No description available.

Systems of polynomial equations appear both in mathematics, as well as in many applications in the sciences, economics and engineering. Solving these systems is at the heart of computational algebraic geometry, a field which is often associated with symbolic computations based on Gr¨obner bases. Over the last thirty years, increasing performance and versatility made numerical algebraic geometry emerge as an alternative. It enables us to solve problems which are infeasible with symbolic methods. The focus of this thesis is the rich interplay between algebraic geometry, numerical computation and optimization in various instances. As a first application of algebraic geometry, we investigate global optimization problems whose objective function and constraints are all described by multivariate polynomials. One of the most important, and also most common, features of real world data is sparsity. We explore the effects of sparsity in global optimization, when exhibited by constraints and objective functions. Exploiting this property can lead to dramatic improvements of computational performance of algorithms. As a second application of geometry we study a particularly structured class of polynomial programs which stems from the optimization of sequencial decision rules. In the framework of partially observable Markov decision rules, an agent manipulates a system in a sequence of events. It selects an action at every time step, which in turn influences the state of the system at the next time step, and depending on the state it receives an instantaneous reward. Optimizing the long term reward has a long-standing history in computer science, economics and statistics. The ability to incorporate nondeterministic effects makes the framework particularly well suited for real world applications. We initiate a novel, geometric perspective on the underlying optimization problem and explore algorithmic consequences. As a third application of geometry we present the usage of tropical geometry in order to numerically compute defining equations of unirational varieties from their parametrization. Tropical geometry is an emerging field in mathematics at the boundary of discrete geometry and algebraic geometry. The tropicalization of a variety is a polyhedral complex which encodes geometric information of the variety. Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its coefficients. We use this as a preprocessing step for numerical computation. Contrary to the above uses of geometry in application, we also employ numerical computation in pure mathematics. When relying on numerical methods, problems can be solved that are infeasible with symbolic methods, but the computational results lack a certificate for correctness. This often hinders the application of numerical computation with the purpose of proving mathematical theorems. With this in mind, we develop interval arithmetic as a practical tool for certification in numerical algebraic geometry.:1. Introduction 2. Certifying zeros of polynomial systems using interval arithmetic 3. Algebraic methods in decision processes 4. Discriminants and tropical implicitization

The aim of this work is to lay down a formalism for parametric models that encapsulates both Classical and Quantum Information Geometry. This will be done introducing parametric models on spaces of normal positive linear functionals on W*-algebras and providing a way of defining a Riemannian structure on this models that comes from the Jordan product of the W*-algebra. This Riemannian structure will have some features that are appealing from the viewpoint of Information Geometry. After introducing this W*-algebraic framework, we will move to Estimation Theory. We will see how and to what extent it is possible to formulate in this framework two well-known statistical bounds: the Cramér-Rao bound and the Helstrom bound. Finally, we will explicitly construct some examples that show how it is possible to reduce this general framework to obtain well-known structures in Classical and Quantum Information Geometry.

In recent years, algebraic geometry (both complex and real) has proven to be useful in numerous applications in optimization, statistics, quantum information, and physics. In this thesis, we concentrate on studying semi-algebraic sets and varieties defined over the real numbers that arise in these applied contexts. We begin with the study of Gibbs manifolds and Gibbs varieties. Gibbs manifolds are images of affine spaces of symmetric matrices under the matrix exponential map. They appear naturally in the context of entropic regularization for semidefinite programming or entropy maximization in quantum information theory. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold. We compute these polynomials and show that the Gibbs variety is low-dimensional. We give an exact formula for this dimension, and an upper bound for the degree of the Gibbs variety. We apply our theory to a range of scenarios: matrix pencils, quantum optimal transport, and sparse matrices. The role of Gibbs manifolds in quantum information theory leads us to consider the notion of quantum conditional independence from an algebraic perspective. We take inspiration from algebraic statistics, where graphical models encoding conditional independence relations can be described as intersections of an algebraic variety with the probability simplex, and study quantum counterparts of such models. We present several ways to associate an algebraic variety to such a model. We study basic properties of these varieties and provide algorithms to compute their defining equations. We also study toric varieties defined by commuting Hamiltonians arising from a graph in the context of stabilizer codes. We give an efficient algorithm to compute the defining equations of such a toric variety. Moreover, we investigate a quantum analog of maximum likelihood estimation for quantum exponential families, the so-called quantum information projection. We continue with studying (semi-)algebraic geometry of minimizing dual volumes of polytopes. Similarly to Gibbs manifolds, this appears in the context of regularization of convex optimization problems. The interior point of a convex polytope that leads to a polar dual of minimal volume is called the Santaló point. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set called the Santaló patchwork. We describe and compute this set. We then investigate several naturally defined algebraic varieties containing the Santaló patchwork. We continue by treating the question of computing the Santaló points of polytopes numerically. We also explore connections to physics and algebraic statistics. Finally, we study Grasstopes. This is yet another class of semi-algebraic sets inspired by physics. These are linear projections of the positive Grassmannian. When the linear projection is given by a totally positive matrix, we recover the definition of the amplituhedron, a semi-algebraic set that computes scattering amplitudes in a certain quantum field theory. We concentrate on the case when the image lives in the projective space, and give a combinatorial characterization of such Grasstopes in terms of sign flips, extending the results of Karp and Williams for the amplituhedron.

Fibril formation of amyloid β (Aβ) peptides is one of the key molecular events connected to Alzheimer's disease. The pathway of formation and mechanism of action of Aβ aggregates in biological systems is still object of very active research. To this end, systematic modifications of the Phe19-Leu34 hydrophobic contact, which has been reported in almost all structural studies of Aβ40 fibrils, helps understanding Aβ folding pathways and the underlying free energy landscape of the amyloid formation process. In our approach, a series of Aβ40 peptide variants with two types of backbone modifications, namely incorporation of (i) a methylene or an ethylene spacer group and (ii) a N-methylation at the amide functional group, of the amino acids at positions 19 or 34 was applied. These mutations are expected to challenge the inter-β-strand side chain contacts as well as intermolecular backbone β-sheet hydrogen bridges. Using a multitude of biophysical methods, it is shown that these backbone modifications lead, in most of the cases, to alterations in the fibril formation kinetics, a higher local structural heterogeneity, and a somewhat modified fibril morphology without generally impairing the fibril formation capacity of the peptides. The toxicological profile found for the variants depend on the type and extent of the modification.

Metapherntheorie betrachtet die Metapher als einen Ort der Bedeutungsentstehung in der Sprache. Die Konzeptuelle Metapherntheorie (CMT), die Teile dieser Überlegungen im Rahmen der kognitiven Linguistik formuliert, sieht in Metaphern die linguistische Ausprägung zugrundeliegender kognitiver Mechanismen, durch die abstrakte Themen durch Übertragungen aus konkreten Domainen strukturiert werden. Diese Arbeit analysiert systematisch das der englischen Sprache zugrunde liegende metaphorische Netzwerk durch die Analyse des MappingMetaphor Datensatzes. Mittels Methoden der Netzwerkanalyse, werden verschiedene Grundannahmen der CMT empirisch untersucht. Darauf aufbauend wird CMT mit kontinentalen Ansätzen zur Metapher zusammengeführt und schematisch erweitert.:1 Introduction 1.1 Metaphor Research in the Context of Computational Science 1.2 Cognitive Linguistics 1.3 Conceptual Metaphor Theory 1.4 Methodological Problems in CMT 1.5 Aims and Structure of the Thesis 2 Materials and Methods 2.1 Mapping Metaphor Data 2.1.1 Limitations 2.2 Data Representation 2.2.1 Directed Graph Representation 2.2.2 Directed Multigraph Representation 2.2.3 Directed Hypergraph Representation 2.2.4 In- and Out-Degree 2.3 Network Models 2.3.1 Erdős-Rényi Model 2.3.2 Directed Configuration Model 2.3.3 Random Graph Model with Edge Preferential Attachment 2.3.4 Linear Hyperedge Growth Model 2.3.5 Preferential Attachment Hyperedge Growth Model 2.3.6 Linear Head to Tail Relation Growth Model 2.4 Properties of Hypergraph Growth Models 2.4.1 Cardinality Distribution of the Linear Hyperedge Growth Model 2.4.2 Cardinality Distribution of the Preferential Attachment Hyperedge Growth Model 2.4.3 Cardinality and Target to Source Ratio for the Linear Relation Growth Model 2.5 Network Analysis 2.5.1 Adjacency Matrix 2.5.2 Motif Analysis 2.5.3 Transitivity 2.5.4 Symmetry 2.5.5 Forman–Ricci Curvature 2.5.6 Ollivier–Ricci Curvature 2.5.7 Anti-community Detection 2.5.8 Hirarchical Cluster Analysis 2.5.9 Fasttext Word Embedding 2.5.10 Time Dependency and Amount of Data 3 Metaphor Mapping Analysis 3.1 Category Activity 3.1.1 Non-Randomness of Metaphoric Activity 3.1.2 Classes of most Divers Sources and Targets 3.1.3 Age of Acquisition, Concreteness and Familiarity 3.1.4 Metaphor Density of Categories 3.2 Global Metaphor Network Form 3.2.1 Diaphoric Tension: Anti-Communities 3.2.2 Curvature 3.3 Conceptual Connections 3.3.1 Persistence of Metaphorical Mappings 3.3.2 Mapping Symmetry 3.3.3 Mapping Transitivity 3.3.4 Conceptual Knots 3.3.5 Semasiological and Onomasiological Evolution 3.4 Word-meaning Dynamics 3.4.1 Highest Flexibility Words 3.4.2 Models of Hyperedge Growth 3.4.3 Target to Source Ratio 3.5 Semantic Role Structure 4 Discussion 4.1 Influence Between Cognition, Experience and Culture 4.2 The Role of Space 4.3 Formal Structure Analogy 4.4 Between Similarity and Tension 4.5 Catalytic Sphere of Concepts 5 Conclusion and Outlook A Category Attributes 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.

No description available.