In 1935, J.G. van der Corput asked a question about how to evenly distribute sequences of points. By 1945, Van Aardenne-Ehrenfest found an answer, which encouraged more mathematical research. The problem they explored was how to place points evenly on a circle and keep this even distribution when adding new points. In other words, van der Corput wondered if points could be placed on a line segment in such a way that any two subintervals of the same length would have almost the same number of points. Van Aardenne showed that perfect equal distribution is impossible, leading to more studies on the irregularities in point distribution in different areas of mathematics.

A lot of mathematical work has been done on studying sequences of points in the interval [0, 1) that are evenly distributed. This report looks at the idea that a sequence is evenly distributed if its first n points divide a circle into intervals that are roughly equal in length. The sequence X_k = log_2(2k − 1) mod 1 (section 2.2) was introduced by De Bruijn and Erdös in this context. We will show that the way the gaps between points in this sequence are structured is uniquely optimal in a certain way.

This thesis aims to uncover new insights through a mix of theoretical and numerical studies. The first sections focus on the work of Nicolaas Govert de Bruijn and Paul Erdös, especially their 1949 pa- per "Sequences of Points on a Circle". Later sections will present numerical findings and compare different methods for achieving the best way to split a circle evenly.

In short, this thesis aims to reproduce mathematical theories about evenly distributed sequences and the best ways to divide intervals. By merging these theories with new numerical techniques, it hopes to provide a new viewpoint on a classic topic, inspiring more investigation and study in this fascinating area of mathematics.

During this research, we investigate if there exists a k(n)-core in the scale-free configuration model, this is a commonly used null model to simulate networks. The scale-free configuration model produces a random graph, where the degree of every vertex is determined using a random variable. In this thesis, the discrete Pareto variable is used with parameter τ ∈ (2, 3). The k-core of a graph is the biggest induced subgraph where every vertex is connected to at least k edges in the subgraph. In this thesis, we let k be dependent on the number of vertices in a graph, this makes it a k(n)-core. By investigating whether k(n)-cores exist in graphs produced by the scale-free configuration model, we hope to to get a better understanding of the structure of these graphs.

In this thesis, the scale-free configuration model is modeled as a death process. This death process will produce the graph and its k(n)-core jointly. First, we remove all edges which cannot be part of the k(n)-core until the point where no such edges exist anymore. This is the moment where we reach the k(n)-core. Using this method, we prove that whenever the number of vertices is sufficiently high a (log(n))^α-core exists with high probability for all constant α >0. For α < (3−τ)/(8τ) , also a n^α-core exists with high probability when the number of vertices is sufficiently high. We also find a lower bound for the number of vertices and edges left in the cores.

Constraint programming solvers provide a generalizable approach to finding solutions for optimization problems. However, when comparing the performance of constraint programming solvers to the performance of a heuristic solver for an optimization problem such as cluster editing, the heuristic solver is able to find near-optimal and optimal solutions much faster. The goal of this research is to understand how the behaviour of such a heuristic solver can assist the performance of constraint programming solvers in optimization problems. In order to achieve this, first Chuffed, a state-of-the-art constraint programming solver was combined with a heuristic approach to cluster editing, with the goal of emulating the performance of the heuristic algorithm, in particular, being able to find near-optimal solutions faster. Continuing, the goal was to generalize the behaviour observed by the modified solver, by emulating the performance observed without the need for the specialized heuristic solver. The generalized approach is tested on a wide variety of different tests. An approach to value selection was developed that performs lookahead propagations for the two values of a boolean variable and selects the value that has the most optimal solution within the domain after performing the lookahead propagation. This approach added a significant time overhead that increased the overall solving time for many problems, with the lookahead configuration having a median increase of 8.7% over the default configuration for the optimization problems of the MiniZinc Challenge 2022. However, it was able to successfully emulate the performance of the heuristic solver, finding near-optimal solutions significantly faster than the default value selection. In particular, for the optimization problems of the MiniZinc Challenge 2022, on average, the lookahead configuration had a definite integral for the time vs objective graph 54.7% lower than the default Chuffed configuration.

In this thesis, we examine the kernel-based spatial random graph (KSRG) model, which is a generalisation of many known models such as long-range percolation, scale-free percolation, the Poisson Boolean model and age-based spatial preferential attachment. We construct a KSRG from a vertex set V = Z^d, assigning each vertex v ∈ V a weight Wv according to a power-law with parameter τ − 1 and connecting each pair of vertices conditionally independently according to
P(u ↔ v|Wu,Wv) = Θ(1 ∧(max {Wu,Wv}^(σ1) min {Wu,Wv}^(σ2)/|u − v|^d))^α.
Our first contribution is to show that, under certain choices of the parameters σ1, σ2, τ and α, the graph distance is at most poly-logarithmic or at most doubly logarithmic when compared to the spatial distance. Furthermore, the parameters σ1 and σ2 allow for extra degrees of freedom when compared to the models mentioned earlier, which yields a new exponent for poly-logarithmic distances and a generalised constant for doubly logarithmic distances. Our second contribution lies in the techniques used to prove these results. In particular, we show that with probability tending to 1 there is a subset of vertices that behaves pseudo-randomly with regards to the expected amount of vertices with a given weight in any radius. The presence of this subset, which we call a net, allows for the avoidance of the use of FKG-like inequalities in proofs. Primarily, it allows us to reveal all relevant weights in advance, which allows us to disregard many correlations that we would otherwise need to take into account during construction of paths.

Out of Home advertising is traditional outdoor advertising. Over the last decade the digital Out of Home advertising possibilities have grown substantially. These possibilities include hour-based advertisement schedules, rapidly implemented changes to advertisements and obtaining location data from consumers being exposed to advertisements. Investigating these possibilities further for improving customer engagement and optimising target group reach is of great importance in marketing and marketing science. This thesis will focus on exposure data obtained from vehicle locations in central London. Using this data we want to model the frequency with which individuals see advertisements and research the overlap in exposures of individuals to multiple media vehicles where advertisements are displayed. We focus on modelling exposures of individuals to two vehicles, comparing different types of bivariate distributions. In this thesis we present the comparison of two models: an adapted version of the Sarmanov bivariate distribution and copulas. We refer to these models as the Danaher and Copula model respectively.

The fitting and simulations of the models are based on bivariate data of two specified vehicles, to be able to comment on the overlap between these vehicles. To investigate the performance of the models all combinations of vehicles are modelled after which the results are combined. Both models simulate small frequencies of exposures of individuals adequate, but for more extreme values both models fail to represent the observed data. Especially the simulation of overlap is done poorly by the models. Several approaches have been tried to improve the models' performance, without much success. The analysis shows that the data at hand requires more sophisticated methods to handle joint exposure and more research needs to be done.

In addition the variable distance is added to the research problem, which intuitively has great influence on the overlap between media vehicles. This is done by using 3-dimensional copulas. For this research we modify the data: instead of first fitting and simulating a copula model on the data of two vehicles and then combining the results of all vehicle combinations, we first combine the data from all the combinations of vehicles and repeat the fitting procedure thereafter. Several options for the modelling of these copulas are presented. Similar analysis shows, again, that the nature of the data requires more complex models to handle the overlap while also accounting for the vehicle distance. Finally, using the modified data we look back at the 2-dimensional copula model and perceive an excellent resemblance of the data for the Student-t copula.

The Game of Cycles, invented by Francis Su (2020, p.51) is an impartial game played on a graph, where players take turns marking an edge according to a set of rules. Together with the game, there also came a conjecture that gives a condition for whether a specific position is winning or losing. Proving or disproving this conjecture is the main focus of this research, which we end up succeeding in by giving a counter-example, thus disproving the conjecture. We do this by first showcasing some relevant background knowledge from game theory in chapter 1. In chapter 2 we then introduce the Game of Cycles and its rules, as well as some of the previous results others have found. We continue in chapter 3 by creating a python script to brute-force the game for us and it is here that we find a counter-example to the main conjecture, of which we prove that it is indeed a counter-example. We close off with chapter 4 by looking at a simplification of the game, where it is played on trees instead of any graph. Here we prove that the main conjecture does hold for a special family of trees and state a conjecture for the solution of any tree.

In this thesis, we consider the threshold metric dimension problem of graphs, related to and motivated by source detection.
We construct a graph G = (V,E) for a given set of sensors of size m: {s1, s2, ..., sm} and a range k > 0. We want that each node v ∈ V has a unique combination of distances (dk (s1, v),dk (s2, v), ...,dk (sm, v)), where dk is the distance function in a graph limited by the range k (the distance is denoted as being ∞ if the distance is larger than k). Our aim in this thesis is to construct such a graph that is extremal in size, that is: the vertex set V is as large as possible. We shall give such constructions with proof for optimality up to k = 3 and general m
and a different construction with incomplete proof for optimality for general k and m. For any construction we will prove that each vertex is uniquely identified.
Furthermore, we will compare our results to another paper with a similar conclusion about the extremal size of graphs with metric dimension m and a given diameter D.