations, we define and study the class of standard complexes associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant under matroid duality. For the lexicographic term order, the standard complexes satisfy a deletion-contraction-type recurrence. We explicitly determine the lexicographic standard complexes for lattice path matroids using classical bijective combinatorics.
We present new results on nonlocal Dirichlet problems established by means of suitable spectral theoretic and variational methods, taking care of the nonlocal feature of the operators. We mainly address: First, we estimate the Morse index of radially symmetric sign changing bounded weak solutions to a semilinear Dirichlet problem involving the fractional Laplacian. In particular, we derive a conjecture due to Bañuelos and Kulczycki on the geometric structure of the second Dirichlet eigenfunctions. Secondly, we study a small order asymptotics with respect to the parameter s of the Dirichlet eigenvalues problem for the fractional Laplacian. Thirdly, we deal with the logarithmic Schrödinger operator. In particular, we provide an alternative to derive the singular integral representation corresponding to the associated Fourier symbol and introduce tools and functional analytic framework for variational studies. Finaly, we study nonlocal operators of order strictly below one. In particular, we investigate interior regularity properties of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side.