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Wed, 26 Apr 2017 14:40:34 +0200Wed, 26 Apr 2017 14:40:34 +0200Einfaches Motion Capturing in MATLAB
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4632
Der vorliegende Artikel befasst sich mit der Realisierung eines einfachen Motion Capturing Verfahrens in MATLAB als Vorschlag für eine Umsetzung in der Schule. Die zugrunde liegende Mathematik kann ab der Mittelstufe leicht vermittelt werden. Je nach technischer Ausstattung können mit einfachen Mitteln farbige Marker in Videos oder Webcam-Streams verfolgt werden. Notwendige Konzepte und Algorithmen werden im Artikel beleuchtet.Andreas Rotharticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4632Wed, 26 Apr 2017 14:40:34 +0200Wir entwickeln einen Synthesizer
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4631
Die Akustik liefert einen interessanten Hintergrund, interdisziplinären und fächerverbindenen Unterricht zwischen Mathematik, Physik und Musik durchzuführen. SchülerInnen können hierbei beispielsweise experimentell tätig sein, indem sie Audioaufnahmen selbst erzeugen und sich mit Computersoftware Frequenzspektren erzeugen lassen. Genauso können die Schüler auch Frequenzspektren vorgeben und daraus Klänge erzeugen. Dies kann beispielsweise dazu dienen, den Begriff der Obertöne im Musikunterricht physikalisch oder mathematisch greifbar zu machen oder in der Harmonielehre Frequenzverhältnisse von Intervallen und Dreiklängen näher zu untersuchen.
Der Computer ist hier ein sehr nützliches Hilfsmittel, da der mathematische Hintergrund dieser Aufgabe -- das Wechseln zwischen Audioaufnahme und ihrem Frequenzbild -- sich in der Fourier-Analysis findet, die für SchülerInnen äußerst anspruchsvoll ist. Indem man jedoch die Fouriertransformation als numerisches Hilfsmittel einführt, das nicht im Detail verstanden werden muss, lässt sich an anderer Stelle interessante Mathematik betreiben und die Zusammenhänge zwischen Akustik und Musik können spielerisch erfahren werden.
Im folgenden Beitrag wird eine Herangehensweise geschildert, wie wir sie bereits bei der Felix-Klein-Modellierungswoche umgesetzt haben: Die SchülerInnen haben den Auftrag erhalten, einen Synthesizer zu entwickeln, mit dem verschiedene Musikinstrumente nachgeahmt werden können. Als Hilfsmittel haben sie eine kurze Einführung in die Eigenschaften der Fouriertransformation erhalten, sowie Audioaufnahmen verschiedener Instrumente.Patrick Capraroarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4631Wed, 26 Apr 2017 14:38:18 +0200Small self-centralizing subgroups in defect groups of finite classical groups
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4617
In this thesis, we consider a problem from modular representation theory of finite groups. Lluís Puig asked the question whether the order of the defect groups of a block \( B \) of the group algebra of a given finite group \( G \) can always be bounded in terms of the order of the vertices of an arbitrary simple module lying in \( B \).
In characteristic \( 2 \), there are examples showing that this is not possible in general, whereas in odd characteristic, no such examples are known. For instance, it is known that the answer to Puig's question is positive in case that \( G \) is a symmetric group, by work of Danz, Külshammer, and Puig.
Motivated by this, we study the cases where \( G \) is a finite classical group in non-defining characteristic or one of the finite groups \( G_2(q) \) or \( ³D_4(q) \) of Lie type, again in non-defining characteristic. Here, we generalize Puig's original question by replacing the vertices occurring in his question by arbitrary self-centralizing subgroups of the defect groups. We derive positive and negative answers to this generalized question.
\[\]
In addition to that, we determine the vertices of the unipotent simple \( GL_2(q) \)-module labeled by the partition \( (1,1) \) in characteristic \( 2 \). This is done using a method known as Brauer construction.Pablo Lukadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4617Thu, 16 Mar 2017 08:27:05 +0100Graph Coloring Applications and Defining Sets in Graph Theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4612
Abstract
The main theme of this thesis is about Graph Coloring Applications and Defining Sets in Graph Theory.
As in the case of block designs, finding defining sets seems to be difficult problem, and there is not a general conclusion. Hence we confine us here to some special types of graphs like bipartite graphs, complete graphs, etc.
In this work, four new concepts of defining sets are introduced:
• Defining sets for perfect (maximum) matchings
• Defining sets for independent sets
• Defining sets for edge colorings
• Defining set for maximal (maximum) clique
Furthermore, some algorithms to find and construct the defining sets are introduced. A review on some known kinds of defining sets in graph theory is also incorporated, in chapter 2 the basic definitions and some relevant notations used in this work are introduced.
chapter 3 discusses the maximum and perfect matchings and a new concept for a defining set for perfect matching.
Different kinds of graph colorings and their applications are the subject of chapter 4.
Chapter 5 deals with defining sets in graph coloring. New results are discussed along with already existing research results, an algorithm is introduced, which enables to determine a defining set of a graph coloring.
In chapter 6, cliques are discussed. An algorithm for the determination of cliques using their defining sets. Several examples are included.
Masoumeh Ahadi Moghaddamdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4612Mon, 06 Mar 2017 14:35:31 +0100Portfolio Optimization with Risk Constraints in the View of Stochastic Interest Rates
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4602
We discuss the portfolio selection problem of an investor/portfolio manager in an arbitrage-free financial market where a money market account, coupon bonds and a stock are traded continuously. We allow for stochastic interest rates and in particular consider one and two-factor Vasicek models for the instantaneous
short rates. In both cases we consider a complete and an incomplete market setting by adding a suitable number of bonds.
The goal of an investor is to find a portfolio which maximizes expected utility
from terminal wealth under budget and present expected short-fall (PESF) risk
constraints. We analyze this portfolio optimization problem in both complete and
incomplete financial markets in three different cases: (a) when the PESF risk is
minimum, (b) when the PESF risk is between minimum and maximum and (c) without risk constraints. (a) corresponds to the portfolio insurer problem, in (b) the risk constraint is binding, i.e., it is satisfied with equality, and (c) corresponds
to the unconstrained Merton investment.
In all cases we find the optimal terminal wealth and portfolio process using the
martingale method and Malliavin calculus respectively. In particular we solve in the incomplete market settings the dual problem explicitly. We compare the
optimal terminal wealth in the cases mentioned using numerical examples. Without
risk constraints, we further compare the investment strategies for complete
and incomplete market numerically.William Ntambaradoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4602Wed, 01 Mar 2017 08:14:31 +0100Asymptotics for change-point tests and change-point estimators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4599
In change-point analysis the point of interest is to decide if the observations follow one model
or if there is at least one time-point, where the model has changed. This results in two sub-
fields, the testing of a change and the estimation of the time of change. This thesis considers
both parts but with the restriction of testing and estimating for at most one change-point.
A well known example is based on independent observations having one change in the mean.
Based on the likelihood ratio test a test statistic with an asymptotic Gumbel distribution was
derived for this model. As it is a well-known fact that the corresponding convergence rate is
very slow, modifications of the test using a weight function were considered. Those tests have
a better performance. We focus on this class of test statistics.
The first part gives a detailed introduction to the techniques for analysing test statistics and
estimators. Therefore we consider the multivariate mean change model and focus on the effects
of the weight function. In the case of change-point estimators we can distinguish between
the assumption of a fixed size of change (fixed alternative) and the assumption that the size
of the change is converging to 0 (local alternative). Especially, the fixed case in rarely analysed
in the literature. We show how to come from the proof for the fixed alternative to the
proof of the local alternative. Finally, we give a simulation study for heavy tailed multivariate
observations.
The main part of this thesis focuses on two points. First, analysing test statistics and, secondly,
analysing the corresponding change-point estimators. In both cases, we first consider a
change in the mean for independent observations but relaxing the moment condition. Based on
a robust estimator for the mean, we derive a new type of change-point test having a randomized
weight function. Secondly, we analyse non-linear autoregressive models with unknown
regression function. Based on neural networks, test statistics and estimators are derived for
correctly specified as well as for misspecified situations. This part extends the literature as
we analyse test statistics and estimators not only based on the sample residuals. In both
sections, the section on tests and the one on the change-point estimator, we end with giving
regularity conditions on the model as well as the parameter estimator.
Finally, a simulation study for the case of the neural network based test and estimator is
given. We discuss the behaviour under correct and mis-specification and apply the neural
network based test and estimator on two data sets.Stefanie Schwaardoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4599Tue, 28 Feb 2017 13:18:02 +0100Having a Plan B for Robust Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4585
We extend the standard concept of robust optimization by the introduction of an alternative solution. In contrast to the classic concept, one is allowed to chose two solutions from which the best can be picked after the uncertain scenario has been revealed. We focus in this paper on the resulting robust problem for combinatorial problems with bounded uncertainty sets. We present a reformulation of the robust problem which decomposes it into polynomially many subproblems. In each subproblem one needs to find two solutions which are connected by a cost function which penalizes if the same element is part of both solutions. Using this reformulation, we show how the robust problem can be solved efficiently for the unconstrained combinatorial problem, the selection problem, and the minimum spanning tree problem. The robust problem corresponding to the shortest path problem turns out to be NP-complete on general graphs. However, for series-parallel graphs, the robust shortest path problem can be solved efficiently. Further, we show how approximation algorithms for the subproblem can be used to compute approximate solutions for the original problem.André Chasseinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4585Wed, 22 Feb 2017 14:40:39 +0100Manifolds
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4529
Lecture notes written to accompany a one semester course introducing to differential manifolds. Beyond the basic notions differential forms including Stokes' theorem are treated, as well as vector fields and flows on a differential manifold.Klaus Wirthmüllerlecturehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4529Tue, 03 Jan 2017 11:57:33 +0100A predictive-control framework to eliminate bus bunching
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4500
Buses not arriving on time and then arriving all at once - this phenomenon is known from
busy bus routes and is called bus bunching.
This thesis combines the well studied but so far separate areas of bus-bunching prediction
and dynamic holding strategies, which allow to modulate buses’ dwell times at stops to
eliminate bus bunching. We look at real data of the Dublin Bus route 46A and present
a headway-based predictive-control framework considering all components like data
acquisition, prediction and control strategies. We formulate time headways as time series
and compare several prediction methods for those. Furthermore we present an analytical
model of an artificial bus route and discuss stability properties and dynamic holding
strategies using both data available at the time and predicted headway data. In a numerical
simulation we illustrate the advantages of the presented predictive-control framework
compared to the classical approaches which only use directly available data.Matthias Andresmasterthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4500Fri, 25 Nov 2016 10:16:51 +0100On a structured multiscale model for acid-mediated tumor invasion: the effects of adhesion and proliferation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4488
We propose a multiscale model for tumor cell migration in a tissue network. The system of equations involves a structured population model for the tumor cell density, which besides time and
position depends on a further variable characterizing the cellular state with respect to the amount
of receptors bound to soluble and insoluble ligands. Moreover, this equation features pH-taxis and
adhesion, along with an integral term describing proliferation conditioned by receptor binding. The
interaction of tumor cells with their surroundings calls for two more equations for the evolution of
tissue fibers and acidity (expressed via concentration of extracellular protons), respectively. The
resulting ODE-PDE system is highly nonlinear. We prove the global existence of a solution and
perform numerical simulations to illustrate its behavior, paying particular attention to the influence
of the supplementary structure and of the adhesion.Christian Engwer; Christian Stinner; Christina Surulescupreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4488Fri, 11 Nov 2016 08:15:27 +0100Modeling Road Roughness with Conditional Random Fields
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4469
A vehicles fatigue damage is a highly relevant figure in the complete vehicle design process.
Long term observations and statistical experiments help to determine the influence of differnt parts of the vehicle, the driver and the surrounding environment.
This work is focussing on modeling one of the most important influence factors of the environment: road roughness. The quality of the road is highly dependant on several surrounding factors which can be used to create mathematical models.
Such models can be used for the extrapolation of information and an estimation of the environment for statistical studies.
The target quantity we focus on in this work ist the discrete International Roughness Index or discrete IRI. The class of models we use and evaluate is a discriminative classification model called Conditional Random Field.
We develop a suitable model specification and show new variants of stochastic optimizations to train the model efficiently.
The model is also applied to simulated and real world data to show the strengths of our approach.Alexander Lemkendoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4469Mon, 17 Oct 2016 14:16:31 +0200Signature Standard Bases over Principal Ideal Rings
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4457
By using Gröbner bases of ideals of polynomial algebras over a field, many implemented algorithms manage to give exciting examples and counter examples in Commutative Algebra and Algebraic Geometry. Part A of this thesis will focus on extending the concept of Gröbner bases and Standard bases for polynomial algebras over the ring of integers and its factors \(\mathbb{Z}_m[x]\). Moreover we implemented two algorithms for this case in Singular which use different approaches in detecting useless computations, the classical Buchberger algorithm and a F5 signature based algorithm. Part B includes two algorithms that compute the graded Hilbert depth of a graded module over a polynomial algebra \(R\) over a field, as well as the depth and the multigraded Stanley depth of a factor of monomial ideals of \(R\). The two algorithms provide faster computations and examples that lead B. Ichim and A. Zarojanu to a counter example of a question of J. Herzog. A. Duval, B. Goeckner, C. Klivans and J. Martin have recently discovered a counter example for the Stanley Conjecture. We prove in this thesis that the Stanley Conjecture holds in some special cases. Part D explores the General Neron Desingularization in the frame of Noetherian local domains of dimension 1. We have constructed and implemented in Singular and algorithm that computes a strong Artin Approximation for Cohen-Macaulay local rings of dimension 1. Adrian Popescudoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4457Tue, 04 Oct 2016 09:49:56 +0200On a coupled SDE-PDE system modeling acid-mediated tumor invasion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4451
We propose and analyze a multiscale model for acid-mediated tumor invasion
accounting for stochastic effects on the subcellular level.
The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density,
the movement being directed towards pH gradients in the local microenvironment,
which is coupled to a PDE-SDE system characterizing the
dynamics of extracellular and intracellular proton concentrations, respectively.
The global well-posedness of the model is shown and
numerical simulations are performed in order to illustrate the solution behavior.Sandesh Athni Hiremath; Anna Zhigun; Stefanie Sonner; Christina Surulescupreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4451Thu, 22 Sep 2016 08:23:05 +0200Mathematische Modellierung eines Segways mit Umsetzung in der Schule als interdisziplinäre Projektarbeit
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4441
Das Ziel dieser Arbeit besteht darin, aufzuzeigen, wie eine mathematische Modellierung, verbunden mit Simulations- und Ansteuerungsaspekten eines Segways im Mathematikunterricht der gymnasialen Oberstufe als interdisziplinäres Projekt umgesetzt werden kann. Dabei werden sowohl Chancen, im Sinne von erreichbaren mathematischen Kompetenzen, als auch Schwierigkeiten eines solchen Projektes mit einer interdisziplinären Umsetzung geschildert.Jean-Marie Lantaumasterthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4441Tue, 06 Sep 2016 13:53:45 +0200Gröbner Bases over Extention Fields of \(\mathbb{Q}\)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4428
Gröbner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field.
In this thesis, we consider Gröbner bases computations over two types of coefficient fields. First, consider a simple extension \(K=\mathbb{Q}(\alpha)\) of \(\mathbb{Q}\), where \(\alpha\) is an algebraic number, and let \(f\in \mathbb{Q}[t]\) be the minimal polynomial of \(\alpha\). Second, let \(K'\) be the algebraic function field over \(\mathbb{Q}\) with transcendental parameters \(t_1,\ldots,t_m\), that is, \(K' = \mathbb{Q}(t_1,\ldots,t_m)\). In particular, we present efficient algorithms for computing Gröbner bases over \(K\) and \(K'\). Moreover, we present an efficient method for computing syzygy modules over \(K\).
To compute Gröbner bases over \(K\), starting from the ideas of Noro [35], we proceed by joining \(f\) to the ideal to be considered, adding \(t\) as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,4,27], that is, by inferring information in characteristic zero from information in characteristic \(p > 0\). For suitable primes \(p\), the minimal polynomial \(f\) is reducible over \(\mathbb{F}_p\). This allows us to apply modular methods once again, on a second level, with respect to the
modular factors of \(f\). The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over \(K\).
On the other hand, to compute Gröbner bases over \(K'\), our new algorithm first specializes the parameters \(t_1,\ldots,t_m\) to reduce the problem from \(K'[x_1,\ldots,x_n]\) to \(\mathbb{Q}[x_1,\ldots,x_n]\). The algorithm then computes a set of Gröbner bases of specialized ideals. From this set of Gröbner bases with coefficients in \(\mathbb{Q}\), it obtains a Gröbner basis of the input ideal using sparse multivariate rational interpolation.
At current state, these algorithms are probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods.Dereje Kifle Bokudoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4428Wed, 10 Aug 2016 15:34:30 +0200Regionalized Assortment Planning for Multiple Chain Stores: Complexity, Approximability, and Solution Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4424
In retail, assortment planning refers to selecting a subset of products to offer that maximizes profit. Assortments can be planned for a single store or a retailer with multiple chain stores where demand varies between stores. In this paper, we assume that a retailer with a multitude of stores wants to specify her offered assortment. To suit all local preferences, regionalization and store-level assortment optimization are widely used in practice and lead to competitive advantages. When selecting regionalized assortments, a tradeoff between expensive, customized assortments in every store and inexpensive, identical assortments in all stores that neglect demand variation is preferable.
We formulate a stylized model for the regionalized assortment planning problem (APP) with capacity constraints and given demand. In our approach, a 'common assortment' that is supplemented by regionalized products is selected. While products in the common assortment are offered in all stores, products in the local assortments are customized and vary from store to store.
Concerning the computational complexity, we show that the APP is strongly NP-complete. The core of this hardness result lies in the selection of the common assortment. We formulate the APP as an integer program and provide algorithms and methods for obtaining approximate solutions and solving large-scale instances.
Lastly, we perform computational experiments to analyze the benefits of regionalized assortment planning depending on the variation in customer demands between stores.Michael Hopf; Clemens Thielen; Benedikt Kasper; Hans Corstenworkingpaperhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4424Tue, 09 Aug 2016 09:43:13 +0200Interest Rate Modeling - The Potential Approach and Multi-Curve Potential Models
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4420
This thesis is concerned with interest rate modeling by means of the potential approach. The contribution of this work is twofold. First, by making use of the potential approach and the theory of affine Markov processes, we develop a general class of rational models to the term structure of interest rates which we refer to as "the affine rational potential model". These models feature positive interest rates and analytical pricing formulae for zero-coupon bonds, caps, swaptions, and European currency options. We present some concrete models to illustrate the scope of the affine rational potential model and calibrate a model specification to real-world market data. Second, we develop a general family of "multi-curve potential models" for post-crisis interest rates. Our models feature positive stochastic basis spreads, positive term structures, and analytic pricing formulae for interest rate derivatives. This modeling framework is also flexible enough to accommodate negative interest rates and positive basis spreads.Anh-The Nguyendoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4420Fri, 05 Aug 2016 12:31:23 +0200The Bootstrap for the Functional Autoregressive Model FAR(1)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4410
Functional data analysis is a branch of statistics that deals with observations \(X_1,..., X_n\) which are curves. We are interested in particular in time series of dependent curves and, specifically, consider the functional autoregressive process of order one (FAR(1)), which is defined as \(X_{n+1}=\Psi(X_{n})+\epsilon_{n+1}\) with independent innovations \(\epsilon_t\). Estimates \(\hat{\Psi}\) for the autoregressive operator \(\Psi\) have been investigated a lot during the last two decades, and their asymptotic properties are well understood. Particularly difficult and different from scalar- or vector-valued autoregressions are the weak convergence properties which also form the basis of the bootstrap theory.
Although the asymptotics for \(\hat{\Psi}{(X_{n})}\) are still tractable, they are only useful for large enough samples. In applications, however, frequently only small samples of data are available such that an alternative method for approximating the distribution of \(\hat{\Psi}{(X_{n})}\) is welcome. As a motivation, we discuss a real-data example where we investigate a changepoint detection problem for a stimulus response dataset obtained from the animal physiology group at the Technical University of Kaiserslautern.
To get an alternative for asymptotic approximations, we employ the naive or residual-based bootstrap procedure. In this thesis, we prove theoretically and show via simulations that the bootstrap provides asymptotically valid and practically useful approximations of the distributions of certain functions of the data. Such results may be used to calculate approximate confidence bands or critical bounds for tests.
Euna Gesare Nyarigedoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4410Wed, 06 Jul 2016 12:30:55 +0200Integrality of representations of finite groups
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4408
Since the early days of representation theory of finite groups in the 19th century, it was known that complex linear representations of finite groups live over number fields, that is, over finite extensions of the field of rational numbers.
While the related question of integrality of representations was answered negatively by the work of Cliff, Ritter and Weiss as well as by Serre and Feit, it was not known how to decide integrality of a given representation.
In this thesis we show that there exists an algorithm that given a representation of a finite group over a number field decides whether this representation can be made integral.
Moreover, we provide theoretical and numerical evidence for a conjecture, which predicts the existence of splitting fields of irreducible characters with integrality properties.
In the first part, we describe two algorithms for the pseudo-Hermite normal form, which is crucial when handling modules over ring of integers.
Using a newly developed computational model for ideal and element arithmetic in number fields, we show that our pseudo-Hermite normal form algorithms have polynomial running time.
Furthermore, we address a range of algorithmic questions related to orders and lattices over Dedekind domains, including computation of genera, testing local isomorphism, computation of various homomorphism rings and computation of Solomon zeta functions.
In the second part we turn to the integrality of representations of finite groups and show that an important ingredient is a thorough understanding of the reduction of lattices at almost all prime ideals.
By employing class field theory and tools from representation theory we solve this problem and eventually describe an algorithm for testing integrality.
After running the algorithm on a large set of examples we are led to a conjecture on the existence of integral and nonintegral splitting fields of characters.
By extending techniques of Serre we prove the conjecture for characters with rational character field and Schur index two.Tommy Hofmanndoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4408Mon, 04 Jul 2016 16:07:15 +0200Hecke algebras of type A: Auslander--Reiten quivers and branching rules
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4386
The thesis consists of two parts. In the first part we consider the stable Auslander--Reiten quiver of a block \(B\) of a Hecke algebra of the symmetric group at a root of unity in characteristic zero. The main theorem states that if the ground field is algebraically closed and \(B\) is of wild representation type, then the tree class of every connected component of the stable Auslander--Reiten quiver \(\Gamma_{s}(B)\) of \(B\) is \(A_{\infty}\). The main ingredient of the proof is a skew group algebra construction over a quantum complete intersection. Also, for these algebras the stable Auslander--Reiten quiver is computed in the case where the defining parameters are roots of unity. As a result, the tree class of every connected component of the stable Auslander--Reiten quiver is \(A_{\infty}\).\[\]
In the second part of the thesis we are concerned with branching rules for Hecke algebras of the symmetric group at a root of unity. We give a detailed survey of the theory initiated by I. Grojnowski and A. Kleshchev, describing the Lie-theoretic structure that the Grothendieck group of finite-dimensional modules over a cyclotomic Hecke algebra carries. A decisive role in this approach is played by various functors that give branching rules for cyclotomic Hecke algebras that are independent of the underlying field. We give a thorough definition of divided power functors that will enable us to reformulate the Scopes equivalence of a Scopes pair of blocks of Hecke algebras of the symmetric group. As a consequence we prove that two indecomposable modules that correspond under this equivalence have a common vertex. In particular, we verify the Dipper--Du Conjecture in the case where the blocks under consideration have finite representation type.Simon Schmiderdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4386Wed, 01 Jun 2016 15:32:16 +0200Global existence for a degenerate haptotaxis model of tumor invasion under the go-or-grow dichotomy hypothesis
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4384
We propose and study a strongly coupled PDE-ODE-ODE system modeling cancer cell invasion through a tissue network
under the go-or-grow hypothesis asserting that cancer cells can either move or proliferate. Hence our setting features
two interacting cell populations with their mutual transitions and involves tissue-dependent degenerate diffusion and
haptotaxis for the moving subpopulation. The proliferating cells and the tissue evolution are characterized by way of ODEs
for the respective densities. We prove the global existence of weak solutions and illustrate the model behaviour by
numerical simulations in a two-dimensional setting.Anna Zhigun; Christina Surulescu; Alexander Huntpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4384Tue, 31 May 2016 08:55:45 +0200New Aspects of Inflation Modeling
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4381
Inflation modeling is a very important tool for conducting an efficient monetary policy. This doctoral thesis reviewed inflation models, in particular the Phillips curve models of inflation dynamics. We focused on a well known and widely used model, the so-called three equation new Keynesian model which is a system of equations consisting of a new Keynesian Phillips curve (NKPC), an investment and saving (IS) curve and an interest rate rule.
We gave a detailed derivation of these equations. The interest rate rule used in this model is normally determined by using a Lagrangian method to solve an optimal control problem constrained by a standard discrete time NKPC which describes the inflation dynamics and an IS curve that represents the output gaps dynamics. In contrast to the real world, this method assumes that the policy makers intervene continuously. This means that the costs resulting from the change in the interest rates are ignored. We showed also that there are approximation errors made, when one log-linearizes non linear equations, by doing the derivation of the standard discrete time NKPC.
We agreed with other researchers as mentioned in this thesis, that errors which result from ignoring such log-linear approximation errors and the costs of altering interest rates by determining interest rate rule, can lead to a suboptimal interest rate rule and hence to non-optimal paths of output gaps and inflation rate.
To overcome such a problem, we proposed a stochastic optimal impulse control method. We formulated the problem as a stochastic optimal impulse control problem by considering the costs of change in interest rates and the approximation error terms. In order to formulate this problem, we first transform the standard discrete time NKPC and the IS curve into their high-frequency versions and hence into their continuous time versions where error terms are described by a zero mean Gaussian white noise with a finite and constant variance. After formulating this problem, we use the quasi-variational inequality approach to solve analytically a special case of the central bank problem, where an inflation rate is supposed to be on target and a central bank has to optimally control output gap dynamics. This method gives an optimal control band in which output gap process has to be maintained and an optimal control strategy, which includes the optimal size of intervention and optimal intervention time, that can be used to keep the process into the optimal control band.
Finally, using a numerical example, we examined the impact of some model parameters on optimal control strategy. The results show that an increase in the output gap volatility as well as in the fixed and proportional costs of the change in interest rate lead to an increase in the width of the optimal control band. In this case, the optimal intervention requires the central bank to wait longer before undertaking another control action.François Sindamubaradoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4381Tue, 24 May 2016 15:03:18 +0200Recursive Utility and Stochastic Differential Utility: From Discrete to Continuous Time
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4380
In this thesis, mathematical research questions related to recursive utility and stochastic differential utility (SDU) are explored.
First, a class of backward equations under nonlinear expectations is investigated: Existence and uniqueness of solutions are established, and the issues of stability and discrete-time approximation are addressed. It is then shown that backward equations of this class naturally appear as a continuous-time limit in the context of recursive utility with nonlinear expectations.
Then, the Epstein-Zin parametrization of SDU is studied. The focus is on specifications with both relative risk aversion and elasitcity of intertemporal substitution greater that one. A concave utility functional is constructed and a utility gradient inequality is established.
Finally, consumption-portfolio problems with recursive preferences and unspanned risk are investigated. The investor's optimal strategies are characterized by a specific semilinear partial differential equation. The solution of this equation is constructed by a fixed point argument, and a corresponding efficient and accurate method to calculate optimal strategies numerically is given.Thomas Seiferlingdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4380Mon, 23 May 2016 10:55:22 +0200Utility-Based Risk Measures and Time Consistency of Dynamic Risk Measures
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4370
This thesis deals with risk measures based on utility functions and time consistency of dynamic risk measures. It is therefore aimed at readers interested in both, the theory of static and dynamic financial risk measures in the sense of Artzner, Delbaen, Eber and Heath [7], [8] and the theory of preferences in the tradition of von Neumann and Morgenstern [134].
A main contribution of this thesis is the introduction of optimal expected utility (OEU) risk measures as a new class of utility-based risk measures. We introduce OEU, investigate its main properties, and its applicability to risk measurement and put it in perspective to alternative risk measures and notions of certainty equivalents. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU reacts in a more sensitive way to slight changes of the probability of a financial loss than value at risk (V@R) and average value at risk.
Further, we propose implied risk aversion as a coherent rating methodology for retail structured products (RSPs). Implied risk aversion is based on optimal expected utility risk measures and, in contrast to standard V@R-based ratings, takes into account both the upside potential and the downside risks of such products. In addition, implied risk aversion is easily interpreted in terms of an individual investor's risk aversion: A product is attractive (unattractive) for an investor if its implied risk aversion is higher (lower) than his individual risk aversion. We illustrate this approach in a case study with more than 15,000 warrants on DAX ® and find that implied risk aversion is able to identify favorable products; in particular, implied risk aversion is not necessarily increasing with respect to the strikes of call warrants.
Another main focus of this thesis is on consistency of dynamic risk measures. To this end, we study risk measures on the space of distributions, discuss concavity on the level of distributions and slightly generalize Weber's [137] findings on the relation of time consistent dynamic risk measures to static risk measures to the case of dynamic risk measures with time-dependent parameters. Finally, this thesis investigates how recursively composed dynamic risk measures in discrete time, which are time consistent by construction, can be related to corresponding dynamic risk measures in continuous time. We present different approaches to establish this link and outline the theoretical basis and the practical benefits of this relation. The thesis concludes with a numerical implementation of this theory.Sebastian Geisseldoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4370Tue, 17 May 2016 10:22:33 +0200Approximation of Ellipsoids Using Bounded Uncertainty Sets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4344
In this paper, we discuss the problem of approximating ellipsoid uncertainty sets with bounded (gamma) uncertainty sets. Robust linear programs with ellipsoid uncertainty lead to quadratically constrained programs, whereas robust linear programs with bounded uncertainty sets remain linear programs which are generally easier to solve.
We call a bounded uncertainty set an inner approximation of an ellipsoid if it is contained in it. We consider two different inner approximation problems. The first problem is to find a bounded uncertainty set which sticks close to the ellipsoid such that a shrank version of the ellipsoid is contained in it. The approximation is optimal if the required shrinking is minimal. In the second problem, we search for a bounded uncertainty set within the ellipsoid with maximum volume. We present how both problems can be solved analytically by stating explicit formulas for the optimal solutions of these problems.
Further, we present in a computational experiment how the derived approximation techniques can be used to approximate shortest path and network flow problems which are affected by ellipsoidal uncertainty.André Chasseinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4344Wed, 23 Mar 2016 16:01:15 +0100