The knowledge of structural properties in microscopic materials contributes to a deeper understanding of macroscopic properties. For the study of such materials, several imaging techniques reaching scales in the order of nanometers have been developed. One of the most powerful and sophisticated imaging methods is focused-ion-beam scanning electron microscopy (FIB-SEM), which combines serial sectioning by an ion beam and imaging by a scanning electron microscope. FIB-SEM imaging reaches extraordinary scales below 5 nm with large representative volumes. However, the complexity of the imaging process results in the addition of artificial distortion and artifacts generating poor-quality images. We introduce a method for the quality evaluation of images by analyzing general characteristics of the images as well as artifacts exclusively for FIB-SEM, namely curtaining and charging. For the evaluation, we propose quality indexes, which are tested on several data sets of porous and non-porous materials with different characteristics and distortions. The quality indexes report objective evaluations in accordance with visual judgment. Moreover, the acquisition of large volumes at high resolution can be time-consuming. An approach to speed up the imaging is by decreasing the resolution and by considering cuboidal voxel configurations. However, non-isotropic resolutions may lead to errors in the reconstructions. Even if the reconstruction is correct, effects are visible in the analysis. We study the effects of different voxel settings on the prediction of material and flow properties of reconstructed structures. Results show good agreement between highly resolved cases and ground truths as is expected. Structural anisotropy is reported as resolution decreases, especially in anisotropic grids. Nevertheless, gray image interpolation remedies the induced anisotropy. These benefits are visible at flow properties as well. For highly porous structures, the structural reconstruction is even more difficult as a consequence of deeper parts of the material visible through the pores. We show as an application example, the reconstruction of two highly porous structures of optical layers, where a typical workflow from image acquisition, preprocessing, reconstruction until a spatial analysis is performed. The study case shows the advantages of 3D imaging for optical porous layers. The analysis reveals geometrical structural properties related to the manufacturing processes.

The main objects of study in this thesis are abelian varieties and their endomorphism rings. Abelian varieties are not just interesting in their own right, they also have numerous applications in various areas such as in algebraic geometry, number theory and information security. In fact, they make up one of the best choices in public key cryptography and more recently in post-quantum cryptography. Endomorphism rings are objects attached to abelian varieties. Their computation plays an important role in explicit class field theory and in the security of some post-quantum cryptosystems. There are subexponential algorithms to compute the endomorphism rings of abelian varieties of dimension one and two. Prior to this work, all these subexponential algorithms came with a probability of failure and additional steps were required to unconditionally prove the output. In addition, these methods do not cover all abelian varieties of dimension two. The objective of this thesis is to analyse the subexponential methods and develop ways to deal with the exceptional cases. We improve the existing methods by developing algorithms that always output the correct endomorphism ring. In addition to that, we develop a novel approach to compute endomorphism rings of some abelian varieties that could not be handled before. We also prove that the subexponential approaches are simply not good enough to cover all the cases. We use some of our results to construct a family of abelian surfaces with which we build post-quantum cryptosystems that are believed to resist subexponential quantum attacks - a desirable property for cryptosystems. This has the potential of providing an efficient non interactive isogeny based key exchange protocol, which is also capable of resisting subexponential quantum attacks and will be the first of its kind.

An increasing number of nowadays tasks, such as speech recognition, image generation, translation, classification or prediction are performed with the help of machine learning. Especially artificial neural networks (ANNs) provide convincing results for these tasks. The reasons for this success story are the drastic increase of available data sources in our more and more digitalized world as well as the development of remarkable ANN architectures. This development has led to an increasing number of model parameters together with more and more complex models. Unfortunately, this yields a loss in the interpretability of deployed models. However, there exists a natural desire to explain the deployed models, not just by empirical observations but also by analytical calculations. In this thesis, we focus on variational autoencoders (VAEs) and foster the understanding of these models. As the name suggests, VAEs are based on standard autoencoders (AEs) and therefore used to perform dimensionality reduction of data. This is achieved by a bottleneck structure within the hidden layers of the ANN. From a data input the encoder, that is the part up to the bottleneck, produces a low dimensional representation. The decoder, the part from the bottleneck to the output, uses this representation to reconstruct the input. The model is learned by minimizing the error from the reconstruction. In our point of view, the most remarkable property and, hence, also a central topic in this thesis is the auto-pruning property of VAEs. Simply speaking, the auto-pruning is preventing the VAE with thousands of parameters from overfitting. However, such a desirable property comes with the risk for the model of learning nothing at all. In this thesis, we look at VAEs and the auto-pruning from two different angles and our main contributions to research are the following: (i) We find an analytic explanation of the auto-pruning. We do so, by leveraging the framework of generalized linear models (GLMs). As a result, we are able to explain training results of VAEs before conducting the actual training. (ii) We construct a time dependent VAE and show the effects of the auto-pruning in this model. As a result, we are able to model financial data sequences and estimate the value-at-risk (VaR) of associated portfolios. Our results show that we surpass the standard benchmarks for VaR estimation.

LinTim is a scientific software toolbox that has been under development since 2007, giving the possibility to solve the various planning steps in public transportation. Although the name originally derives from "Lineplanning and Timetabling", the available functions have grown far beyond this scope. This document is the documentation for version 2021.10. For more information, see https://www.lintim.net

The great interest in robust covering problems is manifold, especially due to the plenitude of real world applications and the additional incorporation of uncertainties which are inherent in many practical issues. In this thesis, for a fixed positive integer \(q\), we introduce and elaborate on a new robust covering problem, called Robust Min-\(q\)-Multiset-Multicover, and related problems. The common idea of these problems is, given a collection of subsets of a ground set, to decide on the frequency of choosing each subset so as to satisfy the uncertain demand of each overall occurring element. Yet, in contrast to general covering problems, the subsets may only cover at most \(q\) of their elements. Varying the properties of the occurring elements leads to a selection of four interesting robust covering problems which are investigated. We extensively analyze the complexity of the arising problems, also for various restrictions to particular classes of uncertainty sets. For a given problem, we either provide a polynomial time algorithm or show that, unless \(\text{P}=\text{NP}\), such an algorithm cannot exist. Furthermore, in the majority of cases, we even give evidence that a polynomial time approximation scheme is most likely not possible for the hard problem variants. Moreover, we aim for approximations and approximation algorithms for these hard variants, where we focus on Robust Min-\(q\)-Multiset-Multicover. For a wide class of uncertainty sets, we present the first known polynomial time approximation algorithm for Robust Min-\(q\)-Multiset-Multicover having a provable worst-case performance guarantee.

Dealing with uncertain structures or data has lately been getting much attention in discrete optimization. This thesis addresses two different areas in discrete optimization: Connectivity and covering. When discussing uncertain structures in networks it is often of interest to determine how many vertices or edges may fail in order for the network to stay connected. Connectivity is a broad, well studied topic in graph theory. One of the most important results in this area is Menger's Theorem which states that the minimum number of vertices needed to separate two non-adjacent vertices equals the maximum number of internally vertex-disjoint paths between these vertices. Here, we discuss mixed forms of connectivity in which both vertices and edges are removed from a graph at the same time. The Beineke Harary Conjecture states that for any two distinct vertices that can be separated with k vertices and l edges but not with k-1 vertices and l edges or k vertices and l-1 edges there exist k+l edge-disjoint paths between them of which k+1 are internally vertex-disjoint. In contrast to Menger's Theorem, the existence of the paths is not sufficient for the connectivity statement to hold. Our main contribution is the proof of the Beineke Harary Conjecture for the case that l equals 2. We also consider different problems from the area of facility location and covering. We regard problems in which we are given sets of locations and regions, where each region has an assigned number of clients. We are now looking for an allocation of suppliers into the locations, such that each client is served by some supplier. The notable difference to other covering problems is that we assume that each supplier may only serve a fixed number of clients which is not part of the input. We discuss the complexity and solution approaches of three such problems which vary in the way the clients are assigned to the suppliers.

Life insurance companies are asked by the Solvency II regime to retain capital requirements against economically adverse developments. This ensures that they are continuously able to meet their payment obligations towards the policyholders. When relying on an internal model approach, an insurer's solvency capital requirement is defined as the 99.5% value-at-risk of its full loss probability distribution over the coming year. In the introductory part of this thesis, we provide the actuarial modeling tools and risk aggregation methods by which the companies can accomplish the derivations of these forecasts. Since the industry still lacks the computational capacities to fully simulate these distributions, the insurers have to refer to suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. We dedicate the first part of this thesis to establishing a theoretical framework of the LSMC method. We start with how LSMC for calculating capital requirements is related to its original use in American option pricing. Then we decompose LSMC into four steps. In the first one, the Monte Carlo simulation setting is defined. The second and third steps serve the calibration and validation of the proxy function, and the fourth step yields the loss distribution forecast by evaluating the proxy model. When guiding through the steps, we address practical challenges and propose an adaptive calibration algorithm. We complete with a slightly disguised real-world application. The second part builds upon the first one by taking up the LSMC framework and diving deeper into its calibration step. After a literature review and a basic recapitulation, various adaptive machine learning approaches relying on least-squares regression and model selection criteria are presented as solutions to the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over GLM and GAM methods to MARS and kernel regression routines. We justify the combinability of the regression ingredients mathematically and compare their approximation quality in slightly altered real-world experiments. Thereby, we perform sensitivity analyses, discuss numerical stability and run comprehensive out-of-sample tests. The scope of the analyzed regression variants extends to other high-dimensional variable selection applications. Life insurance contracts with early exercise features can be priced by LSMC as well due to their analogies to American options. In the third part of this thesis, equity-linked contracts with American-style surrender options and minimum interest rate guarantees payable upon contract termination are valued. We allow randomness and jumps in the movements of the interest rate, stochastic volatility, stock market and mortality. For the simultaneous valuation of numerous insurance contracts, a hybrid probability measure and an additional regression function are introduced. Furthermore, an efficient seed-related simulation procedure accounting for the forward discretization bias and a validation concept are proposed. An extensive numerical example rounds off the last part.

This dissertation consists of three independent parts: The yield curve shapes generated by interest rate models, the yield curve forecasting, and the application of the chance-risk classification to a portfolio of pension products. As a component of the capital market model, the yield curve influences the chance-risk classification which was introduced to improve the comparability of pension products and strengthen consumer protection. Consequently, all three topics have a major impact on this essential safeguard. Firstly, we focus on the obtained yield curve shapes of the Vasicek interest rate models. We extend the existing studies on the attainable yield curve shapes in the one-factor Vasicek model by analysis of the curvature. Further, we show that the two-factor Vasicek model can explain significantly more effects that are observed at the market than its one-factor variant. Among them is the occurrence of dipped yield curves. We further introduce a general change of measure framework for the Monte Carlo simulation of the Vasicek model under a subjective measure. This can be used to avoid the occurrence of a far too high frequency of inverse yield curves with growing time. Secondly, we examine different time series models including machine learning algorithms forecasting the yield curve. For this, we consider statistical time series models such as autoregression and vector autoregression. Their performances are compared with the performance of a multilayer perceptron, a fully connected feed-forward neural network. For this purpose, we develop an extended approach for the hyperparameter optimization of the perceptron which is based on standard procedures like Grid and Random Search but allows to search a larger hyperparameter space. Our investigation shows that multilayer perceptrons outperform statistical models for long forecast horizons. The third part deals with the chance-risk classification of state-subsidized pension products in Germany as well as its relevance for customer consulting. To optimize the use of the chance-risk classes assigned by Produktinformationsstelle Altersvorsorge gGmbH, we develop a procedure for determining the chance-risk class of different portfolios of state-subsidized pension products under the constraint that the portfolio chance-risk class does not exceed the customer's risk preference. For this, we consider a portfolio consisting of two new pension products as well as a second one containing a product already owned by the customer as well as the offer of a new one. This is of particular interest for customer consulting and can include other assets of the customer. We examine the properties of various chance and risk parameters as well as their corresponding mappings and show that a diversification effect exists. Based on the properties, we conclude that the average final contract values have to be used to obtain the upper bound of the portfolio chance-risk class. Furthermore, we develop an approach for determining the chance-risk class over the contract term since the chance-risk class is only assigned at the beginning of the accumulation phase. On the one hand, we apply the current legal situation, but on the other hand, we suggest an approach that requires further simulations. Finally, we translate our results into recommendations for customer consultation.

In diesem Text werden einige wichtige Grundlagen zusammengefasst, mit denen ein schneller Einstieg in das Arbeiten mit Arduino und Raspberry Pi möglich ist. Wir diskutieren nicht die Grundfunktionen der Geräte, weil es dafür zahlreiche Hilfestellungen im Internet gibt. Stattdessen konzentrieren wir uns vor allem auf die Steuerung von Sensoren und Aktoren und diskutieren einige Projektideen, die den MINT-interdisziplinären Projektunterricht bereichern können.

Skript zur Vorlesung "Character Theory of finite groups".

Simulating the flow of water in district heating networks requires numerical methods which are independent of the CFL condition. We develop a high order scheme for networks of advection equations allowing large time steps. With the MOOD technique unphysical oscillations of non smooth solutions are avoided. In numerical tests the applicability to real networks is shown.

Laser-induced interstitial thermotherapy (LITT) is a minimally invasive procedure to destroy liver tumors through thermal ablation. Mathematical models are the basis for computer simulations of LITT, which support the practitioner in planning and monitoring the therapy. In this thesis, we propose three potential extensions of an established mathematical model of LITT, which is based on two nonlinearly coupled partial differential equations (PDEs) modeling the distribution of the temperature and the laser radiation in the liver. First, we introduce the Cattaneo–LITT model for delayed heat transfer in this context, prove its well-posedness and study the effect of an inherent delay parameter numerically. Second, we model the influence of large blood vessels in the heat-transfer model by means of a spatially varying blood-perfusion rate. This parameter is unknown at the beginning of each therapy because it depends on the individual patient and the placement of the LITT applicator relative to the liver. We propose a PDE-constrained optimal-control problem for the identification of the blood-perfusion rate, prove the existence of an optimal control and prove necessary first-order optimality conditions. Furthermore, we introduce a numerical example based on which we demonstrate the algorithmic solution of this problem. Third, we propose a reformulation of the well-known PN model hierarchy with Marshak boundary conditions as a coupled system of second-order PDEs to approximate the radiative-transfer equation. The new model hierarchy is derived in a general context and is applicable to a wide range of applications other than LITT. It can be generated in an automated way by means of algebraic transformations and allows the solution with standard finite-element tools. We validate our formulation in a general context by means of various numerical experiments. Finally, we investigate the coupling of this new model hierarchy with the LITT model numerically.

The construction of number fields with given Galois group fits into the framework of the inverse Galois problem. This problem remains still unsolved, although many partial results have been obtained over the last century. Shafarevich proved in 1954 that every solvable group is realizable as the Galois group of a number field. Unfortunately, the proof does not provide a method to explicitly find such a field. This work aims at producing a constructive version of the theorem by solving the following task: given a solvable group $G$ and a $B\in \mathbf N$, construct all normal number fields with Galois group $G$ and absolute discriminant bounded by $B$. Since a field with solvable Galois group can be realized as a tower of abelian extensions, the main role in our algorithm is played by class field theory, which is the subject of the first part of this work. The second half is devoted to the study of the relation between the group structure and the field through Galois correspondence. In particular, we study the existence of obstructions to embedding problems and some criteria to predict the Galois group of an extension.

We study a multi-scale model for growth of malignant gliomas in the human brain. Interactions of individual glioma cells with their environment determine the gross tumor shape. We connect models on different time and length scales to derive a practical description of tumor growth that takes these microscopic interactions into account. From a simple subcellular model for haptotactic interactions of glioma cells with the white matter we derive a microscopic particle system, which leads to a meso-scale model for the distribution of particles, and finally to a macroscopic description of the cell density. The main body of this work is dedicated to the development and study of numerical methods adequate for the meso-scale transport model and its transition to the macroscopic limit.

Diversification is one of the main pillars of investment strategies. The prominent 1/N portfolio, which puts equal weight on each asset is, apart from its simplicity, a method which is hard to outperform in realistic settings, as many studies have shown. However, depending on the number of considered assets, this method can lead to very large portfolios. On the other hand, optimization methods like the mean-variance portfolio suffer from estimation errors, which often destroy the theoretical benefits. We investigate the performance of the equal weight portfolio when using fewer assets. For this we explore different naive portfolios, from selecting the best Sharpe ratio assets to exploiting knowledge about correlation structures using clustering methods. The clustering techniques separate the possible assets into non-overlapping clusters and the assets within a cluster are ordered by their Sharpe ratio. Then the best asset of each portfolio is chosen to be a member of the new portfolio with equal weights, the cluster portfolio. We show that this portfolio inherits the advantages of the 1/N portfolio and can even outperform it empirically. For this we use real data and several simulation models. We prove these findings from a statistical point of view using the framework by DeMiguel, Garlappi and Uppal (2009). Moreover, we show the superiority regarding the Sharpe ratio in a setting, where in each cluster the assets are comonotonic. In addition, we recommend the consideration of a diversification-risk ratio to evaluate the performance of different portfolios.

In this thesis we study a variant of the quadrature problem for stochastic differential equations (SDEs), namely the approximation of expectations \(\mathrm{E}(f(X))\), where \(X = (X(t))_{t \in [0,1]}\) is the solution of an SDE and \(f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R}\) is a functional, mapping each realization of \(X\) into the real numbers. The distinctive feature in this work is that we consider randomized (Monte Carlo) algorithms with random bits as their only source of randomness, whereas the algorithms commonly studied in the literature are allowed to sample from the uniform distribution on the unit interval, i.e., they do have access to random numbers from \([0,1]\). By assumption, all further operations like, e.g., arithmetic operations, evaluations of elementary functions, and oracle calls to evaluate \(f\) are considered within the real number model of computation, i.e., they are carried out exactly. In the following, we provide a detailed description of the quadrature problem, namely we are interested in the approximation of \begin{align*} S(f) = \mathrm{E}(f(X)) \end{align*} for \(X\) being the \(r\)-dimensional solution of an autonomous SDE of the form \begin{align*} \mathrm{d}X(t) = a(X(t)) \, \mathrm{d}t + b(X(t)) \, \mathrm{d}W(t), \quad t \in [0,1], \end{align*} with deterministic initial value \begin{align*} X(0) = x_0 \in \mathbb{R}^r, \end{align*} and driven by a \(d\)-dimensional standard Brownian motion \(W\). Furthermore, the drift coefficient \(a \colon \mathbb{R}^r \to \mathbb{R}^r\) and the diffusion coefficient \(b \colon \mathbb{R}^r \to \mathbb{R}^{r \times d}\) are assumed to be globally Lipschitz continuous. For the function classes \begin{align*} F_{\infty} = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{\sup}\bigr\} \end{align*} and \begin{align*} F_p = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{L_p}\bigr\}, \quad 1 \leq p < \infty. \end{align*} we have established the following. \[\] \(\textit{Theorem 1.}\) There exists a random bit multilevel Monte Carlo (MLMC) algorithm \(M\) using \[ L = L(\varepsilon,F) = \begin{cases}\lceil{\log_2(\varepsilon^{-2}}\rceil, &\text{if} \ F = F_p,\\ \lceil{\log_2(\varepsilon^{-2} + \log_2(\log_2(\varepsilon^{-1}))}\rceil, &\text{if} \ F = F_\infty \end{cases} \] and replication numbers \[ N_\ell = N_\ell(\varepsilon,F) = \begin{cases} \lceil{(L+1) \cdot 2^{-\ell} \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F = F_p,\\ \lceil{(L+1) \cdot 2^{-\ell} \cdot \max(\ell,1) \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F=f_\infty \end{cases} \] for \(\ell = 0,\ldots,L\), for which exists a positive constant \(c\) such that \begin{align*} \mathrm{error}(M,F) = \sup_{f \in F} \bigl(\mathrm{E}(S(f) - M(f))^2\bigr)^{1/2} \leq c \cdot \varepsilon \end{align*} and \begin{align*} \mathrm{cost}(M,F) = \sup_{f \in F} \mathrm{E}(\mathrm{cost}(M,f)) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases} (\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\ (\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty \end{cases} \end{align*} for every \(\varepsilon \in {]0,1/2[}\). \[\] Hence, in terms of the \(\varepsilon\)-complexity \begin{align*} \mathrm{comp}(\varepsilon,F) = \inf\bigl\{\mathrm{cost}(M,F) \colon M \ \text{is a random bit MC algorithm}, \mathrm{error}(M,F) \leq \varepsilon\bigr\} \end{align*} we have established the upper bound \begin{align*} \mathrm{comp}(\varepsilon,F) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases} (\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\ (\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty \end{cases} \end{align*} for some positive constant \(c\). That is, we have shown the same weak asymptotic upper bound as in the case of random numbers from \([0,1]\). Hence, in this sense, random bits are almost as powerful as random numbers for our computational problem. Moreover, we present numerical results for a non-analyzed adaptive random bit MLMC Euler algorithm, in the particular cases of the Brownian motion, the geometric Brownian motion, the Ornstein-Uhlenbeck SDE and the Cox-Ingersoll-Ross SDE. We also provide a numerical comparison to the corresponding adaptive random number MLMC Euler method. A key challenge in the analysis of the algorithm in Theorem 1 is the approximation of probability distributions by means of random bits. A problem very closely related to the quantization problem, i.e., the optimal approximation of a given probability measure (on a separable Hilbert space) by means of a probability measure with finite support size. Though we have shown that the random bit approximation of the standard normal distribution is 'harder' than the corresponding quantization problem (lower weak rate of convergence), we have been able to establish the same weak rate of convergence as for the corresponding quantization problem in the case of the distribution of a Brownian bridge on \(L_2([0,1])\), the distribution of the solution of a scalar SDE on \(L_2([0,1])\), and the distribution of a centered Gaussian random element in a separable Hilbert space.

Lecture notes for the lecture "Cohomology of Groups" SS 2018

During the last couple of years, there has been a variety of publications on the topic of minimum quantity constraints. In general, a minimum quantity constraint is a lower bound constraint on an entity of an optimization problem that only has to be fulfilled if the entity is “used” in the respective solution. For example, if a minimum quantity \(q_e\) is defined on an edge \(e\) of a flow network, the edge flow on \(e\) may either be \(0\) or at least \(q_e\) units of flow. Minimum quantity constraints have already been applied to problem classes such as flow, bin packing, assignment, scheduling and matching problems. A result that is common to all these problem classes is that in the majority of cases problems with minimum quantity constraints are NP-hard, even if the problem without minimum quantity constraints but with fixed lower bounds can be solved in polynomial time. For instance, the maximum flow problem is known to be solvable in polynomial time, but becomes NP-hard once minimum quantity constraints are added. In this thesis we consider flow, bin packing, scheduling and matching problems with minimum quantity constraints. For each of these problem classes we provide a summary of the definitions and results that exist to date. In addition, we define new problems by applying minimum quantity constraints to the maximum-weight b-matching problem and to open shop scheduling problems. We contribute results to each of the four problem classes: We show NP-hardness for a variety of problems with minimum quantity constraints that have not been considered so far. If possible, we restrict NP-hard problems to special cases that can be solved in polynomial time. In addition, we consider approximability of the problems: For most problems it turns out that, unless P=NP, there cannot be any polynomial-time approximation algorithm. Hence, we consider bicriteria approximation algorithms that allow the constraints of the problem to be violated up to a certain degree. This approach proves to be very helpful and we provide a polynomial-time bicriteria approximation algorithm for at least one problem of each of the four problem classes we consider. For problems defined on graphs, the class of series parallel graphs supports this approach very well. We end the thesis with a summary of the results and several suggestions for future research on minimum quantity constraints.

We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction–diffusion–taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced (Kumar, Li and Surulescu, 2020) description of glioma pseudopalisade formation with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high-grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects.

The famous Mather-Yau theorem in singularity theory yields a bijection of isomorphy classes of germs of isolated hypersurface singularities and their respective Tjurina algebras. This result has been generalized by T. Gaffney and H. Hauser to singularities of isolated singularity type. Due to the fact that both results do not have a constructive proof, it is the objective of this thesis to extract explicit information about hypersurface singularities from their Tjurina algebras. First we generalize the result by Gaffney-Hauser to germs of hypersurface singularities, which are strongly Euler-homogeneous at the origin. Afterwards we investigate the Lie algebra structure of the module of logarithmic derivations of Tjurina algebra while considering the theory of graded analytic algebras by G. Scheja and H. Wiebe. We use the aforementioned theory to show that germs of hypersurface singularities with positively graded Tjurina algebras are strongly Euler-homogeneous at the origin. We deduce the classification of hypersurface singularities with Stanley-Reisner Tjurina ideals. The notion of freeness and holonomicity play an important role in the investigation of properties of the aforementioned singularities. Both notions have been introduced by K. Saito in 1980. We show that hypersurface singularities with Stanley--Reisner Tjurina ideals are holonomic and have a free singular locus. Furthermore, we present a Las Vegas algorithm, which decides whether a given zero-dimensional \(\mathbb{C}\)-algebra is the Tjurina algebra of a quasi-homogeneous isolated hypersurface singularity. The algorithm is implemented in the computer algebra system OSCAR.

Skript zur Vorlesung Elementare Zahlentheorie, SS 2017 and SS 2019

Skript zur Vorlesung Einführung in die Algebra, WS 2016/17

LinTim is a scientific software toolbox that has been under development since 2007, giving the possibility to solve the various planning steps in public transportation. Although the name originally derives from "Lineplanning and Timetabling", the available functions have grown far beyond this scope. This document is the documentation for version 2020.12. For more information, see https://www.lintim.net

Die Felix-Klein-Modellierungswoche ist eine Projektwoche für Schülerinnen und Schüler der gymnasialen Oberstufe. Wir beschreiben hier die Projekte, die während der Modellierungswoche im Februar 2020 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Gesundheit und Medizin".

In this thesis, we deal with the worst-case portfolio optimization problem occuring in discrete-time markets. First, we consider the discrete-time market model in the presence of crash threats. We construct the discrete worst-case optimal portfolio strategy by the indifference principle in the case of the logarithmic utility. After that we extend this problem to general utility functions and derive the discrete worst-case optimal portfolio processes, which are characterized by a dynamic programming equation. Furthermore, the convergence of the discrete worst-case optimal portfolio processes are investigated when we deal with the explicit utility functions. In order to further study the relation of the worst-case optimal value function in discrete-time models to continuous-time models we establish the finite-difference approach. By deriving the discrete HJB equation we verify the worst-case optimal value function in discrete-time models, which satisfies a system of dynamic programming inequalities. With increasing degree of fineness of the time discretization, the convergence of the worst-case value function in discrete-time models to that in continuous-time models are proved by using a viscosity solution method.