This thesis investigates properties of the Narayana sequence, a combinatorial sequence with recursive and morphic structure. We prove several known characteristics of the sequence. We then introduce the Walnut Prover, a tool for automating logical reasoning over automatic sequences. We describe its capabilities and limitations, and apply it to explore additional properties of the Narayana sequence.

This thesis will be about explaining a proof of a theorem about entropy presented in a scientific article by Arstein, Barthe Ball and Naor [1] in detail. The original proof is complex, especially for bachelor-level students. The goal of this thesis is to break down that proof, add mathematics and make the ideas understandable for students at this level. The theorem is about mathematical entropy, a concept in probability theory, which is a measure for uncertainty or chaos in an event or outcome. For instance, consider a coin toss: the outcome is uncertain, and this uncertainty is measured by entropy. The greater the uncertainty, the higher the entropy. There is also a concept of entropy in the world of physics, where it is a measure for describing the uncertainty or randomness in which systems evolve. One of the goals of this theorem is its demonstration that these two types of entropy, though defined in different contexts, exhibit analogous behavior. In particular, it shows that the mathematical entropy behaves like the second law of thermodynamics, which states that entropy in an isolated system increases over time. As an illustrative example, when a glass of water is spilled on a table, the
water gradually spreads out, increasing the disorder of the system. The theorem explains that if the entropy of a normalised sum of independent events is taken, then this entropy will increase with the amount of events that are summed. A normalized sum refers to the average obtained by summing independent events and dividing by their count. This seems logical at first, since the uncertainty of for example two separate events seems bigger than that of one event. However, the proof is complex and requires advanced analysis to prove.

The theorem is thus about the monotonicity of entropy of normalised sums. This thesis will connect the entropy to Fisher information, which enjoys nicer analytical properties to use. The concept is clear, entropy is a measure for uncertainty, while Fisher information quantifies the amount of information a random variable carries. This thesis will thus first connect the Fisher information and entropy and show that proving an increase of entropy can be done by proving a decrease in Fisher information. The proof of the decrease in Fisher information will need another theorem. This theorem is about connecting the Fisher information to the world of analysis. This requires some advanced analysis, like Green-Gauss on infinite surfaces and integrating the divergence of functions over hyperplanes. With this connection to the world of analysis eventually the decrease in Fisher information can be reached with some help from a lemma about commuting orthogonal projections in Hilbert spaces. All supporting theorems and lemmas will also be proved in detail in this thesis. In conclusion, this thesis will show a lot about the behaviour of entropy under normalised summation of independent events and show that these monotonically increase, which implies that the mathematical form of entropy behaves like the second law of thermodynamics.

Contract bridge is a challenging card game that combines strategic bidding, teamwork, and probabilistic outcomes. This thesis develops a rigorous mathematical framework for analyzing duplicate bridge tournaments, with a focus on fair and accurate player ranking. We introduce a threshold ranking model that extends the classic Bradley–Terry model by incorporating a tie parameter, allowing us to model the non-negligible probability of tied scores in bridge results. We derive theoretical results for this model, notably, we prove that the total match-points and number of tied boards are sufficient statistics for the estimation of players’ skill levels and the tie parameter. Using both a reference dataset from literature and the results of the 2024 China National Bridge Championship, we demonstrate that our model can reliably estimate skill parameters: the inferred rankings align with actual tournament outcomes and distinguish the strongest and weakest pairs. To complement the parametric model, we employ a fuzzy logic approach to categorize board-by-board performance into qualitative grades (“Excellent”, “Good”, etc.) and compute aggregate fuzzy performance scores. This provides an additional lens to evaluate each pair’s consistency and tendency for extreme results. The combined analysis yields a richer understanding of performance: while the statistical model emphasizes overall consistency and head-to-head advantages, the fuzzy analysis highlights variability and exceptional highs or lows. Our results show that both methods concur on the identification of top performers, reinforcing confidence in the findings, and together they reveal nuanced differences among closely ranked competitors. These insights further confirm the fairness of the duplicate format (skill prevails in the long run) while quantifying the impact of chance and strategic diversity on interim rankings.

In summary, the contributions of this thesis are: (1) a new tie-aware mathematical model for ranking players in duplicate bridge, supported by theoretical guarantees and validated by empirical data; (2) the integration of fuzzy logic metrics into bridge performance evaluation, offering practical measures of consistency; and (3) a comprehensive case study on championship data illustrating how our methods can be applied to draw meaningful conclusions about player abilities and game dynamics. This work not only advances the methodology for analyzing bridge tournaments but also provides tools and perspectives that can inform the design of fair competition systems and the training of competitive bridge players.

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.

Mahjong enjoys its status as the national game of China. The way it is played is philosophically described as: to create order out of chaos based on random drawings of tiles.

This study focuses on one specific type of mahjong: Sichuan mahjong. Sichuan mahjong is one of the most famous mahjongs in the world, and its unique rules lead to the two-player mahjong situation, which remains a gap in scientific research.

This report presents a Markov chain model for Sichuan mahjong, focusing on the quantitative measure of the game state. We took the combinatorial theory and algorithmic approaches to understand the game states. Based on the game state measure, we calculated the winning probability and expected number of game rounds in the two-player situation, and compared them with the experiment results. The results demonstrated the playability of Sichuan mahjong.

We also investigated the difference between aggressive and conservative players, and simulated the aggressive player’s strategy in the calculation of winning probability and expected number of rounds.

The quantitative methods for game state measure contribute to applications in future mahjong AI, which provides the players with a broader understanding of Sichuan mahjong.

In forensic science, the strength of evidence is calculated mainly by statistical models called likelihood ratio systems. In court cases, the specific-source likelihood ratio system is used by forensic scientists to determine if a trace originates from a known reference, called the trace-reference problem. However, collecting sufficient data to create a specific source model may be time-consuming and costly. If the number of court cases becomes too high this could be problematic. Therefore there is a need for other models that can perform as well as a specific-source model if it is infeasible.

A common-source model could be a solution, as this model can be re-used over cases. To this end, we introduce two common-source systems: a common-source feature-based system and a common-source score-based system. We compare their performance to a specific-source score-based system in a trace-reference setting. The simulations show that the common source feature-based method is the best-performing likelihood ratio system if the dimensionality is not too high, and the sources are equally variable. The analysis shows that the common-source score-based method can work as effectively as a specific-source score-based model in certain scenarios.

Additionally, we researched a preprocessor, known as percentile rank, which aims to consider typicality for score-based methods. For the common-source score-based system, using a percentile-rank preprocessor can improve the performance for large sample sizes, while considering the rarity of the measurements.

In this thesis report, we delve into a contest game within game theory, where agents’ risk-taking capacity, rather than effort, becomes the pivotal variable. High-quality performance in the game is associated with a higher probability of leading to superior scores through the use of risky strategies. The application of this con test game to labor markets prompts considerations for both employees and employers. From the perspective of employees, understanding the balance between risk-taking and payoff can be helpful in decision-making. On the employer’s side, the efficiency of selection mechanisms becomes a critical factor. We assess selection efficiency by examining the winning rates of high-type individuals. We use two parameters—market quality and market size in our analysis. Surprisingly, our theoretical analysis reveals a non-monotonic relationship between these factors and selection efficiency. Contrary to expectations, we find that as market quality improves or the number of agents increases, the winning rates of high types may decrease, resulting in reduced selection efficiency for employers. Simulation experiments inspired by Fictitious Play and evolutionary game theory are conducted to research deeper into these dynamics. Learning rules and replicator dynamics under four scenarios are designed to address the inherent volatility in agents’ strategic choices, test optimal strategies, and enable a comprehensive comparison of selection efficiency. A mechanism is proposed, derived from agents gaining experience from their usual behavior, and attempts to align outcomes more closely with Nash equilibrium, improving the optimal result. The study’s unexpected findings about single-round screening in certain conditions highlight the need for tailored selection processes in different markets. In summary, this research brings a fresh perspective to contest games. It encourages a rethink of traditional ideas and provides practical insights for decision-makers, especially in labor market.

In his paper, Conway presents his remarkable findings, leaving the task of careful examination and verification to us. He unveils a special link between the extraFib series and the Empire State Building, and reveals an unexpected relationship between Fibonacci numbers, Lucas numbers, and trigonometric formulas. This fresh concept, labeled "Fibonometry," underscores the potential connections between diverse mathematical concepts. As we engage with this paper, we take on the responsibility of exploring and confirming these intriguing mathematical relationships.

Temporal networks, like physical contact networks, are networks whose topology changes over time. However, this representation does not account for group interactions, when people gather in groups of more than two people, that can be represented as higher-order events of a temporal network. The prediction of these higher-order events is usually overlooked in traditional temporal network prediction methods, where each higher-order event of a group is regarded as a set of pairwise interactions between every pair of individuals within the group. However, pairwise interactions only allow to partially capture interactions among constituents of a system. Therefore, we want to be able to predict the occurrence of such higher-order interactions one step ahead, based on the higher-order topology observed in the past, and to understand which types of interactions are the most influential for the prediction. We find that the similarity in network topology is relatively high at two time steps with a small time lag between them and that this similarity decreases when the time lag increases. This motivates us to propose a memory-based model that can predict a higherorder temporal network at the next time step based on the network observed in the past. In particular, the occurrence of a group event will be predicted based on the past activity of this target group and of other groups that form a subset or a superset of the target group. Our model is network-based, so it has a relatively low computational cost and allows for a good interpretation of its underlying mechanisms. We propose as a baseline the memory-based method for the traditional pairwise network prediction problem. In this baseline model, the predicted higher-order events at a prediction time step are then deduced from the predicted pairwise network at the same prediction time step. We evaluate the prediction quality of all models in eight real-world physical contact networks and find that our model outperforms the baseline model. We also analyze the contribution of group events of different orders to the prediction quality. We find that the past activity of the target group is the most important factor for the prediction. Moreover, the past activity of groups of a larger size has, in general, a lower impact on the prediction of events of an arbitrary size than groups of a smaller size.

Fog plays a major role in chain collisions. Proper fog detection is essential for the Dutch road authority to anticipate foggy weather conditions. Dozens of stations in the Netherlands can measure fog. However, fog can be a very local phenomenon. Therefore, more local measurements are needed. There are about 5,000 traffic cameras in the Netherlands. Several studies on detecting fog on traffic cameras have been done. The most successful studies used machine learning classification models to detect fog. The biggest challenge they face is the extreme imbalance, limited diversity, and limited accuracy of the dataset. Obtaining adequate precision is one of the primary challenges since the extreme imbalance of the dataset significantly impacts precision. The main objective of this research is to improve the dataset and investigate many machine learning configurations. Another objective is to examine the possibilities of generating synthetic data.

This thesis uses a clever (re)labeling method, significantly improving the dataset's quality. However, it turned out that the dataset still has its limitations. A large portion of false positives are caused by labeling errors. After comparing several machine learning models, it follows that a 9-layer ResNET model is optimal. Adding more layers will not result in better performance. Unexpectedly, initializing ResNET with pre-trained weights actually decreases performance. In addition, the effect of oversampling and/or using a weighted binary cross-entropy loss is investigated. Just oversampling leads to overfitting, but using a weighted binary cross-entropy loss isn't ideal either. The best performance is achieved by combining weighted binary cross-entropy loss with oversampling. Decision threshold optimization substantially improved the results. The experiments allowed for selecting the ideal configuration, which substantially increased performance. The best-performing configuration achieved a strong correlation in the Matthews correlation coefficient.

Finally, the possibilities of generating synthetic data are investigated. ADASYN and SMOTe seem attractive at first sight, but from a recent study, it follows that they don't work better than random oversampling. One of the most promising ideas for generating synthetic data is to add fog to clear images. In this thesis, a conceptual algorithm is designed to add artificial fog to clear images. Most generated images look convincing, but there is much room for improvement.

Almost everyone is familiar with games such as poker and checkers, but games can also be found in non-entertaining settings, such as competing companies in a market or conflict resolutions between countries. However, what happens when players want to avoid working together? This thesis provides two game-theoretical models of boycotts based on existing literature and is complemented with new insights. We discuss the dynamic process of a consumer boycott, where the effectiveness can be determined by the predefined consumers’ and firm’s thresholds or by the maximum boycott duration. Besides, we consider a static model using the Shapley value to analyze the impacts of boycotts, which are balanced for 2-player boycotts but not when three players are involved. We extended the existing models by finding supermodularity as a sufficient requirement for the boycott to never be profitable for the involved players, by introducing a direct formula for the coalition value in a boycott game (the boycott-contraction), and by analyzing real-world data, where we observe the effect of boycotts on trading activities with uninvolved players.

The "Number Hides Game" (NHG) is 2-player game played on a board that consists of a row of p consecutive coins. Player I and player II simultaneously choose subsets of m and n consecutive coins respectively. Player II pays the number of coins that lie in the intersection of the subsets to player I. This thesis introduces basic knowledge from the field of game theory needed to analyse the game. Afterwards, it presents and proves the optimal strategies of the NHG. Finally, a variant of the NHG will be discussed in which the board consist of a general tree instead of a row.

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.

Interest rate products form a large segment of over-the-counter derivatives. When the interest rate became negative, for the first time, in July 2009, interest rate models needed to adjust. Where first a log-normal model, as the Brace Gatarek Musiela (BGM) model, might have seemed logical for interest-rate products, as they were
bounded by zero, now a normally distributed model, as the Hull-White model, could be considered more practical. To our knowledge, no comparison of the Hull-White model and the Brace Gatarek Musiela model has been made on the Expected Positive Exposure (EPE) (and thus Credit Valuation Adjustment (CVA)) of a swap portfolio. Therefore, this thesis compares the Hull-White model with the BGM model on the EPE of a swap portfolio. First, we show how both models can be simulated with Monte Carlo simulation and calibrated to caplets, after which we validate the used simulation. Finally, both models are compared on the convergence, computation time and EPE. It was found that the Hull-White model had a faster convergence and computation time than the BGM model for our implementation. Moreover, it was shown that the Hull-White model and BGM model have significantly different swap EPEs, except for far in-the-money (ITM) swaps and single payment swaps. Therefore, the models used for the EPE of a swap portfolio have a model risk.

In this thesis report we consider a search and rescue problem in which one or multiple targets/objects are hidden in some playing field, and must be rescued/found by a searcher. The targets are for example: earthquake survivors, lost hikers or prisoners held by an adversary, and are hidden in some section of the play field. Searching any of the sections of the play field has a certain probability of failing; the searcher might get lost, trapped or captured herself. The goal is to find the search that maximises the probability of finding all targets, and to find the hiding spot that minimises it. We define and solve the search and rescue game on a play field for which movement between any two section of the play field is always possible. We also define and solve the game played on a tree for which movement is limited by the tree structure. Due to the complexity of this game, we restrict ourselves to one target. Finally, we extend the game by replacing the play field with specific types of graphs that are not trees. For some of these graphs, we have found (partial) solutions.

In financial mathematics, stochastic processes are regularly used to describe observed financial indicators such as stocks, options, futures or interest rates. Identifying the underlying dynamics of observed financial time series is crucial in risk management, as it greatly affects pricing and hedging strategies. The large number of available stochastic processes make selecting the most suitable stochastic process a non-trivial problem. Additionally, realisations of stochastic processes are elements of the path space which is infinite-dimensional and non-locally compact. Given these observations, we find that model selection methods from classical statistics, such as distribution metrics, are inadequate.
In this thesis a signature-based model recognition method is proposed. The goal of this model is to select the most suitable stochastic process to describe an observed financial time series. Signatures are transformations from a path to an infinite-length sequence of properties of that path, which makes signatures a highly interesting approach to construct input features which can be used by a machine learning model. In our evaluation, we use the aforementioned methodology to distinguishing between various classification settings of Arithmetic Brownian Motion, Geometric Brownian Motion and the exponential jump diffusion process, both in a binary and multi-class classification setting. This evaluation shows that the proposed method can adequately distinguish between the same model with different parameters, models with and without jumps and models with different jump sizes and jump intensities.

The Competitive Investor Game from Bell & Cover (1980) and the 푘-Player Ranking Game from Alpern & Howard (2017) are analysed in thesis. Optimal strategies have been derived and the related proofs have been given a new look. The Symmetric Multiplayer Ranking Game is considered as the general interpretation of financial competition among hedge funds. This study focused on the hedge funds that manage a Long/Short U.S. Equity strategy. Some minor evidence has been found to support the hypothesis that the studied hedge funds manage a strategy that has the objective to beat the competition in terms of annual performance in order to achieve the highest ranking.

In deze Bachelor thesis hebben we het gedrag bestudeerd van 3 typen kankercellen die een rol spelen bij prostaatkanker. Dit doen we met behulp van speltheorie met als spelers de dokter tegenover de kankercellen. Met de replicator dynamica hebben we eerst gekeken naar de verdeling van cellen in een tumor zonder dat er medicijn was toegevoegd. We hebben de stabiliteit van de evenwichtspunten hiervan bepaald met behulp van de eigenwaardes, de trace en determinant van het gelineariseerde systeem en de Routh-Hurwitz methode. Vervolgens hebben we de replicator dynamica omgeschreven tot Lotka-Volterra vergelijkingen en medicijn en resistentie toegevoegd aan de differentiaalvergelijkingen. Hiermee hebben we de dynamica van de cel populatie na therapie bekeken. Tot slot hebben we geprobeerd een goede balans te vinden in de dosis medicijn om de tumorcellen zoveel mogelijk te doden, maar de levenskwaliteit van de patiënt zo goed mogelijk te houden. We hebben hierbij drie strategieën vergeleken; maximale getolereerde dosis medicijn (MTD), Nash strategie, Stackelberg strategie. De Stackelberg strategie bleek hierin het meest succesvol. De conclusie is dat er bij de Stackelberg strategie het minste medicijn nodig is om te kunnen leven met de tumor.

In this report, we will look at the connection between the Fibonacci and Euler numbers. By using a combinatorial argument including the Fibonacci and Euler numbers, we will prove our main theorem:  F_n·E_n ≥ n! From the main theorem and the asymptotics of these numbers, we will conclude that π ≤ 2ϕ. We follow the proof in the article of Alejandro H. Morales, Igor Pak & Greta Panova, but we will give a more detailed proof and some extra facts about the Golden Ratio, ϕ, and the Fibonacci and Euler numbers. Finally, the article discusses the number of linear extensions of certain partially ordered sets, or posets. We see that there exist a two-dimensional poset U_n and complement poset Û_n, both with n elements, such that the number of linear extensions are respectively E_n and F_n. We conclude that the Fibonacci and Euler numbers are related to each other.