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Tue, 15 Jun 2021 14:05:47 +0200Tue, 15 Jun 2021 14:05:47 +0200On the construction of number fields with solvable Galois group
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6407
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.Carlo Sircanadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6407Tue, 15 Jun 2021 14:05:47 +0200The generic character table of \({\rm Spin}_8^+(q)\)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6405
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.Emil Rotiliodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6405Fri, 11 Jun 2021 10:39:05 +0200Estimation and Portfolio Optimization with Expert Opinions in Discrete-time Financial Markets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6382
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.Yihua Xudoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6382Mon, 31 May 2021 11:29:43 +0200Improving thermal ablation of liver tumors: Modeling and parameter identification of laser-induced interstitial thermotherapy
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6322
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.Matthias Andresdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6322Fri, 21 May 2021 08:20:53 +0200Implicit finite volume method with a posteriori limiting for transport networks
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6300
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.Matthias Eimer; Raul Borsche; Norbert Siedowpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6300Tue, 16 Mar 2021 11:29:50 +0100Reflected Anticipated Backward Stochastic Differential Equations with Default Risk, Numerical Algorithms and Applications
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6296
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.Jingnan Wangdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6296Fri, 12 Mar 2021 13:26:14 +0100MINT-Projekte mit Arduino & Raspberry Pi - Technische Grundlagen und Projektideen
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6255
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.Patrick Capraroarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6255Tue, 09 Feb 2021 12:05:52 +0100Modular Representation Theory of Finite Groups
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6229
Lecture notes for the lecture "Representation Theory" WS 2019/2020 (Revised version, 2021)Caroline Lassueur; Niamh Farrelllecturehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6229Thu, 21 Jan 2021 13:53:06 +0100Character Theory of Finite Groups
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6228
Caroline Lassueurlecturehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6228Thu, 21 Jan 2021 13:50:05 +0100Mathematische Modellierungswoche Dezember 2020 - Themenschwerpunkt Digitaler Wandel
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6196
Die Felix-Klein-Modellierungswoche ist eine Projektwoche für Schülerinnen und Schüler der gymnasialen Oberstufe. Wir beschreiben hier die Projekte, die während der Modellierungswoche im Dezember 2020 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Digitaler Wandel".Patrick Capraro; Lynn Knippertz; Lisa Schneiderarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6196Thu, 07 Jan 2021 12:11:40 +0100Analysis of Systems of Hyperbolic Partial Differential Equations Coupled to Switched Differential Algebraic Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6205
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 validationDamla Kocogludoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6205Thu, 07 Jan 2021 11:49:29 +0100Documentation for LinTim 2020.12
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6202
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.netAlexander Schiewe; Sebastian Albert; Philine Schiewe; Anita Schöbel; Felix Spühlercoursematerialhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6202Mon, 04 Jan 2021 11:11:45 +0100A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6183
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.Pawan Kumar; Christina Surulescuarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6183Tue, 15 Dec 2020 14:13:14 +0100Produktorientierte mathematische Modellierung am Beispiel eines Schrittzählers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6184
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.Patrick Capraroarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6184Tue, 15 Dec 2020 14:02:10 +0100Numerical methods for multi-scale cell migration models
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6125
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.Gregor Corbindoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6125Fri, 04 Dec 2020 08:55:23 +0100Explicit and effective Mather-Yau correspondence in view of analytic gradings
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6150
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.Raul-Paul Epuredoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6150Tue, 01 Dec 2020 08:23:49 +0100Operator semigroups and infinite dimensional analysis applied to problems from mathematical physics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6128
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.Andreas Nonnenmacherdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6128Fri, 30 Oct 2020 09:39:58 +0100Mathematische Modellierungswoche Februar 2020 - Themenschwerpunkt Gesundheit und Medizin
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6116
Die Felix-Klein-Modellierungswoche ist eine Projektwoche für Schülerinnen und Schüler der gymnasialen Oberstufe. Wir beschreiben hier die Projekte, die während der Modellierungswoche im Februar 2020 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Gesundheit und Medizin".Patrick Capraro; Martin Bracke; Lena Leiß; Jan Boeckmann; Thomas Marx; Stephan Helfrich; Moritz Schäferarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6116Thu, 15 Oct 2020 14:02:45 +0200Mesh Deformation in the Context of Isogeometric Analysis
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6082
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.Alexander Shamanskiydoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6082Mon, 14 Sep 2020 12:45:30 +0200Isogeometric Analysis for Scaled Boundary Parametrizations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6077
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.Clarissa Ariolidoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6077Wed, 09 Sep 2020 10:02:13 +0200Estimating the fibre length distribution in fibre reinforced polymers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6065
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.Jan Niedermeyerdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6065Mon, 31 Aug 2020 12:08:41 +0200On the complexity and approximability of optimization problems with Minimum Quantity Constraints
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6050
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.Alexander Sieberdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/6050Fri, 21 Aug 2020 12:15:45 +0200Cohomology of Groups
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5992
Lecture notes for the lecture "Cohomology of Groups" SS 2018Caroline Lassueurlecturehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5992Tue, 23 Jun 2020 09:01:50 +0200Modular Representation Theory of Finite Groups
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