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Index Insurance for Farmers
(2021)

In this thesis we focus on weather index insurance for agriculture risk. Even though such an index insurance is easily applicable and reduces information asymmetries, the demand for it is quite low. This is in particular due to the basis risk and the lack of knowledge about it’s effectiveness. The basis risk is the difference between the index insurance payout and the actual loss of the insured. We evaluate the performance of weather index insurance in different contexts, because proper knowledge about index insurance will help to use it as a successful alternative for traditional crop insurance. In addition to that, we also propose and discuss methods to reduce the basis risk.
We also analyze the performance of an agriculture loan which is interlinked with a weather index insurance. We show that an index insurance with actuarial fair or subsidized premium helps to reduce the loan default probability. While we first consider an index insurance with a commonly used linear payout function for this analysis, we later design an index insurance payout function which maximizes the expected utility of the insured. Then we show that, an index insurance with that optimal payout function is more appropriate for bundling with an agriculture loan. The optimal payout function also helps to reduce the basis risk. In addition, we show that a lender who issues agriculture loans can be better off by purchasing a weather index insurance in some circumstances.
We investigate the market equilibrium for weather index insurance by assuming risk averse farmers and a risk averse insurer. When we consider two groups of farmers with different risks, we show that the low risk group subsidizes the high risk group when both should pay the same premium for the index insurance. Further, according to the analysis of an index insurance in an informal risk sharing environment, we observe that the demand of the index insurance can be increased by selling it to a group of farmers who informally share the risk based on the insurance payout, because it reduces the adverse effect of the basis risk. Besides of that we analyze the combination of an index insurance with a gap insurance. Such a combination can increase the demand and reduce the basis risk of the index insurance if we choose the correct levels of premium and of gap insurance cover. Moreover our work shows that index insurance can be a good alternative to proportional and excess loss reinsurance when it is issued at a low enough price.

Im Projekt MAFoaM - Modular Algorithms for closed Foam Mechanics - des
Fraunhofer ITWM in Zusammenarbeit mit dem Fraunhofer IMWS wurde eine Methode zur Analyse und Simulation geschlossenzelliger PMI-Hartschäume entwickelt. Die Zellstruktur der Hartschäume wurde auf Basis von CT-Aufnahmen modelliert, um ihr Verformungs- und Versagensverhalten zu simulieren, d.h. wie sich die Schäume unter Belastungen bis hin zum totalen Defekt verhalten.
In der Diplomarbeit wird die
bildanalytische Zellrekonstruktion für PMI-Hartschäume automatisiert. Die Zellrekonstruktion dient der Bestimmung von Mikrostrukturgrößen,
also geometrischer Eigenschaften der Schaumzellen, wie z.B.
Mittelwert und Varianz des Zellvolumens oder der Zelloberfläche.

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 high complexity of civil engineering structures makes it difficult to satisfactorily evaluate their reliability. However, a good risk assessment of such structures is incredibly important to avert dangers and possible disasters for public life. For this purpose, we need algorithms that reliably deliver estimates for their failure probabilities with high efficiency and whose results enable a better understanding of their reliability. This is a major challenge, especially when dynamics, for example due to uncertainties or time-dependent states, must be included in the model.
The contributions are centered around Subset Simulation, a very popular adaptive Monte Carlo method for reliability analysis in the engineering sciences. It particularly well estimates small failure probabilities in high dimensions and is therefore tailored to the demands of many complex problems. We modify Subset Simulation and couple it with interpolation methods in order to keep its remarkable properties and receive all conditional failure probabilities with respect to one variable of the structural reliability model. This covers many sorts of model dynamics with several model constellations, such as time-dependent modeling, sensitivity and uncertainty, in an efficient way, requiring similar computational demands as a static reliability analysis for one model constellation by Subset Simulation. The algorithm offers many new opportunities for reliability evaluation and can even be used to verify results of Subset Simulation by artificially manipulating the geometry of the underlying limit state in numerous ways, allowing to provide correct results where Subset Simulation systematically fails. To improve understanding and further account for model uncertainties, we present a new visualization technique that matches the extensive information on reliability we get as a result from the novel algorithm.
In addition to these extensions, we are also dedicated to the fundamental analysis of Subset Simulation, partially bridging the gap between theory and results by simulation where inconsistencies exist. Based on these findings, we also extend practical recommendations on selection of the intermediate probability with respect to the implementation of the algorithm and derive a formula for correction of the bias. For a better understanding, we also provide another stochastic interpretation of the algorithm and offer alternative implementations which stick to the theoretical assumptions, typically made in analysis.

A significant step to engineering design is to take into account uncertainties and to
develop optimal designs that are robust with respect to perturbations. Furthermore, it
is often of interest to optimize for different conflicting objective functions describing the
quality of a design, leading to a multi-objective optimization problem. In this context,
generating methods for solving multi-objective optimization problems seek to find a
representative set of solutions fulfilling the concept of Pareto optimality. When multiple
uncertain objective functions are involved, it is essential to define suitable measures for
robustness that account for a combined effect of uncertainties in objective space. Many
tasks in engineering design include the solution of an underlying partial differential
equation that can be computationally expensive. Thus, it is of interest to use efficient
strategies for finding optimal designs. This research aims to present suitable measures
for robustness in a multi-objective context, as well as optimization strategies for multi-
objective robust design.
This work introduces new ideas for robustness measures in the context of multi-
objective robust design. Losses and expected losses based on distances in objective space
are used to describe robustness. A direct formulation and a two-phase formulation based
on expected losses are proposed for finding a set of robust optimal solutions.
Furthermore, suitable optimization strategies for solving the resulting multi-objective
robust design problem are formulated and analyzed. The multi-objective optimization
problem is solved with a constraint-based approach that is based on solving several
constrained single-objective optimization problems with a hybrid optimization strategy.
The hybrid method combines a global search method on a surrogate model with adjoint-
based optimization methods. In the context of optimization with an underlying partial
differential equation, a one-shot approach is extended to handle additional constraints.
The developed concepts for multi-objective robust design and the proposed optimiza-
tion strategies are applied to an aerodynamic shape optimization problem. The drag
coefficient and the lift coefficient are optimized under the consideration of uncertain-
ties in the operational conditions and geometrical uncertainties. The uncertainties are
propagated with the help of a non-intrusive polynomial chaos approach. For increasing
the efficiency when considering a higher-dimensional random space, it is made use of a
Karhunen-Loève expansion and a dimension-adaptive sparse grid quadrature.

Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around one or several sites of vaso-occlusion are typical for GBM and hint on poor prognosis of patient survival.
This thesis focuses on studying the establishment and maintenance of these histological patterns specific to GBM with the aim of modeling the microlocal tumor environment under the influence of acidity, tissue anisotropy and hypoxia-induced angiogenesis. This aim is reached with two classes of models: multiscale and multiphase. Each of them features a reaction-diffusion equation (RDE) for the acidity acting as a chemorepellent and inhibitor of growth, coupled in a nonlinear way to a reaction-diffusion-taxis equation (RDTE) for glioma dynamics. The numerical simulations of the resulting systems are able to reproduce pseudopalisade-like patterns. The effect of tumor vascularization on these patterns is studied through a flux-limited model belonging to the multiscale class. Thereby, PDEs of reaction-diffusion-taxis type are deduced for glioma and endothelial cell (EC) densities with flux-limited pH-taxis for the tumor and chemotaxis towards vascular endothelial growth factor (VEGF) for ECs. These, in turn, are coupled to RDEs for acidity and VEGF produced by tumor. The numerical simulations of the obtained system show pattern disruption and transient behavior due to hypoxia-induced angiogenesis. Moreover, comparing two upscaling techniques through numerical simulations, we observe that the macroscopic PDEs obtained via parabolic scaling (directed tissue) are able to reproduce glioma patterns, while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related.

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.

Linear algebra, together with polynomial arithmetic, is the foundation of computer algebra. The algorithms have improved over the last 20 years, and the current state of the art algorithms for matrix inverse, solution of a linear system and determinants have a theoretical sub-cubic complexity. This thesis presents fast and practical algorithms for some classical problems in linear algebra over number fields and polynomial rings. Here, a number field is a finite extension of the field of rational numbers, and the polynomial rings we considered in this thesis are over finite fields.
One of the key problems of symbolic computation is intermediate coefficient swell: the bit length of intermediate results can grow during the computation compared to those in the input and output. The standard strategy to overcome this is not to compute the number directly but to compute it modulo some other numbers, using either the Chinese remainder theorem (CRT) or a variation of Newton-Hensel lifting. Often, the final step of these algorithms is combined with reconstruction methods such as rational reconstruction to convert the integral result into the rational solution. Here, we present reconstruction methods over number fields with a fast and simple vector-reconstruction algorithm.
The state of the art method for computing the determinant over integers is due to Storjohann. When generalizing his method over number field, we encountered the problem that modules generated by the rows of a matrix over number fields are in general not free, thus Strojohann's method cannot be used directly. Therefore, we have used the theory of pseudo-matrices to overcome this problem. As a sub-problem of this application, we generalized a unimodular certification method for pseudo-matrices: similar to the integer case, we check whether the determinant of the given pseudo matrix is a unit by testing the integrality of the corresponding dual module using higher-order lifting.
One of the main algorithms in linear algebra is the Dixon solver for linear system solving due to Dixon. Traditionally this algorithm is used only for square systems having a unique solution. Here we generalized Dixon algorithm for non-square linear system solving. As the solution is not unique, we have used a basis of the kernel to normalize the solution. The implementation is accompanied by a fast kernel computation algorithm that also extends to compute the reduced-row-echelon form of a matrix over integers and number fields.
The fast implementations for computing the characteristic polynomial and minimal polynomial over number fields use the CRT-based modular approach. Finally, we extended Storjohann's determinant computation algorithm over polynomial ring over finite fields, with its sub-algorithms for reconstructions and unimodular certification. In this case, we face the problem of intermediate degree swell. To avoid this phenomenon, we used higher-order lifting techniques in the unimodular certification algorithm. We have successfully used the half-gcd approach to optimize the rational polynomial reconstruction.

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.

In this thesis one considers the periodic homogenization of a linearly coupled magneto-elastic model problem and focuses on the derivation of spectral methods to solve the obtained unit cell problem afterwards. In the beginning, the equations of linear elasticity and magnetism are presented together with the physical quantities used within. After specifying the model assumptions, the system of partial differential equations is rewritten in a weak form for which the existence and uniqueness of solutions is discussed. The model problem then undergoes a homogenization process where the original problem is approximated by a substitute problem with a repeating micro-structural geometry that was generated from a representative volume element (RVE). The following separation of scales, which can be achieved either by an asymptotic expansion or through a two-scale limit process, yields the homogenized problem on the macroscopic scale and the periodic unit cell problem. The latter is further analyzed using Fourier series, leading to periodic Lippmann-Schwinger type equations allowing for the development of matrix-free solvers. It is shown that, while it is possible to craft a scheme for the coupled problem from the purely elastic and magnetic Lippmann-Schwinger equations alone without much additional effort, a more general setting is provided when deriving a Lippmann-Schwinger equation for the coupled system directly. These numerical approaches are then validated with some analytically solvable test problems, before their performance is tested against each other for some more complex examples.

Yield Curves and Chance-Risk Classification: Modeling, Forecasting, and Pension Product Portfolios
(2021)

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.

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.

Deligne-Lusztig theory allows the parametrization of generic character tables of finite groups of Lie type in terms of families of conjugacy classes and families of irreducible characters "independently" of \(q\).
Only in small cases the theory also gives all the values of the table.
For most of the groups the completion of the table must be carried out with ad-hoc methods.
The aim of the present work is to describe one possible computation which avoids Lusztig's theory of "character sheaves".
In particular, the theory of Gel'fand-Graev characters and Clifford theory is used to complete the generic character table of \(G={\rm Spin}_8^+(q)\) for \(q\) odd.
As an example of the computations, we also determine the character table of \({\rm SL}_4(q)\), for \(q\) odd.
In the process of finding character values, the following tools are developed.
By explicit use of the Bruhat decomposition of elements, the fusion of the unipotent classes of \(G\) is determined.
Among others, this is used to compute the 2-parameter Green functions of every Levi subgroup with disconnected centre of \(G\).
Furthermore, thanks to a certain action of the centre \(Z(G)\) on the characters of \(G\), it is shown how, in principle, the values of any character depend on its values at the unipotent elements.
It is important to consider \({\rm Spin}_8^+(q)\) as it is one of the "smallest" interesting examples for which Deligne--Lusztig theory is not sufficient to construct the whole character table.
The reasons is related to the structure of \({\mathbf G}={\rm Spin}_8\), from which \(G\) is constructed.
Firstly, \({\mathbf G}\) has disconnected centre.
Secondly, \({\mathbf G}\) is the only simple algebraic group which has an outer group automorphism of order 3.
And finally, \(G\) can be realized as a subgroup of bigger groups, like \(E_6(q)\), \(E_7(q)\) or \(E_8(q)\).
The computation on \({\rm Spin}_8^+(q)\) serves as preparation for those cases.

Estimation and Portfolio Optimization with Expert Opinions in Discrete-time Financial Markets
(2021)

In this thesis, we mainly discuss the problem of parameter estimation and
portfolio optimization with partial information in discrete-time. In the portfolio optimization problem, we specifically aim at maximizing the utility of
terminal wealth. We focus on the logarithmic and power utility functions. We consider expert opinions as another observation in addition to stock returns to improve estimation of drift and volatility parameters at different times and for the purpose of asset optimization.
In the first part, we assume that the drift term has a fixed distribution, and
the volatility term is constant. We use the Kalman filter to combine the two
types of observations. Moreover, we discuss how to transform this problem
into a non-linear problem of Gaussian noise when the expert opinion is uniformly distributed. The generalized Kalman filter is used to estimate the parameters in this problem.
In the second part, we assume that drift and volatility of asset returns are both driven by a Markov chain. We mainly use the change-of-measure technique to estimate various values required by the EM algorithm. In addition,
we focus on different ways to combine the two observations, expert opinions and asset returns. First, we use the linear combination method. At the same time, we discuss how to use a logistic regression model to quantify expert
opinions. Second, we consider that expert opinions follow a mixed Dirichlet distribution. Under this assumption, we use another probability measure to
estimate the unnormalized filters, needed for the EM algorithm.
In the third part, we assume that expert opinions follow a mixed Dirichlet distribution and focus on how we can obtain approximate optimal portfolio
strategies in different observation settings. We claim the approximate strategies from the dynamic programming equations in different settings and analyze the dependence on the discretization step. Finally we compute different
observation settings in a simulation study.

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.

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.

This thesis consists of two parts, i.e. the theoretical background of (R)ABSDE including basic theorems, theoretical proofs and properties (Chapter 2-4), as well as numerical algorithms and simulations for (R)ABSDES (Chapter 5). For the theoretical part, we study ABSDEs (Chapter 2), RABSDEs with one obstacle (Chapter 3)and RABSDEs with two obstacles (Chapter 4) in the defaultable setting respectively, including the existence and uniqueness theorems, applications, the comparison theorem for ABSDEs, their relations with PDEs and stochastic differential delay equations (SDDE). The numerical algorithm part (Chapter 5) introduces two main algorithms, a discrete penalization scheme and a discrete reflected scheme based on a random walk approximation of the Brownian motion as well as a discrete approximation of the default martingale; we give the convergence results of the algorithms, provide a numerical example and an application in American game options in order to illustrate the performance of the algorithms.

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".

Simplified ODE models describing blood flow rate are governed by the pressure gradient.
However, assuming the orientation of the blood flow in a human body correlates to a positive
direction, a negative pressure gradient forces the valve to shut, which stops the flow through
the valve, hence, the flow rate is zero, whereas the pressure rate is formulated by an ODE.
Presence of ODEs together with algebraic constraints and sudden changes of system characterizations
yield systems of switched differential-algebraic equations (swDAEs). Alternating
dynamics of the heart can be well modelled by means of swDAEs. Moreover, to study pulse
wave propagation in arteries and veins, PDE models have been developed. Connection between
the heart and vessels leads to coupling PDEs and swDAEs. This model motivates
to study PDEs coupled with swDAEs, for which the information exchange happens at PDE
boundaries, where swDAE provides boundary conditions to the PDE and PDE outputs serve
as inputs to swDAE. Such coupled systems occur, e.g. while modelling power grids using
telegrapher’s equations with switches, water flow networks with valves and district
heating networks with rapid consumption changes. Solutions of swDAEs might
include jumps, Dirac impulses and their derivatives of arbitrary high orders. As outputs of
swDAE read as boundary conditions of PDE, a rigorous solution framework for PDE must
be developed so that jumps, Dirac impulses and their derivatives are allowed at PDE boundaries
and in PDE solutions. This is a wider solution class than solutions of small bounded
variation (BV), for instance, used in where nonlinear hyperbolic PDEs are coupled with
ODEs. Similarly, in, the solutions to switched linear PDEs with source terms are
restricted to the class of BV. However, in the presence of Dirac impulses and their derivatives,
BV functions cannot handle the coupled systems including DAEs with index greater than one.
Therefore, hyperbolic PDEs coupled with swDAEs with index one will be studied in the BV
setting and with swDAEs whose index is greater than one will be investigated in the distributional
sense. To this end, the 1D space of piecewise-smooth distributions is extended to a 2D
piecewise-smooth distributional solution framework. 2D space of piecewise-smooth distributions
allows trace evaluations at boundaries of the PDE. Moreover, a relationship between
solutions to coupled system and switched delay DAEs is established. The coupling structure
in this thesis forms a rather general framework. In fact, any arbitrary network, where PDEs
are represented by edges and (switched) DAEs by nodes, is covered via this structure. Given
a network, by rescaling spatial domains which modifies the coefficient matrices by a constant,
each PDE can be defined on the same interval which leads to a formulation of a single
PDE whose unknown is made up of the unknowns of each PDE that are stacked over each
other with a block diagonal coefficient matrix. Likewise, every swDAE is reformulated such
that the unknowns are collected above each other and coefficient matrices compose a block
diagonal coefficient matrix so that each node in the network is expressed as a single swDAE.
The results are illustrated by numerical simulations of the power grid and simplified circulatory
system examples. Numerical results for the power grid display the evolution of jumps
and Dirac impulses caused by initial and boundary conditions as a result of instant switches.
On the other hand, the analysis and numerical results for the simplified circulatory system do
not entail a Dirac impulse, for otherwise such an entity would destroy the entire system. Yet
jumps in the flow rate in the numerical results can come about due to opening and closure of
valves, which suits clinical and physiological findings. Regarding physiological parameters,
numerical results obtained in this thesis for the simplified circulatory system agree well with
medical data and findings from literature when compared for the validation

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

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.

Die Konstruktion eines Schrittzählers mit einem Arduino-Mikrocontroller und einem Bewegungssensor ist ein spannendes Technikprojekt. Wir erläutern den Grundgedanken hinter der produktorientierten Modellierung und die vielfältigen Möglichkeiten, die Fragestellung zu bearbeiten. Darüberhinaus werden die technischen Details der verwendeten Hardware diskutiert, um einen schnellen Einstieg ins Thema zu ermöglichen.

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.

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.

Operator semigroups and infinite dimensional analysis applied to problems from mathematical physics
(2020)

In this dissertation we treat several problems from mathematical physics via methods from functional analysis and probability theory and in particular operator semigroups. The thesis consists thematically of two parts.
In the first part we consider so-called generalized stochastic Hamiltonian systems. These are generalizations of Langevin dynamics which describe interacting particles moving in a surrounding medium. From a mathematical point of view these systems are stochastic differential equations with a degenerated diffusion coefficient. We construct weak solutions of these equations via the corresponding martingale problem. Therefore, we prove essential m-dissipativity of the degenerated and non-sectorial It\^{o} differential operator. Further, we apply results from the analytic and probabilistic potential theory to obtain an associated Markov process. Afterwards we show our main result, the convergence in law of the positions of the particles in the overdamped regime, the so-called overdamped limit, to a distorted Brownian motion. To this end, we show convergence of the associated operator semigroups in the framework of Kuwae-Shioya. Further, we established a tightness result for the approximations which proves together with the convergence of the semigroups weak convergence of the laws.
In the second part we deal with problems from infinite dimensional Analysis. Three different issues are considered. The first one is an improvement of a characterization theorem of the so-called regular test functions and distribution of White noise analysis. As an application we analyze a stochastic transport equation in terms of regularity of its solution in the space of regular distributions. The last two problems are from the field of relativistic quantum field theory. In the first one the $ (\Phi)_3^4 $-model of quantum field theory is under consideration. We show that the Schwinger functions of this model have a representation as the moments of a positive Hida distribution from White noise analysis. In the last chapter we construct a non-trivial relativistic quantum field in arbitrary space-time dimension. The field is given via Schwinger functions. For these which we establish all axioms of Osterwalder and Schrader. This yields via the reconstruction theorem of Osterwalder and Schrader a unique relativistic quantum field. The Schwinger functions are given as the moments of a non-Gaussian measure on the space of tempered distributions. We obtain the measure as a superposition of Gaussian measures. In particular, this measure is itself non-Gaussian, which implies that the field under consideration is not a generalized free field.

This thesis introduces a novel deformation method for computational meshes. It is based on the numerical path following for the equations of nonlinear elasticity. By employing a logarithmic variation of the neo-Hookean hyperelastic material law, the method guarantees that the mesh elements do not become inverted and remain well-shaped. In order to demonstrate the performance of the method, this thesis addresses two areas of active research in isogeometric analysis: volumetric domain parametrization and fluid-structure interaction. The former concerns itself with the construction of a parametrization for a given computational domain provided only a parametrization of the domain’s boundary. The proposed mesh deformation method gives rise to a novel solution approach to this problem. Within it, the domain parametrization is constructed as a deformed configuration of a simplified domain. In order to obtain the simplified domain, the boundary of the target domain is projected in the \(L^2\)-sense onto a coarse NURBS basis. Then, the Coons patch is applied to parametrize the simplified domain. As a range of 2D and 3D examples demonstrates, the mesh deformation approach is able to produce high-quality parametrizations for complex domains where many state-of-the-art methods either fail or become unstable and inefficient. In the context of fluid-structure interaction, the proposed mesh deformation method is applied to robustly update the computational mesh in situations when the fluid domain undergoes large deformations. In comparison to the state-of-the-art mesh update methods, it is able to handle larger deformations and does not result in an eventual reduction of mesh quality. The performance of the method is demonstrated on a classic 2D fluid-structure interaction benchmark reproduced by using an isogeometric partitioned solver with strong coupling.

In this thesis, we present the basic concepts of isogeometric analysis (IGA) and we consider Poisson's equation as model problem. Since in IGA the physical domain is parametrized via a geometry function that goes from a parameter domain, e.g. the unit square or unit cube, to the physical one, we present a class of parametrizations that can be viewed as a generalization of polar coordinates, known as the scaled boundary parametrizations (SB-parametrizations). These are easy to construct and are particularly attractive when only the boundary of a domain is available. We then present an IGA approach based on these parametrizations, that we call scaled boundary isogeometric analysis (SB-IGA). The SB-IGA derives the weak form of partial differential equations in a different way from the standard IGA. For the discretization projection
on a finite-dimensional space, we choose in both cases Galerkin's method. Thanks to this technique, we state an equivalence theorem for linear elliptic boundary value problems between the standard IGA, when it makes use of an SB-parametrization,
and the SB-IGA. We solve Poisson's equation with Dirichlet boundary conditions on different geometries and with different SB-parametrizations.

Fibre reinforced polymers(FRPs) are one the newest and modern materials. In FRPs a light polymer matrix holds but weak polymer matrix is strengthened by glass or carbon fibres. The result is a material that is light and compared to its weight, very strong.\par
The stiffness of the resulting material is governed by the direction and the length of the fibres. To better understand the behaviour of FRPs we need to know the fibre length distribution in the resulting material. The classic method for this is ashing, where a sample of the material is burned and destroyed. We look at CT images of the material. In the first part we assumed that we have a full fibre segmentation, we can fit an a cylinder to each individual fibre. In this setting we identified two problems, sampling bias and censoring.\par
Sampling bias occurs since a longer fibre has a higher probability to be visible in the observation window. To solve this problem we used a reweighed fibre length distribution. The weight depends on the used sampling rule.\par
For the censoring we used an EM algorithm. The EM algorithm is used to get a Maximum Likelihood estimator in cases of missing or censored data.\par
For this setting we deduced conditions such that the EM algorithm converges to at least a stationary point of the underlying likelihood function. We further found conditions such that if the EM converges to the correct ML estimator, the estimator is consistent and asymptotically normally distributed.\par
Since obtaining a full fibre segmentation is hard we further looked in the fibre endpoint process. The fibre end point process can be modelled as a Neymann-Scott cluster process. Using this model we can find a formula for the reduced second moment measure for this process. We use this formula to get an estimator for the fibre length distribution.\par
We investigated all estimators using simulation studies. We especially investigated their performance in the case of non overlapping fibres.

On the complexity and approximability of optimization problems with Minimum Quantity Constraints
(2020)

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.

Cohomology of Groups
(2020)

Einführung in die Algebra
(2020)

Synapses are connections between different nerve cells that form an essential link in neural signal transmission. It is generally distinguished between electrical and chemical synapses, where chemical synapses are more common in the human brain and are also the type we deal with in this work.
In chemical synapses, small container-like objects called vesicles fill with neurotransmitter and expel them from the cell during synaptic transmission. This process is vital for communication between neurons. However, to the best of our knowledge no mathematical models that take different filling states of the vesicles into account have been developed before this thesis was written.
In this thesis we propose a novel mathematical model for modeling synaptic transmission at chemical synapses which includes the description of vesicles of different filling states. The model consists of a transport equation (for the vesicle growth process) plus three ordinary differential equations (ODEs) and focuses on the presynapse and synaptic cleft.
The well-posedness is proved in detail for this partial differential equation (PDE) system. We also propose a few different variations and related models. In particular, an ODE system is derived and a delay differential equation (DDE) system is formulated. We then use nonlinear optimization methods for data fitting to test some of the models on data made available to us by the Animal Physiology group at TU Kaiserslautern.

Elementare Zahlentheorie
(2020)

In a recent paper, G. Malle and G. Robinson proposed a modular anologue to Brauer's famous \( k(B) \)-conjecture. If \( B \) is a \( p \)-block of a finite group with defect group \( D \), then they conjecture that \( l(B) \leq p^r \), where \( r \) is the sectional \( p \)-rank of \( D \). Since this conjecture is relatively new, there is obviously still a lot of work to do. This thesis is concerned with proving their conjecture for the finite groups of exceptional Lie type.

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.

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 2020.02.
For more information, see https://www.lintim.net

A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations (PDEs) and switched differential algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modeled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.

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.

Model uncertainty is a challenge that is inherent in many applications of mathematical models in various areas, for instance in mathematical finance and stochastic control. Optimization procedures in general take place under a particular model. This model, however, might be misspecified due to statistical estimation errors and incomplete information. In that sense, any specified model must be understood as an approximation of the unknown "true" model. Difficulties arise since a strategy which is optimal under the approximating model might perform rather bad in the true model. A natural way to deal with model uncertainty is to consider worst-case optimization.
The optimization problems that we are interested in are utility maximization problems in continuous-time financial markets. It is well known that drift parameters in such markets are notoriously difficult to estimate. To obtain strategies that are robust with respect to a possible misspecification of the drift we consider a worst-case utility maximization problem with ellipsoidal uncertainty sets for the drift parameter and with a constraint on the strategies that prevents a pure bond investment.
By a dual approach we derive an explicit representation of the optimal strategy and prove a minimax theorem. This enables us to show that the optimal strategy converges to a generalized uniform diversification strategy as uncertainty increases.
To come up with a reasonable uncertainty set, investors can use filtering techniques to estimate the drift of asset returns based on return observations as well as external sources of information, so-called expert opinions. In a Black-Scholes type financial market with a Gaussian drift process we investigate the asymptotic behavior of the filter as the frequency of expert opinions tends to infinity. We derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. Our convergence results carry over to convergence of the value function in a portfolio optimization problem with logarithmic utility.
Lastly, we use our observations about how expert opinions improve drift estimates for our robust utility maximization problem. We show that our duality approach carries over to a financial market with non-constant drift and time-dependence in the uncertainty set. A time-dependent uncertainty set can then be defined based on a generic filter. We apply this to various investor filtrations and investigate which effect expert opinions have on the robust strategies.

In this dissertation we apply financial mathematical modelling to electricity markets. Electricity is different from any other underlying of financial contracts: it is not storable. This means that electrical energy in one time point cannot be transferred to another. As a consequence, power contracts with disjoint delivery time spans basically have a different underlying. The main idea throughout this thesis is exactly this two-dimensionality of time: every electricity contract is not only characterized by its trading time but also by its delivery time.
The basis of this dissertation are four scientific papers corresponding to the Chapters 3 to 6, two of which have already been published in peer-reviewed journals. Throughout this thesis two model classes play a significant role: factor models and structural models. All ideas are applied to or supported by these two model classes. All empirical studies in this dissertation are conducted on electricity price data from the German market and Chapter 4 in particular studies an intraday derivative unique to the German market. Therefore, electricity market design is introduced by the example of Germany in Chapter 1. Subsequently, Chapter 2 introduces the general mathematical theory necessary for modelling electricity prices, such as Lévy processes and the Esscher transform. This chapter is the mathematical basis of the Chapters 3 to 6.
Chapter 3 studies factor models applied to the German day-ahead spot prices. We introduce a qualitative measure for seasonality functions based on three requirements. Furthermore, we introduce a relation of factor models to ARMA processes, which induces a new method to estimate the mean reversion speed.
Chapter 4 conducts a theoretical and empirical study of a pricing method for a new electricity derivative: the German intraday cap and floor futures. We introduce the general theory of derivative pricing and propose a method based on the Hull-White model of interest rate modelling, which is a one-factor model. We include week futures prices to generate a price forward curve (PFC), which is then used instead of a fixed deterministic seasonality function. The idea that we can combine all market prices, and in particular futures prices, to improve the model quality also plays the major role in Chapter 5 and Chapter 6.
In Chapter 5 we develop a Heath-Jarrow-Morton (HJM) framework that models intraday, day-ahead, and futures prices. This approach is based on two stochastic processes motivated by economic interpretations and separates the stochastic dynamics in trading and delivery time. Furthermore, this framework allows for the use of classical day-ahead spot price models such as the ones of Schwartz and Smith (2000), Lucia and Schwartz (2002) and includes many model classes such as structural models and factor models.
Chapter 6 unifies the classical theory of storage and the concept of a risk premium through the introduction of an unobservable intrinsic electricity price. Since all tradable electricity contracts are derivatives of this actual intrinsic price, their prices should all be derived as conditional expectation under the risk-neutral measure. Through the intrinsic electricity price we develop a framework, which also includes many existing modelling approaches, such as the HJM framework of Chapter 5.

Dieser Beitrag beschreibt eine Lernumgebung für Schülerinnen und Schüler der Unter- und Mittelstufe mit einem Schwerpunkt im Fach Mathematik. Das Thema dieser Lernumgebung ist die Simulation von Entfluchtungsprozessen im Rahmen von Gebäudeevakuierungen. Dabei wird das Konzept eines zellulären Automaten vermittelt, ohne dabei Programmierkenntnisse vorauszusetzen oder anzuwenden. Anhand dieses speziellen Simulationswerkzeugs des zellulären Automaten werden Eigenschaften, Kenngrößen sowie Vor- und Nachteile von Simulationen im Allgemeinen thematisiert. Dazu gehören unter anderem die experimentelle Datengewinnung, die Festlegung von Modellparametern, die Diskretisierung des zeitlichen und räumlichen Betrachtungshorizonts sowie die zwangsläufig auftretenden (Diskretisierungs-)Fehler, die algorithmischen Abläufe einer Simulation in Form elementarer Handlungsanweisungen, die Speicherung und Visualisierung von Daten aus einer Simulation sowie die Interpretation und kritische Diskussion von Simulationsergebnissen. Die vorgestellte Lernumgebung ermöglicht etliche Variationen zu weiteren Aspekten des Themas „Evakuierungssimulation“ und bietet dadurch auch vielfältige Differenzierungsmöglichkeiten.

Die MINT-EC-Girls-Camp: Math-Talent-School ist eine vom Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM) initiierte Veranstaltung, die regelmäßig als Kooperation zwischen dem Felix-Klein-Zentrum für Mathematik und dem Verein mathematisch-naturwissenschaftlicher Excellence-Center an Schulen e.V. (Verein MINT-EC) durchgeführt wird. Die methodisch-didaktische Konzeption der Math-Talent-Schools erfolgt durch das Kompetenzzentrum für Mathematische Modellierung in MINT-Projekten in der Schule (KOMMS), einer wissenschaftlichen Einrichtung des Fachbereichs Mathematik der Technischen Universität Kaiserslautern. Die inhaltlich-organisatorische Ausführung übernimmt das Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM in enger Abstimmung und Kooperation von Wissenschaftlern der Technischen Universität und des Fraunhofer ITWM. Die MINT-EC-Girls-Camp: Math-Talent-School hat zum Ziel, Mathematik-interessierten Schülerinnen einen Einblick in die Arbeitswelt von Mathematikerinnen und Mathematikern zu geben. In diesem Artikel stellen wir die Math-Talent-School vor. Hierfür werden die fachlichen und fachdidaktischen Hintergründe der Projekte beleuchtet, der Ablauf der Veranstaltung erläutert und ein Fazit gezogen.

Wir zeigen an einigen Beispielen, wie man numerische Simulationen in Tabellenkalkulationsprogrammen (hier speziell in Excel) erzeugen kann. Diese können beispielsweise im Kontext von mathematischer Modellierung verwendet werden.
Die Beispiele umfassen ein Modell zur Ausbreitung von Krankheiten, die Flugkurve eines Fußballs unter Berücksichtigung von Luftreibung, eine Monte-Carlo-Simulation zur experimentellen Bestimmung von pi, eine Monte-Carlo-Simulation eines gemischten Kartenstapels und die Modellierung von Benzinpreisen mit einem Preistrend und Rauschen

Many loads acting on a vehicle depend on the condition and quality of roads
traveled as well as on the driving style of the motorist. Thus, during vehicle development,
good knowledge on these further operations conditions is advantageous.
For that purpose, usage models for different kinds of vehicles are considered. Based
on these mathematical descriptions, representative routes for multiple user
types can be simulated in a predefined geographical region. The obtained individual
driving schedules consist of coordinates of starting and target points and can
thus be routed on the true road network. Additionally, different factors, like the
topography, can be evaluated along the track.
Available statistics resulting from travel survey are integrated to guarantee reasonable
trip length. Population figures are used to estimate the number of vehicles in
contained administrative units. The creation of thousands of those geo-referenced
trips then allows the determination of realistic measures of the durability loads.
Private as well as commercial use of vehicles is modeled. For the former, commuters
are modeled as the main user group conducting daily drives to work and
additional leisure time a shopping trip during workweek. For the latter, taxis as
example for users of passenger cars are considered. The model of light-duty commercial
vehicles is split into two types of driving patterns, stars and tours, and in
the common traffic classes of long-distance, local and city traffic.
Algorithms to simulate reasonable target points based on geographical and statistical
data are presented in detail. Examples for the evaluation of routes based
on topographical factors and speed profiles comparing the influence of the driving
style are included.

In this thesis we consider the directional analysis of stationary point processes. We focus on three non-parametric methods based on second order analysis which we have defined as Integral method, Ellipsoid method, and Projection method. We present the methods in a general setting and then focus on their application in the 2D and 3D case of a particular type of anisotropy mechanism called geometric anisotropy. We mainly consider regular point patterns motivated by our application to real 3D data coming from glaciology. Note that directional analysis of 3D data is not so prominent in the literature.
We compare the performance of the methods, which depends on the relative parameters, in a simulation study both in 2D and 3D. Based on the results we give recommendations on how to choose the methods´ parameters in practice.
We apply the directional analysis to the 3D data coming from glaciology, which consist in the locations of air-bubbles in polar ice cores. The aim of this study is to provide information about the deformation rate in the ice and the corresponding thinning of ice layers at different depths. This information is substantial for the glaciologists in order to build ice dating models and consequently to give a correct interpretation of the climate information which can be found by analyzing ice cores. In this thesis we consider data coming from three different ice cores: the Talos Dome core, the EDML core and the Renland core.
Motivated by the ice application, we study how isotropic and stationary noise influences the directional analysis. In fact, due to the relaxation of the ice after drilling, noise bubbles can form within the ice samples. In this context we take two classification algorithms into consideration, which aim to classify points in a superposition of a regular isotropic and stationary point process with Poisson noise.
We introduce two methods to visualize anisotropy, which are particularly useful in 3D and apply them to the ice data. Finally, we consider the problem of testing anisotropy and the limiting behavior of the geometric anisotropy transform.

Various physical phenomenons with sudden transients that results into structrual changes can be modeled via
switched nonlinear differential algebraic equations (DAEs) of the type
\[
E_{\sigma}\dot{x}=A_{\sigma}x+f_{\sigma}+g_{\sigma}(x). \tag{DAE}
\]
where \(E_p,A_p \in \mathbb{R}^{n\times n}, x\mapsto g_p(x),\) is a mapping, \(p \in \{1,\cdots,P\}, P\in \mathbb{N}
f \in \mathbb{R} \rightarrow \mathbb{R}^n , \sigma: \mathbb{R} \rightarrow \{1,\cdots, P\}\).
Two related common tasks are:
Task 1: Investigate if above (DAE) has a solution and if it is unique.
Task 2: Find a connection among a solution of above (DAE) and solutions of related
partial differential equations.
In the linear case \(g(x) \equiv 0\) the task 1 has been tackeled already in a
distributional solution framework.
A main goal of the dissertation is to give contribution to task 1 for the
nonlinear case \(g(x) \not \equiv 0\) ; also contributions to the task 2 are given for
switched nonlinear DAEs arising while modeling sudden transients in water
distribution networks. In addition, this thesis contains the following further
contributions:
The notion of structured switched nonlinear DAEs has been introduced,
allowing also non regular distributions as solutions. This extend a previous
framework that allowed only piecewise smooth functions as solutions. Further six mild conditions were given to ensure existence and uniqueness of the solution within the space of piecewise smooth distribution. The main
condition, namely the regularity of the matrix pair \((E,A)\), is interpreted geometrically for those switched nonlinear DAEs arising from water network graphs.
Another contribution is the introduction of these switched nonlinear DAEs
as a simplication of the PDE model used classically for modeling water networks. Finally, with the support of numerical simulations of the PDE model it has been illustrated that this switched nonlinear DAE model is a good approximation for the PDE model in case of a small compressibility coefficient.

In modern algebraic geometry solutions of polynomial equations are studied from a qualitative point of view using highly sophisticated tools such as cohomology, \(D\)-modules and Hodge structures. The latter have been unified in Saito’s far-reaching theory of mixed Hodge modules, that has shown striking applications including vanishing theorems for cohomology. A mixed Hodge module can be seen as a special type of filtered \(D\)-module, which is an algebraic counterpart of a system of linear differential equations. We present the first algorithmic approach to Saito’s theory. To this end, we develop a Gröbner basis theory for a new class of algebras generalizing PBW-algebras.
The category of mixed Hodge modules satisfies Grothendieck’s six-functor formalism. In part these functors rely on an additional natural filtration, the so-called \(V\)-filtration. A key result of this thesis is an algorithm to compute the \(V\)-filtration in the filtered setting. We derive from this algorithm methods for the computation of (extraordinary) direct image functors under open embeddings of complements of pure codimension one subvarieties. As side results we show how to compute vanishing and nearby cycle functors and a quasi-inverse of Kashiwara’s equivalence for mixed Hodge modules.
Describing these functors in terms of local coordinates and taking local sections, we reduce the corresponding computations to algorithms over certain bifiltered algebras. It leads us to introduce the class of so-called PBW-reduction-algebras, a generalization of the class of PBW-algebras. We establish a comprehensive Gröbner basis framework for this generalization representing the involved filtrations by weight vectors.

Cell migration is essential for embryogenesis, wound healing, immune surveillance, and
progression of diseases, such as cancer metastasis. For the migration to occur, cellular
structures such as actomyosin cables and cell-substrate adhesion clusters must interact.
As cell trajectories exhibit a random character, so must such interactions. Furthermore,
migration often occurs in a crowded environment, where the collision outcome is deter-
mined by altered regulation of the aforementioned structures. In this work, guided by a
few fundamental attributes of cell motility, we construct a minimal stochastic cell migration
model from ground-up. The resulting model couples a deterministic actomyosin contrac-
tility mechanism with stochastic cell-substrate adhesion kinetics, and yields a well-defined
piecewise deterministic process. The signaling pathways regulating the contractility and
adhesion are considered as well. The model is extended to include cell collectives. Numer-
ical simulations of single cell migration reproduce several experimentally observed results,
including anomalous diffusion, tactic migration, and contact guidance. The simulations
of colliding cells explain the observed outcomes in terms of contact induced modification
of contractility and adhesion dynamics. These explained outcomes include modulation
of collision response and group behavior in the presence of an external signal, as well as
invasive and dispersive migration. Moreover, from the single cell model we deduce a pop-
ulation scale formulation for the migration of non-interacting cells. In this formulation,
the relationships concerning actomyosin contractility and adhesion clusters are maintained.
Thus, we construct a multiscale description of cell migration, whereby single, collective,
and population scale formulations are deduced from the relationships on the subcellular
level in a mathematically consistent way.

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.

Magnetoelastic coupling describes the mutual dependence of the elastic and magnetic fields and can be observed in certain types of materials, among which are the so-called "magnetostrictive materials". They belong to the large class of "smart materials", which change their shape, dimensions or material properties under the influence of an external field. The mechanical strain or deformation a material experiences due to an externally applied magnetic field is referred to as magnetostriction; the reciprocal effect, i.e. the change of the magnetization of a body subjected to mechanical stress is called inverse magnetostriction. The coupling of mechanical and electromagnetic fields is particularly observed in "giant magnetostrictive materials", alloys of ferromagnetic materials that can exhibit several thousand times greater magnitudes of magnetostriction (measured as the ratio of the change in length of the material to its original length) than the common magnetostrictive materials. These materials have wide applications areas: They are used as variable-stiffness devices, as sensors and actuators in mechanical systems or as artificial muscles. Possible application fields also include robotics, vibration control, hydraulics and sonar systems.
Although the computational treatment of coupled problems has seen great advances over the last decade, the underlying problem structure is often not fully understood nor taken into account when using black box simulation codes. A thorough analysis of the properties of coupled systems is thus an important task.
The thesis focuses on the mathematical modeling and analysis of the coupling effects in magnetostrictive materials. Under the assumption of linear and reversible material behavior with no magnetic hysteresis effects, a coupled magnetoelastic problem is set up using two different approaches: the magnetic scalar potential and vector potential formulations. On the basis of a minimum energy principle, a system of partial differential equations is derived and analyzed for both approaches. While the scalar potential model involves only stationary elastic and magnetic fields, the model using the magnetic vector potential accounts for different settings such as the eddy current approximation or the full Maxwell system in the frequency domain.
The distinctive feature of this work is the analysis of the obtained coupled magnetoelastic problems with regard to their structure, strong and weak formulations, the corresponding function spaces and the existence and uniqueness of the solutions. We show that the model based on the magnetic scalar potential constitutes a coupled saddle point problem with a penalty term. The main focus in proving the unique solvability of this problem lies on the verification of an inf-sup condition in the continuous and discrete cases. Furthermore, we discuss the impact of the reformulation of the coupled constitutive equations on the structure of the coupled problem and show that in contrast to the scalar potential approach, the vector potential formulation yields a symmetric system of PDEs. The dependence of the problem structure on the chosen formulation of the constitutive equations arises from the distinction of the energy and coenergy terms in the Lagrangian of the system. While certain combinations of the elastic and magnetic variables lead to a coupled magnetoelastic energy function yielding a symmetric problem, the use of their dual variables results in a coupled coenergy function for which a mixed problem is obtained.
The presented models are supplemented with numerical simulations carried out with MATLAB for different examples including a 1D Euler-Bernoulli beam under magnetic influence and a 2D magnetostrictive plate in the state of plane stress. The simulations are based on material data of Terfenol-D, a giant magnetostrictive materials used in many industrial applications.

For some optimization problems on a graph \(G=(V,E)\), one can give a general formulation: Let \(c\colon E \to \mathbb{R}_{\geq 0}\) be a cost function on the edges and \(X \subseteq 2^E\) be a set of (so-called feasible) subsets of \(E\), one aims to minimize \(\sum_{e\in S} c(e)\) among all feasible \(S\in X\). This formulation covers, for instance, the shortest path problem by choosing \(X\) as the set of all paths between two vertices, or the minimum spanning tree problem by choosing \(X\) to be the set of all spanning trees. This bachelor thesis deals with a parametric version of this formulation, where the edge costs \(c_\lambda\colon E \to \mathbb{R}_{\geq 0}\) depend on a parameter \(\lambda\in\mathbb{R}_{\geq 0}\) in a concave and piecewise linear manner. The goal is to investigate the worst case minimum size of a so-called representation system \(R\subseteq X\), which contains for each scenario \(\lambda\in\mathbb{R}_{\geq 0}\) an optimal solution \(S(\lambda)\in R\). It turns out that only a pseudo-polynomial size can be ensured in general, but smaller systems have to exist in special cases. Moreover, methods are presented to find such small systems algorithmically. Finally, the notion of a representation system is relaxed in order to get smaller (i.e. polynomial) systems ensuring a certain approximation ratio.

Cutting-edge cancer therapy involves producing individualized medicine for many patients at the same time. Within this process, most steps can be completed for a certain number of patients simultaneously. Using these resources efficiently may significantly reduce waiting times for the patients and is therefore crucial for saving human lives. However, this involves solving a complex scheduling problem, which can mathematically be modeled as a proportionate flow shop of batching machines (PFB). In this thesis we investigate exact and approximate algorithms for tackling many variants of this problem. Related mathematical models have been studied before in the context of semiconductor manufacturing.

Destructive diseases of the lung like lung cancer or fibrosis are still often lethal. Also in case of fibrosis in the liver, the only possible cure is transplantation.
In this thesis, we investigate 3D micro computed synchrotron radiation (SR\( \mu \)CT) images of capillary blood vessels in mouse lungs and livers. The specimen show so-called compensatory lung growth as well as different states of pulmonary and hepatic fibrosis.
During compensatory lung growth, after resecting part of the lung, the remaining part compensates for this loss by extending into the empty space. This process is accompanied by an active vessel growing.
In general, the human lung can not compensate for such a loss. Thus, understanding this process in mice is important to improve treatment options in case of diseases like lung cancer.
In case of fibrosis, the formation of scars within the organ's tissue forces the capillary vessels to grow to ensure blood supply.
Thus, the process of fibrosis as well as compensatory lung growth can be accessed by considering the capillary architecture.
As preparation of 2D microscopic images is faster, easier, and cheaper compared to SR\( \mu \)CT images, they currently form the basis of medical investigation. Yet, characteristics like direction and shape of objects can only properly be analyzed using 3D imaging techniques. Hence, analyzing SR\( \mu \)CT data provides valuable additional information.
For the fibrotic specimen, we apply image analysis methods well-known from material science. We measure the vessel diameter using the granulometry distribution function and describe the inter-vessel distance by the spherical contact distribution. Moreover, we estimate the directional distribution of the capillary structure. All features turn out to be useful to characterize fibrosis based on the deformation of capillary vessels.
It is already known that the most efficient mechanism of vessel growing forms small torus-shaped holes within the capillary structure, so-called intussusceptive pillars. Analyzing their location and number strongly contributes to the characterization of vessel growing. Hence, for all three applications, this is of great interest. This thesis provides the first algorithm to detect intussusceptive pillars in SR\( \mu \)CT images. After segmentation of raw image data, our algorithm works automatically and allows for a quantitative evaluation of a large amount of data.
The analysis of SR\( \mu \)CT data using our pillar algorithm as well as the granulometry, spherical contact distribution, and directional analysis extends the current state-of-the-art in medical studies. Although it is not possible to replace certain 3D features by 2D features without losing information, our results could be used to examine 2D features approximating the 3D findings reasonably well.

Numerical Godeaux surfaces are minimal surfaces of general type with the smallest possible numerical invariants. It is known that the torsion group of a numerical Godeaux surface is cyclic of order \(m\leq 5\). A full classification has been given for the cases \(m=3,4,5\) by the work of Reid and Miyaoka. In each case, the corresponding moduli space is 8-dimensional and irreducible.
There exist explicit examples of numerical Godeaux surfaces for the orders \(m=1,2\), but a complete classification for these surfaces is still missing.
In this thesis we present a construction method for numerical Godeaux surfaces which is based on homological algebra and computer algebra and which arises from an experimental approach by Schreyer. The main idea is to consider the canonical ring \(R(X)\) of a numerical Godeaux surface \(X\) as a module over some graded polynomial ring \(S\). The ring \(S\) is chosen so that \(R(X)\) is finitely generated as an \(S\)-module and a Gorenstein \(S\)-algebra of codimension 3. We prove that the canonical ring of any numerical Godeaux surface, considered as an \(S\)-module, admits a minimal free resolution whose middle map is alternating. Moreover, we show that a partial converse of this statement is true under some additional conditions.
Afterwards we use these results to construct (canonical rings of) numerical Godeaux surfaces. Hereby, we restrict our study to surfaces whose bicanonical system has no fixed component but 4 distinct base points, in the following referred to as marked numerical Godeaux surfaces.
The particular interest of this thesis lies on marked numerical Godeaux surfaces whose torsion group is trivial. For these surfaces we study the fibration of genus 4 over \(\mathbb{P}^1\) induced by the bicanonical system. Catanese and Pignatelli showed that the general fibre is non-hyperelliptic and that the number \(\tilde{h}\) of hyperelliptic fibres is bounded by 3. The two explicit constructions of numerical Godeaux surfaces with a trivial torsion group due to Barlow and Craighero-Gattazzo, respectively, satisfy \(\tilde{h} = 2\).
With the method from this thesis, we construct an 8-dimensional family of numerical Godeaux surfaces with a trivial torsion group and whose general element satisfy \(\tilde{h}=0\).
Furthermore, we establish a criterion for the existence of hyperelliptic fibres in terms of a minimal free resolution of \(R(X)\). Using this criterion, we verify experimentally the
existence of a numerical Godeaux surface with \(\tilde{h}=1\).

SDE-driven modeling of phenotypically heterogeneous tumors: The influence of cancer cell stemness
(2018)

We deduce cell population models describing the evolution of a tumor (possibly interacting with its
environment of healthy cells) with the aid of differential equations. Thereby, different subpopulations
of cancer cells allow accounting for the tumor heterogeneity. In our settings these include cancer
stem cells known to be less sensitive to treatment and differentiated cancer cells having a higher
sensitivity towards chemo- and radiotherapy. Our approach relies on stochastic differential equations
in order to account for randomness in the system, arising e.g., by the therapy-induced decreasing
number of clonogens, which renders a pure deterministic model arguable. The equations are deduced
relying on transition probabilities characterizing innovations of the two cancer cell subpopulations,
and similarly extended to also account for the evolution of normal tissue. Several therapy approaches
are introduced and compared by way of tumor control probability (TCP) and uncomplicated tumor
control probability (UTCP). A PDE approach allows to assess the evolution of tumor and normal
tissue with respect to time and to cell population densities which can vary continuously in a given set
of states. Analytical approximations of solutions to the obtained PDE system are provided as well.

Optimal control of partial differential equations is an important task in applied mathematics where it is used in order to optimize, for example, industrial or medical processes. In this thesis we investigate an optimal control problem with tracking type cost functional for the Cattaneo equation with distributed control, that is, \(\tau y_{tt} + y_t - \Delta y = u\). Our focus is on the theoretical and numerical analysis of the limit process \(\tau \to 0\) where we prove the convergence of solutions of the Cattaneo equation to solutions of the heat equation.
We start by deriving both the Cattaneo and the classical heat equation as well as introducing our notation and some functional analytic background. Afterwards, we prove the well-posedness of the Cattaneo equation for homogeneous Dirichlet boundary conditions, that is, we show the existence and uniqueness of a weak solution together with its continuous dependence on the data. We need this in the following, where we investigate the optimal control problem for the Cattaneo equation: We show the existence and uniqueness of a global minimizer for an optimal control problem with tracking type cost functional and the Cattaneo equation as a constraint. Subsequently, we do an asymptotic analysis for \(\tau \to 0\) for both the forward equation and the aforementioned optimal control problem and show that the solutions of these problems for the Cattaneo equation converge strongly to the ones for the heat equation. Finally, we investigate these problems numerically, where we examine the different behaviour of the models and also consider the limit \(\tau \to 0\), suggesting a linear convergence rate.

Certain brain tumours are very hard to treat with radiotherapy due to their irregular shape caused by the infiltrative nature of the tumour cells. To enhance the estimation of the tumour extent one may use a mathematical model. As the brain structure plays an important role for the cell migration, it has to be included in such a model. This is done via diffusion-MRI data. We set up a multiscale model class accounting among others for integrin-mediated movement of cancer cells in the brain tissue, and the integrin-mediated proliferation. Moreover, we model a novel chemotherapy in combination with standard radiotherapy.
Thereby, we start on the cellular scale in order to describe migration. Then we deduce mean-field equations on the mesoscopic (cell density) scale on which we also incorporate cell proliferation. To reduce the phase space of the mesoscopic equation, we use parabolic scaling and deduce an effective description in the form of a reaction-convection-diffusion equation on the macroscopic spatio-temporal scale. On this scale we perform three dimensional numerical simulations for the tumour cell density, thereby incorporating real diffusion tensor imaging data. To this aim, we present programmes for the data processing taking the raw medical data and processing it to the form to be included in the numerical simulation. Thanks to the reduction of the phase space, the numerical simulations are fast enough to enable application in clinical practice.

Composite materials are used in many modern tools and engineering applications and
consist of two or more materials that are intermixed. Features like inclusions in a matrix
material are often very small compared to the overall structure. Volume elements that
are characteristic for the microstructure can be simulated and their elastic properties are
then used as a homogeneous material on the macroscopic scale.
Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated
trigonometric polynomials on a tensor product grid as ansatz functions.
In this thesis, we generalize their approach to anisotropic lattices and extend it to
anisotropic translation invariant spaces. We discretize the partial differential equation
on these spaces and prove the convergence rate. The speed of convergence depends on
the smoothness of the coefficients and the regularity of the ansatz space. The spaces of
translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and
periodic Box splines, including the constant finite element discretization of Brisard and
Dormieux [1].
For finely resolved images, sampling on a coarser lattice reduces the computational
effort. We introduce mixing rules as the means to transfer fine-grid information to the
smaller lattice.
Finally, we show the effect of the anisotropic pattern, the space of translates, and the
convergence of the method, and mixing rules on two- and three-dimensional examples.
References
[1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites:
A general variational framework”. In: Computational Materials Science 49.3 (2010),
pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009.
[2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response
of nonlinear composites with complex microstructure”. In: Computer Methods in
Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.

Multiphase materials combine properties of several materials, which makes them interesting for high-performing components. This thesis considers a certain set of multiphase materials, namely silicon-carbide (SiC) particle-reinforced aluminium (Al) metal matrix composites and their modelling based on stochastic geometry models.
Stochastic modelling can be used for the generation of virtual material samples: Once we have fitted a model to the material statistics, we can obtain independent three-dimensional “samples” of the material under investigation without the need of any actual imaging. Additionally, by changing the model parameters, we can easily simulate a new material composition.
The materials under investigation have a rather complicated microstructure, as the system of SiC particles has many degrees of freedom: Size, shape, orientation and spatial distribution. Based on FIB-SEM images, that yield three-dimensional image data, we extract the SiC particle structure using methods of image analysis. Then we model the SiC particles by anisotropically rescaled cells of a random Laguerre tessellation that was fitted to the shapes of isotropically rescaled particles. We fit a log-normal distribution for the volume distribution of the SiC particles. Additionally, we propose models for the Al grain structure and the Aluminium-Copper (\({Al}_2{Cu}\)) precipitations occurring on the grain boundaries and on SiC-Al phase boundaries.
Finally, we show how we can estimate the parameters of the volume-distribution based on two-dimensional SEM images. This estimation is applied to two samples with different mean SiC particle diameters and to a random section through the model. The stereological estimations are within acceptable agreement with the parameters estimated from three-dimensional image data
as well as with the parameters of the model.

Using valuation theory we associate to a one-dimensional equidimensional semilocal Cohen-Macaulay ring \(R\) its semigroup of values, and to a fractional ideal of \(R\) we associate its value semigroup ideal. For a class of curve singularities (here called admissible rings) including algebroid curves the semigroups of values, respectively the value semigroup ideals, satisfy combinatorial properties defining good semigroups, respectively good semigroup ideals. Notably, the class of good semigroups strictly contains the class of value semigroups of admissible rings. On good semigroups we establish combinatorial versions of algebraic concepts on admissible rings which are compatible with their prototypes under taking values. Primarily we examine duality and quasihomogeneity.
We give a definition for canonical semigroup ideals of good semigroups which characterizes canonical fractional ideals of an admissible ring in terms of their value semigroup ideals. Moreover, a canonical semigroup ideal induces a duality on the set of good semigroup ideals of a good semigroup. This duality is compatible with the Cohen-Macaulay duality on fractional ideals under taking values.
The properties of the semigroup of values of a quasihomogeneous curve singularity lead to a notion of quasihomogeneity on good semigroups which is compatible with its algebraic prototype. We give a combinatorial criterion which allows to construct from a quasihomogeneous semigroup \(S\) a quasihomogeneous curve singularity having \(S\) as semigroup of values.
As an application we use the semigroup of values to compute endomorphism rings of maximal ideals of algebroid curves. This yields an explicit description of the intermediate rings in an algorithmic normalization of plane central arrangements of smooth curves based on a criterion by Grauert and Remmert. Applying this result to hyperplane arrangements we determine the number of steps needed to compute the normalization of a the arrangement in terms of its Möbius function.

In the present master’s thesis we investigate the connection between derivations and
homogeneities of complete analytic algebras. We prove a theorem, which describes a specific set of generators
for the module of derivations of an analytic algebra, which map the maximal ideal of R into itself. It turns out, that this set has a structure similar to a Cartan subalgebra and contains
information regarding multi-homogeneity. In order to prove
this theorem, we extend the notion of grading by Scheja and Wiebe to projective systems and state the connection between multi-gradings and pairwise
commuting diagonalizable derivations. We prove a theorem similar to Cartan’s Conjugacy Theorem in the setup of infinite-dimensional Lie algebras, which arise as projective limits of finite-dimensional Lie algebras. Using this result, we can show that the structure of the aforementioned set of generators is an intrinsic property of the analytic algebra. At the end we state an algorithm, which is theoretically able to compute the maximal multi-homogeneity of a complete analytic algebra.

In this thesis we integrate discrete dividends into the stock model, estimate
future outstanding dividend payments and solve different portfolio optimization
problems. Therefore, we discuss three well-known stock models, including
discrete dividend payments and evolve a model, which also takes early
announcement into account.
In order to estimate the future outstanding dividend payments, we develop a
general estimation framework. First, we investigate a model-free, no-arbitrage
methodology, which is based on the put-call parity for European options. Our
approach integrates all available option market data and simultaneously calculates
the market-implied discount curve. We illustrate our method using stocks
of European blue-chip companies and show within a statistical assessment that
the estimate performs well in practice.
As American options are more common, we additionally develop a methodology,
which is based on market prices of American at-the-money options.
This method relies on a linear combination of no-arbitrage bounds of the dividends,
where the corresponding optimal weight is determined via a historical
least squares estimation using realized dividends. We demonstrate our method
using all Dow Jones Industrial Average constituents and provide a robustness
check with respect to the used discount factor. Furthermore, we backtest our
results against the method using European options and against a so called
simple estimate.
In the last part of the thesis we solve the terminal wealth portfolio optimization
problem for a dividend paying stock. In the case of the logarithmic utility
function, we show that the optimal strategy is not a constant anymore but
connected to the Merton strategy. Additionally, we solve a special optimal
consumption problem, where the investor is only allowed to consume dividends.
We show that this problem can be reduced to the before solved terminal wealth
problem.

In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.

A popular model for the locations of fibres or grains in composite materials
is the inhomogeneous Poisson process in dimension 3. Its local intensity function
may be estimated non-parametrically by local smoothing, e.g. by kernel
estimates. They crucially depend on the choice of bandwidths as tuning parameters
controlling the smoothness of the resulting function estimate. In this
thesis, we propose a fast algorithm for learning suitable global and local bandwidths
from the data. It is well-known, that intensity estimation is closely
related to probability density estimation. As a by-product of our study, we
show that the difference is asymptotically negligible regarding the choice of
good bandwidths, and, hence, we focus on density estimation.
There are quite a number of data-driven bandwidth selection methods for
kernel density estimates. cross-validation is a popular one and frequently proposed
to estimate the optimal bandwidth. However, if the sample size is very
large, it becomes computational expensive. In material science, in particular,
it is very common to have several thousand up to several million points.
Another type of bandwidth selection is a solve-the-equation plug-in approach
which involves replacing the unknown quantities in the asymptotically optimal
bandwidth formula by their estimates.
In this thesis, we develop such an iterative fast plug-in algorithm for estimating
the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed
asymptotics of the estimators of the intensity function and of its second
derivatives and integrals of second derivatives which appear in the formulae
for asymptotically optimal bandwidths. These asymptotics are utilised to determine
the exact number of iteration steps and some tuning parameters. For
both global and local case, fewer than 10 iterations suffice. Simulation studies
show that the estimated intensity by local bandwidth can better indicate
the variation of local intensity than that by global bandwidth. Finally, the
algorithm is applied to two real data sets from test bodies of fibre-reinforced
high-performance concrete, clearly showing some inhomogeneity of the fibre
intensity.

In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.

Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε.

In this article a new numerical solver for simulations of district heating networks is presented. The numerical method applies the local time stepping introduced in [11] to networks of linear advection equations. In combination with the high order approach of [4] an accurate and very efficient scheme is developed. In several numerical test cases the advantages for simulations of district heating networks are shown.

In this paper, we demonstrate the power of functional data models for a statistical analysis of stimulus-response experiments which is a quite natural way to look at this kind of data and which makes use of the full information available. In particular, we focus on the detection of a change in the mean of the response in a series of stimulus-response curves where we also take into account dependence in time.

Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.

In this thesis we address two instances of duality in commutative algebra.
In the first part, we consider value semigroups of non irreducible singular algebraic curves
and their fractional ideals. These are submonoids of Z^n closed under minima, with a conductor and which fulfill special compatibility properties on their elements. Subsets of Z^n
fulfilling these three conditions are known in the literature as good semigroups and their ideals, and their class strictly contains the class of value semigroup ideals. We examine
good semigroups both independently and in relation with their algebraic counterpart. In the combinatoric setting, we define the concept of good system of generators, and we
show that minimal good systems of generators are unique. In relation with the algebra side, we give an intrinsic definition of canonical semigroup ideals, which yields a duality
on good semigroup ideals. We prove that this semigroup duality is compatible with the Cohen-Macaulay duality under taking values. Finally, using the duality on good semigroup ideals, we show a symmetry of the Poincaré series of good semigroups with special properties.
In the second part, we treat Macaulay’s inverse system, a one-to-one correspondence
which is a particular case of Matlis duality and an effective method to construct Artinian k-algebras with chosen socle type. Recently, Elias and Rossi gave the structure of the inverse system of positive dimensional Gorenstein k-algebras. We extend their result by establishing a one-to-one correspondence between positive dimensional level k-algebras and certain submodules of the divided power ring. We give several examples to illustrate
our result.

We continue in this paper the study of k-adaptable robust solutions for combinatorial optimization problems with bounded uncertainty sets. In this concept not a single solution needs to be chosen to hedge against the uncertainty. Instead one is allowed to choose a set of k different solutions from which one can be chosen after the uncertain scenario has been revealed. We first show how the problem can be decomposed into polynomially many subproblems if k is fixed. In the remaining part of the paper we consider the special case where k=2, i.e., one is allowed to choose two different solutions to hedge against the uncertainty. We decompose this problem into so called coordination problems. The study of these coordination problems turns out to be interesting on its own. We prove positive results for the unconstrained combinatorial optimization problem, the matroid maximization problem, the selection problem, and the shortest path problem on series parallel graphs. The shortest path problem on general graphs turns out to be NP-complete. Further, we present for minimization problems how to transform approximation algorithms for the coordination problem to approximation algorithms for the original problem. We study the knapsack problem to show that this relation does not hold for maximization problems in general. We present a PTAS for the corresponding coordination problem and prove that the 2-adaptable knapsack problem is not at all approximable.

This paper presents a case study of duty rostering for physicians at a department of orthopedics and trauma surgery. We provide a detailed description of the rostering problem faced and present an integer programming model that has been used in practice for creating duty rosters at the department for more than a year. Using real world data, we compare the model output to a manually generated roster as used previously by the department and analyze the quality of the rosters generated by the model over a longer time span. Moreover, we demonstrate how unforeseen events such as absences of scheduled physicians are handled.

Order-semi-primal lattices
(1994)

A nonequilibrium situation governed by kinetic equations with strongly contrasted Knudsen numbers in different subdomains is discussed. We consider a domain decomposition problem for Boltzmann- and Euler equations, establish the correct coupling conditions and prove the validity of the obtained coupled solution . Moreover numerical examples comparing different types of coupling conditions are presented.

We are concerned with a parameter choice strategy for the Tikhonov regularization \((\tilde{A}+\alpha I)\tilde{x}\) = T* \(\tilde{y}\)+ w where \(\tilde{A}\) is a (not necessarily selfadjoint) approximation of T*T and T*\(\tilde y\)+ w is a perturbed form of the (not exactly computed) term T*y. We give conditions for convergence and optimal convergence rates.

A polynomial function \(f : L \to L\) of a lattice \(\mathcal{L}\) = \((L; \land, \lor)\) is generated by the identity function id \(id(x)=x\) and the constant functions \(c_a (x) = a\) (for every \(x \in L\)), \(a \in L\) by applying the operations \(\land, \lor\) finitely often. Every polynomial function in one or also in several variables is a monotone function of \(\mathcal{L}\).
If every monotone function of \(\mathcal{L}\)is a polynomial function then \(\mathcal{L}\) is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices
and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice \(\mathcal{L}\) with a finite interval is order-polynomially complete if and only if \(\mathcal{L}\) is finite projective geometry. If \(\mathcal{L}\) is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of \(\mathcal{L}\). In the last sections we show the descriptive power of polynomial functions of
lattices and present several applications in geometry.

On derived varieties
(1996)

Derived varieties play an essential role in the theory of hyperidentities. In [11] we have shown that derivation diagrams are a useful tool in the analysis of derived algebras and varieties. In this paper this tool is developed further in order to use it for algebraic constructions of derived algebras. Especially the operator \(S\) of subalgebras, \(H\) of homomorphic irnages and \(P\) of direct products are studied. Derived groupoids from the groupoid \(N or (x,y)\) = \(x'\wedge y'\) and from abelian groups are considered. The latter class serves as an example for fluid algebras and varieties. A fluid variety \(V\) has no derived variety as a subvariety and is introduced as a counterpart for solid varieties. Finally we use a property of the commutator of derived algebras in order to show that solvability and nilpotency are preserved under derivation.

It is shown that Tikhonov regularization for ill- posed operator equation
\(Kx = y\) using a possibly unbounded regularizing operator \(L\) yields an orderoptimal algorithm with respect to certain stability set when the regularization parameter is chosen according to the Morozov's discrepancy principle. A more realistic error estimate is derived when the operators \(K\) and \(L\) are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also the estimates available under the Hilbert scale approach.

The article provides an asymptotic probabilistic analysis of the variance of the number of pivot steps required by phase II of the "shadow vertex algorithm" - a parametric variant of the simplex algorithm, which has been proposed by Borgwardt [1] . The analysis is done for data which satisfy a rotationally
invariant distribution law in the \(n\)-dimensional unit ball.

Let \(a_i i:= 1,\dots,m.\) be an i.i.d. sequence taking values in \(\mathbb{R}^n\). Whose convex hull is interpreted as a stochastic polyhedron \(P\). For a special class of random variables which decompose additively relative to their boundary simplices, eg. the volume of \(P\), integral representations of their first two moments are given which lead to asymptotic estimations of variances for special "additive variables" known from stochastic approximation theory in case of rotationally symmetric distributions.

Let \(a_1,\dots,a_m\) be independent random points in \(\mathbb{R}^n\) that are independent and identically distributed spherically symmetrical in \(\mathbb{R}^n\). Moreover, let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_m\) and let \(L_k\) be an arbitrary \(k\)-dimensional
subspace of \(\mathbb{R}^n\) with \(2\le k\le n-1\). Let \(X_k\) be the orthogonal projection image of \(X\) in \(L_k\). We call those vertices of \(X\), whose projection images in \(L_k\) are vertices of \(X_k\)as well shadow vertices of \(X\) with respect to the subspace \(L_k\) . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of \(X\) in \(L_k\).

Let (\(a_i)_{i\in \bf{N}}\) be a sequence of identically and independently distributed random vectors drawn from the \(d\)-dimensional unit ball \(B^d\)and let \(X_n\):= convhull \((a_1,\dots,a_n\)) be the random polytope generated by \((a_1,\dots\,a_n)\). Furthermore, let \(\Delta (X_n)\) : = (Vol \(B^d\) \ \(X_n\)) be the deviation of the polytope's volume from the volume of the ball. For uniformly distributed \(a_i\) and \(d\ge2\), we prove that tbe limiting distribution of \(\frac{\Delta (X_n)} {E(\Delta (X_n))}\) for \(n\to\infty\) satisfies a 0-1-law. Especially, we provide precise information about the asymptotic behaviour of the variance of \(\Delta (X_n\)). We deliver analogous results for spherically symmetric distributions in \(B^d\) with regularly varying tail.

Let \(a_1,\dots,a_m\) be i.i .d. vectors uniform on the unit sphere in \(\mathbb{R}^n\), \(m\ge n\ge3\) and let \(X\):= {\(x \in \mathbb{R}^n \mid a ^T_i x\leq 1\)} be the random polyhedron generated by. Furthermore, for linearly independent vectors \(u\), \(\bar u\) in \(\mathbb{R}^n\), let \(S_{u, \bar u}(X)\) be the number of shadow vertices of \(X\) in \(span (u, \bar u\)). The paper provides an asymptotic expansion of the expectation value \(E (S_{u, \bar u})\) for fixed \(n\) and \(m\to\infty\). The first terms of the expansion are given explicitly. Our investigation of \(E (S_{u, \bar u})\) is closely connected to Borgwardt's probabilistic analysis of the shadow vertex algorithm - a parametric variant of the simplex algorithm. We obtain an improved asymptotic upper bound for the number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data.

Let \(A\):= {\(a_i\mid i= 1,\dots,m\)} be an i.i.d. random sample in (\mathbb{R}^n\), which we consider a random polyhedron, either as the convex hull of the \(a_i\) or as the intersection of halfspaces {\(x \mid a ^T_i x\leq 1\)}. We introduce a class of polyhedral functionals we will call "additive-type functionals", which covers a number of polyhedral functionals discussed in different mathematical fields, where the emphasis in our contribution will be on those, which arise in linear optimization theory. The class of additive-type functionals is a suitable setting in order to unify and to simplify the asymptotic probabilistic analysis of first and second moments of polyhedral functionals. We provide examples of asymptotic results on expectations and on variances.

Let \(a_1,\dots,a_n\) be independent random points in \(\mathbb{R}^d\) spherically symmetrically but not necessarily identically distributed. Let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_n\) and for any \(k\)-dimensional subspace \(L\subseteq \mathbb{R}^d\) let \(Vol_L(X) :=\lambda_k(L\cap X)\) be the volume of \(X\cap L\) with respect to the \(k\)-dimensional Lebesgue measure \(\lambda_k, k=1,\dots,d\). Furthermore, let \(F^{(i)}\)(t):= \(\bf{Pr}\) \(\)(\(\Vert a_i \|_2\leq t\)),
\(t \in \mathbb{R}^+_0\) , be the radial distribution function of \(a_i\). We prove that the expectation
functional \(\Phi_L\)(\(F^{(1)}, F^{(2)},\dots, F^{(n)})\) := \(E(Vol_L(X)\)) is strictly decreasing in
each argument, i.e. if \(F^{(i)}(t) \le G^{(i)}(t)t\), \(t \in {R}^+_0\), but \(F^{(i)} \not\equiv G^{(i)}\), we show \(\Phi\) \((\dots, F^{(i)}, \dots\)) > \(\Phi(\dots,G^{(i)},\dots\)). The proof is clone in the more general framework
of continuous and \(f\)- additive polytope functionals.