Open Theses

Bachelor Theses

Master/Diploma Theses

A Mathematical Model with Delays for the Interaction of Transcription Factors GATA-1 and PU.1

Announcement: 15.11.2010
Supervisor: PD Dr. Peter Massopust, massopustma.tum.de
Topic and Description: Hematopoietic stem cells guarantee the continuous supply of fully differentiated, functional cells of various types in the peripheral blood. The process that controls the differentiation into the different lineages of the hematopoietic system (erythroid, myeloid, lymphoid) is referred to as lineage specification. This process requires a potentially multi-step decision sequence which determines the fate of the cells and their successors. It is generally accepted that lineage specification is regulated by a complex system of interacting transcription factors. However, the underlying principles controlling this regulation are currently unknown.
In a recent paper entitled Towards an understanding of lineage specification in hematopoietic stem cells: A mathematical model for the interaction of transcription factors GATA-1 and PU.1 by I. Roeder and I. Glauche (Journal of Theoretical Biology, 241, 2006), a quantitative model describing the interaction of two transcription factors is proposed and the stability behavior of the solution of the underlying system of differential equations investigated. In this model, any time delay due to the transcription and translation processes are neglected.

The aim of this thesis is to modify the system of differential equations given in the afore-mentioned paper by incorporating time delays, to consider the associated delay differential system and to discuss the stability behavior of the solution as a function of the delay parameters.

$\boldsymbol{q}$ -Splines

Announcement: 17.06.2011
Supervisor: PD Dr. Peter Massopust, massopustma.tum.de
Topic and Description:Der -Analog, auch -Erweiterung oder -Verallgemeinerung genannt, eines mathematischen Konzepts oder Ausdrucks ist deren Parametrisierung durch einen Parameter , die sich im Limes $q\to 1-$ auf die ursprüngliche Form reduzieren. So gibt es zum Beispiel -Analoge der Binomialkoeffizienten, der Ableitung, des Integrals und der Fourier Analysis. Als Beispiele dienen folgende Ausdrücke:

$\textrm{Setze:\;}\textrm{fac}(k,q) := 1 (1+q)(1+q+q^2)\cdots (1+q+\cdots+q^{k-1})$

$\textrm{$q$-\&quot;{A}quivalent von $\pi$:} \quad \pi_q = q^{1/4}\,\textrm{fac}^2(-1/2, q^2)$

$\textrm{$q$-Integral:} \quad \int_0^1 f(x) d(q,x) = (1-q) \sum_{i=0}^\infty f(q^i) q^i$

$\textrm{$q$-Ableitung:} \quad (D_q f)(x) = \frac{f(q x) - f(x)}{q x - x}$

Ziel dieser Arbeit ist die Entwicklung des -Analogs von Splines und B-Splines, sowie die Erarbeitung deren Eigenschaften mit eventueller Anwendung auf Wavelets und Frames. Die Untersuchung kann sich auch auf fraktionelle Splines und/oder Splines komplexer Ordnung ausdehnen.
Anforderungen: Grundkenntnisse aus der Theorie der Splines, sowie der Analysis und eventuell Funktionentheorie.
Literatur:
- Quantum Calculus by V. Kac und P. Cheung, Springer Verlag, 2001.
- Lecture Notes for an Introductory Minicourse on -Series by G. Casper.