Intermittent Convex Integration is a technique for constructing weak solutions to non-linear partial differential equations. It originates from Buckmaster and Vicol's celebrated result about non-uniqueness of distributional solutions to the three-dimensional Navier-Stokes equation. Their construction uses highly concentrated functions as building blocks in a recursively defined infinite series. The recursive definition stems from the method De Lellis and Székelyhidi developed for the Euler equation which later led to the proof of Onsager's conjecture by Isett. Using methods from but other building blocks, Modena and Székelyhidi proved the non-uniqueness of solutions to the transport equation with incompressible velocity fields with Sobolev regularity, relying on a much simpler construction than in the first instance of Intermittent Convex Integration mentioned above. In a similar manner Luo proved the existence of stationary solutions to the Navier-Stokes equation in dimension 4. Another step in the development was the introduction of temporal intermittency by Cheskidov and Luo - earlier constructions were highly concentrated in the spatial variable but homogeneous in time. This innovation admitted results on the two-dimensional Navier-Stokes equation as well as the transport equation with almost Lipschitz velocity field and almost smooth density. In a series of works which combine the iterative ansatz from the proof of Onsager's conjecture with methods and building blocks from Intermittent Convex Integration Novack et al. were able to prove an intermittent analog of the conjecture. The contrary approach, in some sense, was taken by the author of this thesis and Székelyhidi by showing that in most results which use Intermittent Convex Integration, iterations are unnecessary and can be replaced by a simple perturbation argument and applying the Baire category theorem. This allows for stronger results since not only existence but also genericity (in the Baire category sense) of solutions can be concluded. In this work all these developments are presented in an accessible and transparent manner; to this end we will not follow the historically correct order (which is outlined above) but the didactically optimal one: starting from the proof of Onsager's conjecture (which can be considered classical by now) we introduce 'concentrated Mikado flows' and show how they can be applied to the transport equation and the Navier-Stokes equation. In the next step we present building blocks which are entirely localised in space and therefore feature optimal concentration properties, and showcase their use in the transport equation. Then we introduce temporal intermittency as described in and show that it can be used in convex integration independently from spatial intermittency in order to give an elementary proof for non-uniqueness of solutions to the hypodissipative Navier-Stokes equation. The final step is the introduction of the 'Baire category method' and its application to transport and Navier-Stokes equations.:Contents Chapter I. Introduction Chapter II. Turbulent Energy Cascade and Onsager’s Conjecture 1. Observations and Heuristics: Richardson and Kolmogorov 2. Onsager’s conjecture on dissipation of energy 3. Proof of Conservation of energy and why it fails for low regularity 4. A proof of Onsager’s Conjecture by Convex Integration Chapter III. Intermittency in Turbulence and Intermittent Onsager Conjecture 5. Deviation from Homogeneity in Experiments and Modelling 6. Excursion into dyadic energy cascade models 7. Intermittent Energy Cascade and Onsager’s Conjecture Chapter IV. Concentrated Mikado Flows and Applications 8. Technical Prerequisites 9. Transport equation with Sobolev fields 10. Navier Stokes equation in dimension four and higher 11. Convex Integration for the Intermittent Onsager Conjecture Chapter V. Full Dimensional Concentration 12. Building blocks and methods 13. Transport equation with Sobolev fields 14. Three-dimensional Navier-Stokes equation Chapter VI. Temporal Intermittency 15. Hypodissipative Navier-Stokes equations 16. Two-dimensional Navier-Stokes equation and sharp non-uniqueness 17. Transport with almost Lipschitz fields and almost smooth density Chapter VII. Baire Category Method for Intermittent Convex Integration 18. Outline of the Baire category method 19. Genericity of three-dimensional Navier-Stokes solutions 20. Genericity of solutions to the transport equation with Sobolev fields Bibliography

The geometrical limits of the deep drawing process of paper to advanced shapes are not known. This report examines the technological limits of convex elements of the base shape in relation to the drawing height and shows the material behavior in the bottom radius of 3D shapes with regard to special material properties. In the bottom radius, non-compressed wrinkles occurred due to the in-plane compression, but wrinkles were reduced by an increased blank holder force or tool temperatures and improved extensibility or in-plane compressive strain. The forming ratio during deep drawing (drawing height related to base diameter) was increased to a value of more than 1 by a blank holder force, which increased with the drawing height such that the initial blank holder force was reduced concurrently. Straight sections in the base shape reduced the risk for ruptures in the edge radii of rectangular shapes, producing a forming ratio in these radii of 2.5. The forming ratio was further supported by a pattern of creasing lines at the blanks with a radial orientation and a number near the expected maximum number of wrinkles. The spring-back at rectangular shapes mainly depended on the drawing height and edge radius.

This thesis studies effective resistances of finite and infinite weighted graphs. Classical results state that it is a metric on the set of vertices of the graph and that it can be expressed completely in terms of the graph’s random walk. The first goal of this thesis is to provide a concise and accessible starting point for new scholars interested in the topic. In that spirit, we reproduce existing results and review different approaches to effective resistances using tools from several fields such as linear algebra, probability theory, geometry and functional analysis. The second goal is to characterize which metric spaces are given by the effective resistance of a graph. For the finite case, we begin by reconstructing the associated graph from the effective resistance. This leads to a complete algebraic characterization in terms of triangle inequality defects. A more geometric condition is given by showing that a metric space can only be an effective resistance if its minimal graph realization contains no incomplete cycles. We also show that our algebraic characterization can be applied to the more general theory of resistance forms as defined by Kigami. The third goal of this thesis is to investigate probabilistic representations of effective resistances. Building on the work of Tetali and Barlow, we characterize under which conditions known representations for finite graphs can be extended to infinite graphs.

The failure of β cells to secrete sufficient amounts of insulin is a key feature of diabetes mellitus. Each β cell secretes only a small amount of insulin upon stimulation in a highly regulated fashion: young insulin is preferentially released, whereas old insulin is mainly degraded within the β cell. How this process is regulated in vivo and likely altered in diabetes is currently unknown. We present here a transgenic pig model that allows the in vivo fluorescent labeling of age-distinct insulin secretory granule pools, hence providing a close-to-life readout of insulin turnover. This will enable the study of alterations in β cell function in an animal model close to humans.

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In this thesis, we undertake a combinatorial study of certain aspects of triangulations of cartesian products of simplices, particularly in relation to their relevance in toric algebra and to their underlying product structure. The first chapter reports joint work with Samu Potka. The object of study is a class of homogeneous toric ideals called cut ideals of graphs, that were introduced by Sturmfels and Sullivant 2006. Apart from their inherent appeal to combinatorial commutative algebra, these ideals also generalize graph statistical models for binary data and are related to some statistical models for phylogenetic trees. Specifically, we consider minimal free resolutions for the cut ideals of trees. We propose a method to combinatorially estimate the Betti numbers of the ideals in this class. Using this method, we derive upper bounds for some of the Betti numbers, given by formulas exponential in the number of vertices of the tree. Our method is based on a common technique in commutative algebra whereby arbitrary homogeneous ideals are deformed to initial monomial ideals, which are easier to analyze while conserving some of the information of the original ideals. The cut ideal of a tree on n vertices turns out to be isomorphic to the Segre product of the cut ideals of its n-1 edges (in particular, its algebraic properties do not depend on its shape). We exploit this product structure to deform the cut ideal of a tree to an initial monomial ideal with a simple combinatorial description: it coincides with the edge ideal of the incomparability graph of the power set of the edges of the tree. The vertices of the incomparability graph are subsets of the edges of the tree, and two subsets form an edge whenever they are incomparable. In order to obtain algebraic information about these edge ideals, we apply an idea introduced by Dochtermann and Engström in 2009 that consists in regarding the edge ideal of a graph as the (monomial) Stanley-Reisner ideal of the independence complex of the graph. Using Hochster\''s formula for computting Betti numbers of Stanley-Reisner ideals by means of simplicial homology, the computation of the Betti numbers of these monomial ideals is turned to the enumeration of induced subgraphs of the incomparability graph. That the resulting values give upper bounds for the Betti numbers of the cut ideals of trees is an important well-known result in commutative algebra. In the second chapter, we focus on some combinatorial features of triangulations of the point configuration obtained as the cartesian product of two standard simplices. These were explored in collaboration with César Ceballos and Arnau Padrol, and had a two-fold motivation. On the one hand, we intended to understand the influence of the product structure on the set of triangulations of the cartesian product of two point configurations; on the other hand, the set of all triangulations of the product of two simplices is an intricate and interesting object that has attracted attention both in discrete geometry and in other fields of mathematics such as commutative algebra, algebraic geometry, enumerative geometry or tropical geometry. Our approach to both objectives is to examine the circumstances under which a triangulation of the polyhedral complex given by the the product of an (n-1)-simplex times the (k-1)-skeleton of a (d-1)-simplex extends to a triangulation of an (n-1)-simplex times a (d-1)-simplex. We refer to the former as a partial triangulation of the product of two simplices. Our main result says that if d >= k > n, a partial triangulation always extends to a uniquely determined triangulation of the product of two simplices. A somewhat unexpected interpretation of this result is as a finiteness statement: it asserts that if d is sufficiently larger than n, then all partial triangulations are uniquely determined by the (compatible) triangulations of its faces of the form “(n-1)-simplex times n-simplex”. Consequently, one can say that in this situation ‘\''triangulations of an (n-1)-simplex times a (d-1)-simplex are not much more complicated than triangulations of an (n-1)-simplex times an n-simplex\''\''. The uniqueness assertion of our main result holds already when d>=k>=n. However, the same is not true for the existence assertion; namely, there are non extendable triangulations of an (n-1)-simplex times the boundary of an n-simplex that we explicitly construct. A key ingredient towards this construction is a triangulation of the product of two (n-1)-simplices that can be seen as its ``second simplest triangulation\''\'' (the simplest being its staircase triangulation). It seems to be knew, and we call it the Dyck path triangulation. This triangulation displays symmetry under the cyclic group of order n that acts by simultaneously cycling the indices of the points in both factors of the product. Next, we exhibit a natural extension of the Dyck path triangulation to a triangulation of an (n-1)-simplex times an n-simplex that, in a sense, enjoys some sort of ‘\''rigidity\''\'' (it also seems new). Performing a ‘\''local modification\''\'' on the restriction of this extended triangulation to the polyhedral complex given by (n-1)-simplex times the boundary of an n-simplex yields the non-extendable partial triangulation. The thesis includes two appendices on basic commutative algebra and triangulations of point configuration, included to make it slightly self-contained.

Das eHealth Service Engineering für seltene Erkrankungen erörtert die systematische Entwicklung von wissensbasierten medizinischen Dienstleistungen (eHealth-Services). Im Fokus der Forschungsarbeit steht das komplexe Problem der Informationsversorgung von Patienten mit degenerativen Nervenerkrankungen (hier: Amyotrophe Lateralsklerose, kurz: ALS). Der unvorhersehbare Krankheitsverlauf, der mit schwerwiegenden Symptomen einhergeht, stellt hohe Anforderungen an die behandelnden Ärzte und beteiligten Akteure, um den betroffenen Patienten eine umfassende Beratung anzubieten. Die Herausforderung ist es, das persönliche Beratungsgespräch IT-basiert zu unterstützen und Patienten eine bedarfsgerechte Informationsbasis für die partizipative Entscheidungsfindung zu bieten. Der exemplarische Anwendungsfall wird anhand umfangreicher Quellenanalysen multidimensional konzeptualisiert. Die extrahierten Konzepte und Relationen werden strukturiert und zu einem semiformalen Gesamtgraphen aggregiert. Dieser Wissensgraph verdeutlicht die fünf systemischen Entwicklungsbereiche Informationen über Patienten (1), Informationen für Patienten (2), Dokumentation bisher getroffener Entscheidungen (3) und dementsprechende Offerten von medizinischen Produkten und Dienstleistungen (4) sowie Management und Koordination der Versorgung (5). Für die Teilbereiche (1-4), welche die Informationsversorgung betreffen, wird eine formale Repräsentation in Form einer RDF-basierten Ontologie (Dispedia) entwickelt und als Linked Data veröffentlicht. Die Nutzung der Dispedia Ontologie wird durch die prototypische Implementierung einer eHealth Anwendung demonstriert. Das Ergebnis ist eine bedarfsgerechte Informationsversorgung für Patienten mit seltenen Erkrankungen.

Diffusion MRI (dMRI) measurements allow us to infer the microstructural properties of white matter and to reconstruct fiber pathways in-vivo. High angular diffusion imaging (HARDI) allows for the creation of more and more complex local models connecting the microstructure to the measured signal. One of the challenges is the derivation of meaningful metrics describing the underlying structure from the local models. The aim hereby is to increase the specificity of the widely used metric fractional anisotropy (FA) by using the additional information contained within the HARDI data. A local model which is connected directly to the underlying microstructure through the model of a single fiber population is spherical deconvolution. It produces a fiber orientation density function (fODF), which can often be interpreted as superposition of multiple peaks, each associated to one relatively coherent fiber population (bundle). Parameterizing these peaks one is able to disentangle and characterize these bundles. In this work, the fODF peaks are approximated by Bingham distributions, capturing first and second order statistics of the fiber orientations, from which metrics for the parametric quantification of fiber bundles are derived. Meaningful relationships between these measures and the underlying microstructural properties are proposed. The focus lies on metrics derived directly from properties of the Bingham distribution, such as peak length, peak direction, peak spread, integral over the peak, as well as a metric derived from the comparison of the largest peaks, which probes the complexity of the underlying microstructure. These metrics are compared to the conventionally used fractional anisotropy (FA) and it is shown how they may help to increase the specificity of the characterization of microstructural properties. Visualization of the micro-structural arrangement is another application of dMRI. This is done by using tractography to propagate the fiber layout, extracted from the local model, in each voxel. In practice most tractography algorithms use little of the additional information gained from HARDI based local models aside from the reconstructed fiber bundle directions. In this work an approach to tractography based on the Bingham parameterization of the fODF is introduced. For each of the fiber populations present in a voxel the diffusion signal and tensor are computed. Then tensor deflection tractography is performed. This allows incorporating the complete bundle information, performing local interpolation as well as using multiple directions per voxel for generating tracts. Another aspect of this work is the investigation of the spherical harmonic representation which is used most commonly for the fODF by means of the parameters derived from the Bingham distribution fit. Here a strong connection between the approximation errors in the spherical representation of the Dirac delta function and the distribution of crossing angles recovered from the fODF was discovered. The final aspect of this work is the application of the metrics derived from the Bingham fit to a number of fetal datasets for quantifying the brain’s development. This is done by introducing the Gini-coefficient as a metric describing the brain’s age.

As Semantic Web technology evolves many open areas emerge, which attract more research focus. In addition to quickly expanding Linked Open Data (LOD) cloud, various embeddable metadata formats (e.g. RDFa, microdata) are becoming more common. Corporations are already using existing Web of Data to create new technologies that were not possible before. Watson by IBM an artificial intelligence computer system capable of answering questions posed in natural language can be a great example. On the other hand, ubiquitous devices that have a large number of sensors and integrated devices are becoming increasingly powerful and fully featured computing platforms in our pockets and homes. For many people smartphones and tablet computers have already replaced traditional computers as their window to the Internet and to the Web. Hence, the management and presentation of information that is useful to a user is a main requirement for today’s smartphones. And it is becoming extremely important to provide access to the emerging Web of Data from the ubiquitous devices. In this thesis we investigate how ubiquitous devices can interact with the Semantic Web. We discovered that there are five different approaches for bringing the Semantic Web to ubiquitous devices. We have outlined and discussed in detail existing challenges in implementing this approaches in section 1.2. We have described a conceptual framework for ubiquitous semantic applications in chapter 4. We distinguish three client approaches for accessing semantic data using ubiquitous devices depending on how much of the semantic data processing is performed on the device itself (thin, hybrid and fat clients). These are discussed in chapter 5 along with the solution to every related challenge. Two provider approaches (fat and hybrid) can be distinguished for exposing data from ubiquitous devices on the Semantic Web. These are discussed in chapter 6 along with the solution to every related challenge. We conclude our work with a discussion on each of the contributions of the thesis and propose future work for each of the discussed approach in chapter 7.