In this bachelor thesis, we investigate some strategies for the firefighting problem on the random geometric graph. The firefighting problem is a problem on graphs where fire breaks out on a set of vertices. Each subsequent turn we can protect vertices using firefighters, after which the fire spreads to all unprotected neighbours of vertices on fire. The process ends when the fire cannot spread anymore. The protected vertices and the vertices not on fire are the vertices saved. A random geometric graph is a graph of which the vertices are obtained by a homogeneous Poisson Point Process of intensity λ. Vertices are connected by edges if they are distance at most 1 apart. We explain the proof from [3] that we can bound the number of firefighters needed to contain the fire from above by 2λ. We do so by translating the firefighting problem to the problem of surrounding a continuously growing blob with a fence built at rate ρ. We explain the proof from [3] that the blob can be surrounded by a teardrop-shaped fence built at rate ρ > 2 and that the blob cannot be surrounded by a fence built at rate ρ ≤ 1. Finally, we consider a blob with wind. In the real world fire propagation is influenced by wind. We therefore assume that the blob grows faster in the direction of the wind, we call this region the fire front and slower in opposite direction, obtaining a blob with wind. We find that for a blob with wind with an ellipse-shaped fire front with maximum propagation speed vmax, we can surround the blob with a teardrop-shaped fence built at rate ρ > 2.

This thesis explores the improvement of computational efficiency in simulating two-dimensional conveyor belt systems by applying model order reduction (MOR) techniques. Conveyor belts, crucial for material handling in various industries, are traditionally modeled using finite element methods (FEM), which can be computationally demanding, particularly for long-term simulations with a high number of grid elements. To address this, MOR techniques aim to reduce high-dimensional models into lowerdimensional models.
The study investigates three MOR approaches—modal decomposition, Proper Orthogonal Decomposition (POD), and Dynamic Mode Decomposition (DMD)—to apply on existing software built by VORtech that simulates two-dimensional conveyor belt systems. When considering MOR to reduce the two-dimensional system, challenges arise related to nonlinearities, differential algebraic equations (DAEs), and complex modeling steps. Among the methods tested, DMD proved to be the most effective, offering significant reductions in computational time while maintaining accuracy. POD also demonstrated accuracy but had less impact on speed due to the time-consuming complex modeling steps in the simulation software. These complex modeling steps are not investigated in detail in this thesis and therefore not reducible with the intrusive POD. Because of the non-intrusive nature of DMD, this method was able to incorporate these extra processes in the reduced order model.
The study concludes with recommendations for future research, emphasizing the need for optimization of the code segments that handle the complex modeling steps. In addition, conventional modeling approaches as alternatives to the complex interpolation step could be explored, to enhance the applicability of MOR. Furthermore, DMD with control or parametric DMD could be explored to obtain a reduced order model by interpreting misalignments of rollers as controls or parameters. Finally, a method is proposed to make modal decomposition useful for models, where the solutions depend highly on the external forces. Although this method is not applied to the model considered in this thesis, it would be interesting to explore this modified modal decomposition method on other models that are significantly influenced by the external forces.

Various problems related to Random Walk in Random Environments have been researched intensively over the past 40 years by both the mathematics and physics communities. The setup of these problems is quite simple: one defines a Random Walk of choice and allows it to traverse a Random Environment that influences the Random Walk in a predetermined way.

In this thesis, we study the survival probability of a Random Walk in various random environments of traps. These Random Environments of traps are divided into two distinct classes: static and dynamic. In the static environment, the trap configuration is randomly chosen at time zero and remains fixed thereafter. In the dynamic case, the trap configuration evolves over time according to some predefined dynamics.

The problem in the static setting has a long history, and the situation is fairly well understood. For the dynamic environments, however, the state of the art is less developed, as there is no unified approach to deriving survival probabilities for different dynamic environments. Additionally, almost nothing is known about the interpolation between the models of random walk in static and dynamic environments. The interpolation model should exhibit exponential decay of survival probability in a fast environment, corresponding to the dynamic environment, and show subexponential decay of survival probability when the environment is slow, corresponding to the static environment.

One of the goals of this thesis is to compile various results of survival probabilities in different dynamic environments and to create a model that allows interpolation between exponential and subexponential decay of survival probability.

Gang-related violence poses a significant challenge in numerous countries across the world. While previous research have explored modeling formation of gang territories, none of them incorporated a city layout such as roads, water bodies and parks while doing so. Hence, this research explores the effects of city layout in the formation of gang territories. First the effects of a single vertical boundary of varying permeability is observed. Following this, the model is applied to real world where a part of Chicago's city layout is incorporated to the model. The results are then compared to gang territory maps shared by Chicago Police Department. The experiments show that city layout significantly affects the shape and size of the territories. By fine tuning the permeability for different terrain features, we can observe that the territories produced by the model show resemblance to their real world counterparts.