# Research and Science

During my education and studies, I already worked more or less
intensively on different topics, in particular **mathematical
image processing** and related **optimisation topics**,
also for my PhD about
**shape
optimisation**. Besides that, I'm also interested in
**economics** and **social sciences** (see
my Master's thesis) and several other topics:

**Computer Science**and algorithms**Cryptography**and**Steganography****Blockchain**technology and P2P networks**Artificial Intelligence**and neural networks**Particle Physics**and related topics of modern physics

For some of the works and publications below, I wrote
**computer code**
and have additional data. I'm usually willing to release all
of that under a free license, please contact me
if you are interested.

## Selected Papers

- A Hopf-Lax Formula for the Time Evolution of the Level-Set Equation and a New Approach to Shape Sensitivity Analysis
- This
paper discusses how the viscosity solution of the classical
**level-set equation**can be found by solving an Eikonal equation. This yields a theoretical justification for the application of the**Fast-Marching Method**. Furthermore, my representation of the solution also allows to draw further theoretical conclusions, including the formation of a new**shape-sensitivity calculus**. First published in Interfaces and Free Boundaries 18(3), published by the European Mathematical Society. - Self-Consistent Gradient Flow for Shape Optimisation
- A new idea
for shape optimisation on the example of
**image segmentation**, which bypasses the inefficient convergence of gradient descent and is at the same time not dependent on second-order derivatives. It is based on a construction of the**gradient flow**that can be efficiently evaluated in practice. Open access publication thanks to FWF. - Geometric Constraints in Descent Methods for Shape Optimisation
- In
this paper, I discuss different projection methods for
**shape optimisation with geometric constraints**. The original publication is available at www.esaim-m2an.org, copyright by EDP Sciences and SMAI. - Measure-Theoretic Properties of Level Sets of Distance Functions
- Geometric
analysis of the
**surface measure**of evolving**level sets**for the case of a constant normal speed. The final publication is available at link.springer.com. - Game Channels for Trustless Off-Chain Interactions in Decentralized Virtual Worlds
- This
paper describes a new protocol that can be applied to
**blockchain-based game worlds**(such as Huntercoin) to scale them in theory to**infinite size**and enable**near real-time interactions**. Since the original publication, I've been able to improve the design further. - Difficulty Control for Blockchain-Based Consensus Systems
- A paper
with statistical analysis of
**Bitcoin's difficulty control**and possible (academic) improvements to it. The final publication is available at link.springer.com.

## Theses

- A Level-Set Framework for Shape Optimisation
- My PhD thesis
in applied mathematics that treats the use of
**level-set methods**in the context of**shape optimisation**. While such a use is not new in applications, my main contribution is a new**theoretical framework**for shape-sensitivity analysis and other important questions. - Political Power and Socio-Economic Inequality
- My master's
thesis in mathematics, where a model for uneven power distribution and
**social inequality**shows a first-order**phase transition**between equal and inequal societies. See the page for more information. - Stochastic Variational Approaches to Non-Hermitian Quantum-Mechanical Problems
- My
master's thesis in theoretical physics. I consider, how quantum-mechanical
**resonances**can be treated using the**complex scaling method**and other related techniques. For solving the resulting**non-Hermitian eigenvalue problems**, I propose a generalisation of the**stochastic variational method**and Rayleigh-Ritz principle to**complex eigenvalues**. The thesis' content was presented by my supervisor Willibald Plessas at**IC-MSquare 2016**, with a short**overview paper**published in JPCS. - A Measure Theoretic Approach to Image Segmentation Framed in Terms of Intensities
- This
is my bachelor's thesis in mathematics. I developed a method
for
**image segmentation**(on gray-level images) that is based on a**measure-theoretic representation of the image**and results in the K means algorithm. - Jules Verne's Journey Through Interplanetary Space
- My bachelor's
thesis in physics. I considered several aspects
of Jules Verne's
**Off on a Comet**from the point of view of (today's) science, modelled and simulated them and interpeted how much Verne was scientifically correct. The book contains a lot of numerical figures. Note that during the Modellierungswoche mit Mathematik 2012 I lead a group where we worked on a similar topic, investigating partially the same and partially different aspects of Jules Verne's books. - Allgemeine Algebra
- My
Fachbereichsarbeit
in mathematics, dealing with
**defining numbers and operations on them**. In German.

## Various Other Publications

- Tip Optimization
- This is the
report about a project done during a course in mathematical modelling together
with my co-student
**Doris Koinegg**about optimal strategy for assigning customers of a restaurant to tables. We did theoretic and numerical analysis based on stochastics, and got some nice results. - Calling the GPU from GNU Octave
- This is a seminar
report, where I use a basic image denoising problem as example to
demonstrate (and test) how to do
**calculations on a GPU**(specifically based on Nvidia's CUDA system)**from within GNU Octave**via compiling the CUDA code into an Oct-File (special kind of shared library which allows definition of Octave functions in native C++ code). - Some Mathematical Aspects of Fairness
- A seminar paper dealing
with some approaches to fairness. In particular, I treat
**inequity aversion**(Fehr & Schmidt 1999) in game theory, matching problems and the**Gale & Shapley algorithm**, and voting systems and in particular**Arrow's impossibility theorem**as well as some ideas to escape it. - Automatic Differentiation with ADOL-C and Complex Numbers
- My seminar
report describing how
**automatic differentiation**with ADOL-C can be used also for**complex numbers**, in the particular case for my work on a**stochastic variational method**for problems of**non-Hermitian quantum mechanics**. (Which became my master's thesis and is listed above.)

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