This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes. Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part? In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part. Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4. In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane. Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope $P_G$ of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of $P_G$, and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction 1. Background 2. Tropical Positivity and Determinantal Varieties 3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes 4. Combinatorics of Correlated Equilibria

Attempts to convince humanities scholars of digital approaches are met with resistance, often. The so-called Digitization Anxiety is the phenomenon that describes the fear of many traditional scientists of being replaced by digital processes. This hinders not only the progress of the scientific domains themselves – since a lot of digital potential is missing – but also makes the everyday work of researchers unnecessarily difficult. Over the past eight years, we have made various attempts to walk the tightrope between 'How can we help traditional humanities to exploit their digital potential?' and 'How can we make them understand that their expertise is not replaced by digital means, but complemented?' We will present our successful interdisciplinary collaborations: How they came about, how they developed, and the problems we encountered. In the first step, we will look at the theoretical basics, which paint a comprehensive picture of the digital humanities and introduces us to the topic of visualization. The field of visualization has shown a special ability: It manages to walk the tightrope and thus keeps digitization anxiety at bay, while not only making it easier for scholars to access their data, but also enabling entirely new research questions. After an introduction to our interdisciplinary collaborations with the Musical Instrument Museum of Leipzig University, as well as with the Bergen-Belsen Memorial, we will present a series of user scenarios that we have collected in the course of 13 publications. These show our cooperation partners solving different research tasks, which we classify using Brehmer and Munzner’s Task Classification. In this way, we show that we provide researchers with a wide range of opportunities: They can answer their traditional research questions – and in some cases verify long-standing hypotheses about the data for the first time – but also develop their own interest in previously impossible, new research questions and approaches. Finally, we conclude our insights on individual collaborative ideas with perspectives on our newest projects. These have risen from the growing interest of collaborators in the methods we deliver. For example, we get insights into the music of real virtuosos of the 20th century. The necessary music storage media can be heard for the first time through digital tools without risking damage to the old material. In addition, we can provide computer-aided analysis capabilities that help musicologists in their work. In the course of the visualization project at the Bergen-Belsen memorial, we will see that what was once a small diary project has grown into a multimodal and international project with institutions of culture and science from eight countries. This is dedicated not only to the question of preserving cultural objects from Nazi persecution contexts but also to modern ways of disseminating and processing knowledge around this context. Finally, we will compile our experience and accumulated knowledge in the form of problems and challenges at the border between computer science and traditional humanities. These will serve as preparation and assistance for future and current interested parties of such interdisciplinary collaborative projects

The goal of this thesis is to understand the deviatoric decomposition of tensors of higher order in 2 and 3 dimensions. In the first chapter an introduction to tensor algebra will be given. Chapter 2 and 3 concentrate on establishing a recursive formula for the deviatoric decomposition in 2D and 3D, respectively. This recursive formula is the key to prove by induction the existense of a deviatoric decomposition for any tensor. Useful examples will also be given at the end of each chapter.:Introduction 1. Introduction to Tensor Algebra 2. Orthogonal Irreducible Decomposition for 2D Tensors 3. Orthogonal Irreducible Decomposition for 3D Tensors 4. Conclusion Bibliography 5. Declaration of Originality

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry.

No description available.

This thesis studies various aspects of differential algebra, from fundamental concepts to practical computations. A characteristic feature of this work is the use of combinatorial techniques, which offer a unique and original perspective on the subject matter. First, we establish the connection between the n-jet space of the fat point defined by xm and the stable set polytope of a perfect graph. We prove that the dimension of the coordinate ring of the scheme defined by polynomial arcs of degree less than or equal to n is a polynomial in m of degree n + 1. This is based on Zobnin’s result which states that the set {x^m} is a differential Gr ̈obner basis for its differential ideal. We generalize this statement to the case of two independent variables and link the dimensions in this case to some triangulations of the p × q rectangle, where the pair (p, q) now plays the role of n. Second, we study the arc space of the fat point x^m on a line from the point of view of filtration by finite-dimensional differential algebras. We prove that the generating series of the dimensions of these differential algebras is m/(1 -mt) . Based on this we propose a definition of the multiplicity of a solution of an algebraic differential equation as the growth of the dimensions of these differential algebras. This generalizes the concept of the multiplicity of an ideal in a polynomial ring. Furthermore, we determine a full description of the set of standard monomials of the differential ideal generated by x^m. This description proves a conjecture by Afsharijoo concerning a new version of the Roger-Ramanujan identities. Every homogeneous linear system of partial differential equations with constant coef- ficients can be encoded by a submodule of the ring of polynomials. We develop practical methods for computing the space of solutions to these PDEs. These spaces are typically infinite dimensional, and we use the Ehrenpreis–Palamodov Theorem for finite encoding. We apply this finite encoding to the solutions of the PDEs associated with the arc spaces of a double point. We prove that these vector spaces are spanned by determinants of some special Wronskians, and we relate them to differentially homogeneous polynomials. Finally, we introduce D-algebraic functions: they are solutions to algebraic differential equations. We study closure properties of these functions. We present practical algorithms and their implementations for carrying out arithmetic operations on D-algebraic functions. This amounts to solving elimination problems for differential ideals.

The Semantic Web has ushered in a vast repository of openly available data across various domains, resulting in over ten thousand Knowledge Graphs (KGs) published under the Linked Open Data (LOD) cloud. However, the exploration of these KGs can be time-consuming and resource-intensive, compounded by issues of availability and duplication across distributed and decentralized databases. Addressing these challenges, this thesis investigates methods for improving information retrieval over Linked Data (LD) through conceptual approaches facilitating access via formal and natural language queries. First, RDFSlice is introduced to efficiently select relevant fragments of RDF data from distributed KGs, demonstrating superior performance compared to conventional methods. Second, a novel distributed and decentralized publishing architecture is proposed to simplify data sharing and querying, enhancing reliability and efficiency. Third, a benchmark for evaluating ranking functions for RDF data is created, leading to the development of new ranking functions such as DBtrends and MIXED-RANK. Fourth, a scoring function based on Term Networks is proposed for interpreting factual queries, outperforming traditional information retrieval methods. Lastly, user interface patterns are discussed, and an extension for semantic search is proposed to improve information access in the face of the vast amounts of data available on the LOD cloud. These contributions collectively address key challenges in accessing and utilizing RDF data, offering insights and solutions to facilitate efficient information retrieval and exploration in the Semantic Web era.

No description available.

Recent advances in tractography allow for connectomes to be constructed in vivo. These have applications for example in brain tumor surgery and understanding of brain development and diseases. The large size of the data produced by these methods lead to a variety problems, including how to evaluate tractography outputs, development of faster processing algorithms for tractography and clustering, and the development of advanced visualization methods for verification and exploration. This thesis presents several advances in these fields. First, an evaluation is presented for the robustness to noise of multiple commonly used tractography algorithms. It employs a Monte–Carlo simulation of measurement noise on a constructed ground truth dataset. As a result of this evaluation, evidence for obustness of global tractography is found, and algorithmic sources of uncertainty are identified. The second contribution is a fast clustering algorithm for tractography data based on k–means and vector fields for representing the flow of each cluster. It is demonstrated that this algorithm can handle large tractography datasets due to its linear time and memory complexity, and that it can effectively integrate interrupted fibers that would be rejected as outliers by other algorithms. Furthermore, a visualization for the exploration of structural connectomes is presented. It uses illustrative rendering techniques for efficient presentation of connecting fiber bundles in context in anatomical space. Visual hints are employed to improve the perception of spatial relations. Finally, a visualization method with application to exploration and verification of probabilistic tractography is presented, which improves on the previously presented Fiber Stippling technique. It is demonstrated that the method is able to show multiple overlapping tracts in context, and correctly present crossing fiber configurations.

No description available.