Die Thematik der Arbeit ist in der Schuranalysis angesiedelt. Angeregt durch die Bedurfnisse von Elektrotechnik und Signalubertragungstechnik entwickelte sich dieses mathematische Gebiet, welches Matrix- und Operatorversionen von Interpolations- und Momentenproblemen behandelt. Die vorliegende Dissertation beschafgt sich mit dem "J{Potapovproblem\, einem Interpolationsproblem furFu ntionen der J{Potapovklasse, welche in einer Umgebung von Null holomorph sind. Eine nahere Untersuchung dieser Funktionenklasse zeigt, dass deren Taylorkoe zientenfolgen zur Klasse der J{Potapovfolgen gehoren. Diese Folgen sind der Ausgangspunkt fur die Konstruktion von Matrixpolynomen, mit deren Hilfe sich die Losungsmenge des J{Potapovproblems als gebrochenlineare Transformation darstellen lasst. In Kapitel 1 werden zunachst J{Potapovfolgen und J{Potapovfunktion eingefuhrt sowie das J{Potapovproblem formuliert. Dabei werden wichtige Eigenschaften und Resulate vorgestellt sowie verdeutlicht, dass das J{Potapovproblem eine Verallgemeinerung des in der Literatur ausgiebig behandelten Schurproblems ist. Daraus entsteht die Zielstellung, wohlbekannte, aus Schurfolgen gebildete, Matrixpolynome auf den J{Potapovfall zu ubertragen. Eine besondere Rolle in der Losungsmenge des J{Potapovproblems spielt die zentrale J{ Potapovfunktion, denn eine Quotientendarstellung zentraler J{Potapovfunktionen ermoglicht auch eine Analyse der allgemeinen Losung. In Kapitel 2 erfolgt die Herleitung einer solchen Darstellung sowie die Untersuchung der damit verbundenen Folgen und Matrixpolynome. Dabei werden wichtige Identitaten und Beziehungen bewiesen, welche bei der Behandlung des J{Potapovproblems eine Schlusselrolle spielen. In Kapitel 3 erfolgt eine Verallgemeinerung der Arov{Krein{Matrixpolynome des Schurproblems auf die J{Potapovklasse. Dabei wird zunachst der nichtdegenerierte Fall untersucht, bevor eine Erweiterung der Ergebnisse auf den degenerierten Fall vorgenommen wird. Ausgangspunkt fur die Untersuchungen des vierten Kapitels ist die Beobachtung, dass mit den gegebenen Daten eines J{Potapovproblems ein spezielles Stein-Tripel und damit im nichtdegenerierten Fall ein J (:= diag(J;

Prosopagnosia is defined as a profound deficit in facial identification which can be either acquired due to brain damage or is present from birth, i.e. congenital. Normally, faces and objects are processed in different parts of the inferotemporal cortex by distinct cortical systems for face vs. object recognition, an association of function and location. Accordingly, in acquired prosopagnosia locally restricted damage can lead to specific deficits in face recognition. However, in congenital prosopagnosia faces and objects are also processed in spatially separated areas. Accordingly, the face recognition deficit in congenital prosopagnosia can not be solely explained by the association of function and location. Rather, this observation raises the question why and how such an association evolves at all. So far, no quantitative or computational model of congenital prosopagnosia has been proposed and models of acquired prosopagnosia have focused on changes in the information processing taking place after in icting some kind of \damage" to the system. To model congenital prosopagnosia, it is thus necessary to understand how face processing in congenital prosopagnosia differs from normal face processing, how differences in neuroanatomical development can give rise to differences in processing and last but not least why facial identification requires a specialized cortical processing system in the first place. In this work, a computational model of congenital prosopagnosia is derived from formal considerations, implemented in artificial neural network models of facial information encoding, and tested in experiments with prosopagnosic subjects. The main hypothesis is that the deficit in congenital prosopagnosia is caused by a failure to obtain adequate descriptions of individual faces: A predisposition towards a reduced structural connectivity in visual cortical areas enforces descriptions of visual stimuli that lack the amount of detail necessary to distinguish a specific exemplar from its population, i.e. achieve a successful identification. Formally recognition tasks can be divided into identification tasks (separating a single individual from its sampling population) and classification tasks (partitioning the full object space into distinct classes). It is shown that a high-dimensionality in the sensory representation facilitates individuation (\blessing of dimensionality"), but complicates estimation of object class representations (\curse of dimensionality"). The dimensionality of representations is then studied explicitly in a neural network model of facial encoding. Whereas optimal encoding entails a \holistic" (high-dimensional) representation, a constraint on the network connectivity induces a decomposition of faces into localized, \featural" (low-dimensional) parts. In an experimental validation, the perceptual deficit in congenital prosopagnosia was limited to holistic face manipulations and didn''t extend to featural manipulations. Finally, an extensive and detailed investigation of face and object recognition in congenital prosopagnosia enabled a better behavioral characterization and the identification of subtypes of the deficit. In contrast to previous models of prosopagnosia, here the developmental aspect of congenital prosopagnosia is incorporated explicitly into the model, quantitative arguments for a deficit that is task specific (identification) - and not necessarily domain specific (faces) - are provided for synthetic as well as real data (face images), and the model is validated empirically in experiments with prosopagnosic subjects.

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No description available.

This thesis is divided into two parts, each part exploring a different topic within the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the so-called quantum Wielandt inequality, solving an open problem regarding the higher-dimensional version of matrix product states. Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which could potentially be used to prove new upper bounds on the complexity of matrix multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional algebras. Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian discriminant in terms of fundamental invariants. This answers Question 15 of the 27 questions on the cubic surface posed by Bernd Sturmfels. In the second part of this thesis, we apply algebro-geometric methods to study matroids and flag matroids. We review a geometric interpretation of the Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By generalizing Grassmannians to partial flag varieties, we obtain a new invariant of flag matroids: the flag-geometric Tutte polynomial. We study this invariant in detail, and prove several interesting combinatorial properties.

The Linked Data concept relies on a collection of best practices to publish and link structured web-based data. However, the number of available datasets has been growing significantly over the last decades. These datasets are interconnected and now represent the well-known Web of Data, which stands for an extensive collection of concise and detailed interlinked data sets from multiple domains with large datasets. Thus, linking entries across heterogeneous data sources such as databases or knowledge bases becomes an increasing challenge. However, connections between datasets play a leading role in significant activities such as cross-ontology question answering, large-scale inferences, and data integration. In Linked Data, the Linksets are well known for executing the task of generating links between datasets. Due to the heterogeneity of the datasets, this uniqueness is reflected in the structure of the dataset, making a hard task to find relations among those datasets, i.e., to identify how similar they are. In this way, we can say that Linked Data involves Datasets and Linksets and those Linksets needs to be maintained. Such lack of information directed us to the current issues addressed in this thesis, which are: How to Identify and query datasets from a huge heterogeneous collection of RDF (Resource Description Framework) datasets. To address this issue, we need to assure the consistency and to know how the datasets are related and how similar they are. As results, to deal with the need for identifying LOD (Linked Open Data) Datasets, we created an approach called WIMU, which is a regularly updated database index of more than 660K datasets from LODStats and LOD Laundromat, an efficient, low cost and scalable service on the web that shows which dataset most likely defines a URI and various statistics of datasets indexed from LODStats and LOD Laundromat. To integrate and to query LOD datasets, we provide a hybrid SPARQL query processing engine that can retrieve results from 559 active SPARQL endpoints (with a total of 163.23 billion triples) and 668,166 datasets (with a total of 58.49 billion triples) from LOD Stats and LOD Laundromat. To assure consistency of semantic web Linked repositories where these LOD datasets are located we create an approach for the mitigation of the identifier heterogeneity problem and implement a prototype where the user can evaluate existing links, as well as suggest new links to be rated and a time-efficient algorithm for the detection of erroneous links in large-scale link repositories without computing all closures required by the property axiom. To know how the datasets are related and how similar they are we provide a String similarity algorithm called Most Frequent K Characters, in which is based in two nested filters, (1) First Frequency Filter and (2) Hash Intersection filter, that allows discarding candidates before calculating the actual similarity value, thus giving a considerable performance gain, allowing to build a LOD Dataset Relation Index, in which provides information about how similar are all the datasets from LOD cloud, including statistics about the current state of those datasets. The work in this thesis showed that to identify and query LOD datasets, we need to know how those datasets are related, assuring consistency. Our analysis demonstrated that most of the datasets are disconnected from others needing to pass through a consistency and linking process to integrate them, providing a way to query a large number of datasets simultaneously. There is a considerable step towards totally queryable LOD datasets, where the information contained in this thesis is an essential step towards Identifying, Relating, and Querying datasets on the Web of Data.:1 introduction and motivation 1 1.1 The need for identifying and querying LOD datasets . 1 1.2 The need for consistency of semantic web Linked repositories . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The need for Relation and integration of LOD datasets 2 1.4 Research Questions and Contributions . . . . . . . . . . 3 1.5 Methodology and Contributions . . . . . . . . . . . . . 3 1.6 General Use Cases . . . . . . . . . . . . . . . . . . . . . 6 1.6.1 The Heloise project . . . . . . . . . . . . . . . . . 6 1.7 Chapter overview . . . . . . . . . . . . . . . . . . . . . . 7 2 preliminaries 8 2.1 Semantic Web . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 URIs and URLs . . . . . . . . . . . . . . . . . . . 8 2.1.2 Linked Data . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Resource Description Framework . . . . . . . . 10 2.1.4 Ontologies . . . . . . . . . . . . . . . . . . . . . . 11 2.2 RDF graph . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Transitive property . . . . . . . . . . . . . . . . . . . . . 12 2.4 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Linkset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 RDF graph partitioning . . . . . . . . . . . . . . . . . . 13 2.7 Basic Graph Pattern . . . . . . . . . . . . . . . . . . . . . 13 2.8 RDF Dataset . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.9 SPARQL . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.10 Federated Queries . . . . . . . . . . . . . . . . . . . . . . 14 3 state of the art 15 3.1 Identifying Datasets in Large Heterogeneous RDF Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Relating Large amount of RDF datasets . . . . . . . . . 19 3.2.1 Obtaining Similar Resources using String Similarity . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Consistency on Large amout of RDF sources . . . . . . 21 3.3.1 Heterogeneity in DBpedia Identifiers . . . . . . 21 3.3.2 Detection of Erroneous Links in Large-Scale RDF Datasets . . . . . . . . . . . . . . . . . . . . 22 3.4 Querying Large Heterogeneous RDF Datasets . . . . . 25 4 relation among large amount of rdf sources 29 4.1 Identifying Datasets in Large Heterogeneous RDF sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.1 The WIMU approach . . . . . . . . . . . . . . . . 29 4.1.2 The approach . . . . . . . . . . . . . . . . . . . . 30 4.1.3 Use cases . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.4 Evaluation: Statistics about the Datasets . . . . 35 4.2 Relating RDF sources . . . . . . . . . . . . . . . . . . . . 38 4.2.1 The ReLOD approach . . . . . . . . . . . . . . . 38 4.2.2 The approach . . . . . . . . . . . . . . . . . . . . 40 4.2.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Relating Similar Resources using String Similarity . . . 50 4.3.1 The MFKC approach . . . . . . . . . . . . . . . . 50 4.3.2 Approach . . . . . . . . . . . . . . . . . . . . . . 51 4.3.3 Correctness and Completeness . . . . . . . . . . 55 4.3.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . 57 5 consistency in large amount of rdf sources 67 5.1 Consistency in Heterogeneous DBpedia Identifiers . . 67 5.1.1 The DBpediaSameAs approach . . . . . . . . . . 67 5.1.2 Representation of the idea . . . . . . . . . . . . . 68 5.1.3 The work-flow . . . . . . . . . . . . . . . . . . . 69 5.1.4 Methodology . . . . . . . . . . . . . . . . . . . . 69 5.1.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . 70 5.1.6 Normalization on DBpedia URIs . . . . . . . . . 70 5.1.7 Rate the links . . . . . . . . . . . . . . . . . . . . 71 5.1.8 Results . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1.9 Discussion . . . . . . . . . . . . . . . . . . . . . . 72 5.2 Consistency in Large-Scale RDF sources: Detection of Erroneous Links . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.1 The CEDAL approach . . . . . . . . . . . . . . . 73 5.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2.3 Error Types and Quality Measure for Linkset Repositories . . . . . . . . . . . . . . . . . . . . . 78 5.2.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . 80 5.2.5 Experimental setup . . . . . . . . . . . . . . . . . 80 5.3 Detecting Erroneous Link candidates in Educational Link Repositories . . . . . . . . . . . . . . . . . . . . . . 85 5.3.1 The CEDAL education approach . . . . . . . . . 85 5.3.2 Research questions . . . . . . . . . . . . . . . . . 86 5.3.3 Our contributions . . . . . . . . . . . . . . . . . . 86 5.3.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . 86 6 querying large amount of heterogeneous rdf datasets 89 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.3 The WimuQ . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.1 Identifying Datasets in Large Heterogeneous RDF Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2 Relating Large Amount of RDF Datasets . . . . . . . . 101 7.3 Obtaining Similar Resources Using String Similarity . . 102 7.4 Heterogeneity in DBpedia Identifiers . . . . . . . . . . . 102 7.5 Detection of Erroneous Links in Large-Scale RDF Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.7 Querying Large Heterogeneous RDF Datasets . . . . . 104

Der Freistaat Sachsen kann bei der Datenerhebung auf eine über 100-jährige Tradition zurückblicken. Beispiele dafür sind das landwirtschaftliche Untersuchungswesen in Leipzig-Möckern seit 1852, das meteorologische Observatorium in Radebeul-Wahnsdorf seit 1916 oder der systematische Aufbau des sächsischen Pegelmessnetzes seit 1908. Dieser Tradition fühlt sich die Staatliche Betriebsgesellschaft für Umwelt und Landwirtschaft (BfUL) verpflichtet. Unser Auftrag ist es, Daten über den Zustand von Boden, Wasser und Luft sowie zu Umweltradioaktivität, Gewässerökologie und Meteorologie zu erheben, Messnetze zum Naturschutz zu betreiben sowie vielfältige Daten im Rahmen des landwirtschaftlichen Untersuchungswesens zur Verfügung zu stellen. Redaktionsschluss: 17.06.2019

This thesis is about statistical models and algebraic varieties. Algebraic Statistics unites these two concepts, turning algebraic structure into statistical insight. Featured here are three types of models that have such an algebraic structure. Linear Gaussian covariance models are continuous models which are simple to define but hard to analyze. We compute their maximum likelihood degree in dimension two and find it equal to $2n-3$ generically if the model has $n$ covariates. Discrete models with rational MLE are those discrete models for which likelihood estimation is easiest. We characterize them geometrically by building on the work of Huh and Kapranov on Horn uniformization. Algebraic manifolds are a more general kind of object which is used to encode continuous data. We introduce a new method for computing integrals and sampling from distributions on them, based on intersecting with random linear spaces. A brief report on mathematics in the sciences featuring case studies from soil ecology and nonparametric statistics closes the thesis.

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In this thesis we study qualitative as well as quantitative stability aspects of isometric and conformal maps from S^(n-1) to R^n, when n is greater or equal to 2 or 3 respectively. Starting from the classical theorem of Liouville, according to which the isometry group of S^(n-1) is the group of its rigid motions and the conformal group of S^(n-1) is the one of its Möbius transformations, we obtain stability results for these classes of mappings among maps from S^(n-1) to R^n in terms of appropriately defined deficits. Unlike classical geometric rigidity results for maps defined on domains of R^n and mapping into R^n, not only an isometric\ conformal deficit is necessary in this more flexible setting, but also a deficit measuring how much the maps in consideration distort S^(n-1) in a generalized sense. The introduction of the latter is motivated by the classical Euclidean isoperimetric inequality.