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- Fachbereich Mathematik (1086) (remove)

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.