"Waarom is de ruimte waarin we leven driedimensionaal?” is een vraag die men zich al sinds de Griekse oudheid stelt. Lange tijd was het antwoord op deze vraag dat er drie coördinaten nodig zijn om een punt in de ruimte te beschrijven, maar in 1877 bewees Georg Cantor het opmerkelijke resultaat dat hier maar één coördinaat voor nodig is. Deze ontdekking was het startschot van de moderne dimensietheorie.

Er stonden twee vraagstukken centraal toen de dimensietheorie in het begin van de 20e eeuw werd ontwikkeld. Men was op zoek naar een goede definitie voor het begrip dimensie, want “het aantal coördinaten” voldeed klaarblijkelijk niet aan de verwachtingen. Een andere sterke motivator was het aantonen van de invariantie van dimensie. Hiermee bedoelen we dat als 𝑛 en 𝑚 verschillende natuurlijke getallen zijn, de ruimten ℝ^𝑛 en ℝ^𝑚 ook topologisch verschillend zijn. De ruimten die zo worden aangeduid heten de euclidische ruimten. Als we voor 𝑛 de getallen 1, 2 en 3 invullen, krijgen we respectievelijk de lijn, het vlak en de ruimte om ons heen. Het bleek dat deze stelling niet heelmakkelijk aan te tonen was, maar in 1911 was het de Nederlander L.E.J. Brouwer als eerst gelukt.

Uiteindelijk is er niet één beste definitie van dimensie gevonden. Over het algemeen worden er drie verschillende definities gebruikt die goed werken. Dit zijn de kleine inductieve dimensie, de groteinductieve dimensie en de overdekkingsdimensie. Volgens alle drie deze definities is de euclidischeruimte ℝ^𝑛 𝑛-dimensionaal, precies zoals we verwachten.

 In deze scriptie bewijzen we dat deze drie dimensies overeenstemmen voor een bepaalde klasse ruimten, namelijk de separabele metrizeerbare ruimten, hieronder vallen ook de euclidische ruimten. Daarentegen geven we een voorbeeld van een andere ruimte waarbij de drie dimensies niet overeenstemmen.

The world can’t keep up with its own data and therefore needs new ways to store it. Research is being done into new types of experimental data storage like DNA sampling or racetrack memory, which could completely revolutionize the way storage works. However, such storage media are prone to certain type of errors like deletion errors where information is lost. Consequently, there has been a surge in interest in developing deletion correcting codes to address this challenge. We are interested in the maximal size of a code correcting up to a certain number of deletions. Nevertheless, constructing these codes can be quite intricate and therefore we are often limited to working with bounds.
In this thesis, we will study and evaluate upper bounds for these codes. We will compare bounds that were derived analytically to bounds that have been computed with the use of linear programs. We will compute upper bounds for 1-, 2- and 3-deletion correcting codes. Moreover, we will show how to compute these bounds with a linear program, by posing the problem of finding a largest deletion correcting code as finding the transversal number on a hypergraph. It will become clear that the bounds computed through linear programming are much stronger than any bound derived analytically.

The four-colour conjecture (4CC) is a question that asks whether any map can be coloured using only four colours, with the constraint that neighboring countries must have distinct colours. This conjecture has re- mained unanswered for over 170 years, with a rich history of various attempts to prove it. In 1879, Kempe put forward a proof, but it was invalidated by Heawood after 11 years. Heawood did succeed, however, in prov- ing the weaker five-colour theorem. It wasn’t until 1976 that the first genuine proof was discovered by Appel and Haken. This proof sparked controversy due to its reliance on approximately 1200 hours of computer computation, making it unverifiable by hand.
The purpose of this report is to provide a comprehensive overview of the historical background and re- cent advancements in the four-colour theorem. It will delve into the numerous failed attempts to prove the conjecture and discuss the groundbreaking proof by Appel and Haken. Additionally, it will explore a recent endeavor by Dr. Xiang, who approached the problem from a different perspective but also encountered a fallacy in his work. The main contribution of this thesis involves a new proof that builds upon the research of Dr. Yeh, who attempted to prove the four-colour theorem by transforming it into a system of linear equa- tions. The missing crucial steps will be addressed for whose correction new ideas will be introduced, using an integral version of Farkas’ lemma and superadditivity. The ideas being researched do not finalize the proof but contribute towards a possible elegant proof in the future. Finally, related topics such as maps on different surfaces and maps with disconnected regions will be considered, broadening the scope of the discussion.

Every day an extremely large amount of messages get sent over the internet, radio, WiFi, etc. These messages need to be encoded in order to protect the important data. However just like many of these systems messages can get corrupted over a certain channel due to noise, fluctuations in power or temperature etc. [2] Therefore cyclic redundancy checks (CRC), which are bits that are appended to message words to protect the data, are used to detect these errors. This is done by using CRC polynomials. An occurring undetected error is possible in any case, therefore we need to study closely the probability of an undetected error occurring, that is the probability that the received, erroneous data is accepted as the transmitted message. The code words that are the results of adding cyclic redunancy checks (CRC) can also be combined with an error-correcting codes like BCH codes. As shown in [3] combining CRC polynomials with error-correcting can reduce the previously mentioned probability, but one CRC polynomial can perform better. In this thesis these CRC polynomials are studied in combination with certain different codes and are evaluated based on their performance with respects to their undetected error probability. Two examples of codes are given, in the first example a CRC polynomial is found that does not perform well in the system without using an error-correcting code, but performs the best in the system with an error-correcting code. An analysis is done which shows that by looking at consecutive roots of the generator polynomial, optimal CRC polynomials can be found. The second example uses a different code and finds another optimal CRC polynomial

One of the first systems developed to provide online security is what is now known as RSA encryption. This method has been first published by Rivest, Shamir and Adleman in 1978. Even on this day, a modification of the first method of RSA encryption is being used and considered safe. This paper explores a number of mathematical structures and describes whether or not it could be possible to extend RSA encryption. In this paper we will focus on verifying whether the RSA encryption and decryption algorithms are well defined on the mathematical structures.

In this report we examine the dual space of $\ell^\infty$. If $p \in [1,\infty)$ and $q \in [1,\infty]$ satisfy $\frac{1}{p}+\frac{1}{q}=1$, then one can identify the spaces $\ell^q$ and $(\ell^p)'$ in a natural way via an isometric isomorphism. This identification does not extend to the case $p=\infty$ and $q=1$. We prove that the obvious candidate for an isometric isomorphism from $\ell^1$ into $(\ell^\infty)'$ fails to be surjective, and moreover, that an isometric isomorphism (even a homeomorphism) between these spaces does not exist at all.

We introduce a space that we can identify with $(\ell^\infty)'$ via an isometric isomorphism. This is the space of bounded finitely additive measures on $\mathbb{N}$, denoted by $\ba(\mathbb{N}, \mathcal{P}(\mathbb{N}))$. Having found this characterization of $(\ell^\infty)'$, we examine what kinds of finitely additive measures on $\mathbb{N}$ exist. These include $\sigma$-additive measures that are induced by $\ell^1$, diffuse measures, shift-invariant and more general invariant measures, measures that extend the asymptotic density, $0,1$-valued measures and stretchable, thinnable and elastic measures. Elastic measures can be considered the nicest measures on $\mathbb{N}$, from an intuitive point of view.

We also describe the functionals that correspond to particular types of measures and vice versa. Moreover, we prove that the collection of ultrafilters on $\mathbb{N}$ can be identified with the collection of $0,1$-valued measures on $\mathbb{N}$, which, in turn, can be identified with the collection of multiplicative functionals on $\ell^\infty$.

Nowadays, we produce an enormous amount of data and all this data needs to be stored somewhere. In order to store all this data in an efficient and environmentally friendly way, solutions need to be found. A storage technique that has great potential to be a solution is synthetic DNA based storage.
A DNA-strand is made up of a sequence of four nucleotides, Adenine (A), Thymine (T), Guanine (G) and Cytosine (C). To save our data using DNA, our binary data is encoded to quaternary data, using this quaternary data DNA strands are made.
During this process, of saving our data in DNA, errors can occur, therefore two constraints are considered to minimise the number of errors in our codes.
The GC-weight, which is the number of G and C nucleotides in a DNA word and the maximal run-length, which is the maximum number of identical nucleotides in a row. In addition, the Hamming distance between words is also considered to be able to detect and even correct errors that still occur.

This research tries to find lower and upper bounds for the maximum size of DNA codes.
We continue the work of Van Leeuwen [1], Vermeer [2] and Laseur [3], who considered codes with fixed GC-weight. In this thesis we will relax this GC-weight constraint and we will look at codes where we have a set of GC-weights and all the DNA words in the code need to satisfy one of these weights. Furthermore, the codes we consider have a run-length of r=1. We will determine the maximum size of codes where the minimum Hamming distance d is equal to the length n of the DNA words. We determine lower and upper bounds for the case where d=n-1 and for specific sets of GC-weights we will determine the maximum size. Finally we will present nine algorithms, these algorithms create DNA-codes. The size of these codes give us lower bounds for the maximum size of DNA codes.

While the Minimum Euclidean Distance detection is known to be optimal for channels affected by Gaussian noise, it has been shown that Minimum Pearson Distance detection (MPD) may perform better when the channel is also affected by an unknown offset, though for a good performance some adaptations for classical binary block codes are necessary. It is shown for cosets of first order Reed-Muller codes R(1,m) containing words of weight d/2, where d is the code's distance, that the minimum Pearson distance is always low for m≤4. However, it is possible to find cosets where the minimum Pearson distance is higher for m≥5.

Dit onderzoek gaat over de mogelijkheden en onmogelijkheden met betrekking tot constructies met passer en ongemarkeerde liniaal. Hier werken we met de euclidische meetkunde en zullen we dus ook beginnen met de vijf postulaten van Euclides en een aantal toepassingen ervan. Er zal besproken worden wat het betekent voor een punt of constructie om construeerbaar te zijn, hiervoor zal gedefinieerd moeten worden wat het betekent voor een getal om construeerbaar te zijn. Vervolgens komen de volgende meetkundige problemen aan bod: de driedeling van een hoek, de verdubbeling van een kubus, de kwadratuur van een cirkel en de construeerbaarheid van regelmatige veelhoeken. Om te laten zien wat wel en niet mogelijk is wordt er gebruik gemaakt van moderne algebra, zo kan een constructie probleem omgeschreven worden naar een concreet algebraïsch probleem. Hiervoor maken we gebruik van de theorie rondom lichaamsuitbreidingen en de Galois theorie.

Humanity is producing data at exponential rates. Solutions need to be found in order to accommodate the bytes of today and the future in an efficient and environmentally friendly way. DeoxyriboNucleic Acid, well-known as DNA, is a suitable destination for this major collection of data, as it is a high-density storage medium with long storage endurance [1]. In the field of research about DNA-Based Data Storage Systems, people examine quaternary codes. The considered DNA-codes consist of equi-length DNA-words over the Alphabet {0, 1, 2, 3} representing the available DNA building blocks or nucleotides {A, T,C,G} standing for Adenine, Thymine, Cytosine and Guanine respectively. Combinatorial constrained coding is introduced to avoid DNA patterns prone to sequencing errors, that might occur when archival data is read from DNA-blocks [2]. If an error occurs anyway, an imposed minimum Hamming distance d that a DNA-code satisfies, enforces error correction and detection capabilities [3]. Methods are considered to optimize the number of DNA-words that can be stored in a DNA-code satisfying these restrictions. This research explores DNA-codes of which the DNA-words adhere to a fixed GC content referred to as the GC-weight w, as well as the no-runlength constraint r = 1 that will not allow adjacent symbol repetition. By increasing distance d, we build on the findings of Van Leeuwen [4] and Vermeer [5], who created DNA-codes satisfying d = 2 for r = 1 and r ≥ 1 respectively. The maximum size of a DNA-code satisfying d = 3 among other constraints is proven to be 12. Furthermore, we cover the case d = n where the length n of the DNA-words and the minimum distance d the DNA-code satisfies are equal. We derive maximum sizes of these DNA-codes for all possible GC-weights w. Finally, we present algorithms that create DNA-d codes satisfying d = 3 and d = 4 and succeed in improving sizes of the DNA-codes created by the algorithms designed by Limbachiya et. al. [6] and Van Leeuwen [4].

The amount of data being produced is growing exponentially. An im-
portant challenge is to find methods to store this data efficiently and in an
environmentally friendly way. One idea that is a growing research topic in-
volves using synthetic DNA. DNA has the potential to be more efficient and
environmentally friendly than current methods. DNA is made of a sequence of
four nucleotides, Adenine (A), Cytosince (C), Guanine (G), and Thymine (T).
To store data, DNA strands can be created with specic nucleotide sequences.
In the process of reading and storing data substitution errors can occur. Two
constraints are introduced to minimise the number of errors. The GC-weight
constraint which states that every DNA sequence must have a fixed number
of G and C nucleotides, and the runlength constraint, which states the maxi-
mum number of repeating nucleotides possible in every DNA sequence.

Every day we produce an extremely high amount of data and a significant portion of this data is archival data. It is an important challenge to save this data in a cheap and environmentally friendly way. The current methods to save archival data are sufficient, but improvements can be made. Synthetic DNA based data storage is a great choice to do so. DNA is (roughly) made out of four nucleotides, Adine (A), Cytosine (C), Guanine (G) and Thymine (T). To save data using DNA the binary data is encoded to quaternary data and later DNA strands are created using this quaternary data. During the process of saving the data errors can occur. Previous research [3] has found that some errors can be prevented by taking two constraints into account.The no run-length constraint, which states that no DNA word can have two repeated symbolsand GC-weight constraint, which states that every DNA word must have a fixed number of G and C nucleotides. These constraints reduce the number of quaternary data sequences that can be used to save data. The aim of this thesis is to improve the lower bound on the maximum number of quaternary data sequences that satisfy the no run-length constraint, the fixed GC-weight and a minimum distance. Limbachiya et. al. [1] gave an algorithm to compute a set with a given minimum distance of quaternary data sequences that satisfies the constraints. The size of this set is the lower bound that we aim to improve. We do so by introducing two other algorithms that also compute a quaternary data sequences that satisfy the constraints and a minimum distance. The size of each computed set gives a lower bound on the maximum size. The lower bound computed with these two algorithms is always better or equal to the lower bound computed by Limbachiya et. al. for certain parameters. When the minimum distance of a code is 2 we know this maximum size. The formula for this maximum size is given in this thesis together with a proof for this formula.

In dit verslag bekijken we zes topologische tegenvoorbeelden. Eerst bestuderen we de topologie van Appert. Deze topologie is gedefinieerd op een aftelbare verzameling, maar heeft overaftelbaar veel open verzamelingen. In de ruimte van Appert zijn de triviale rijtjes de enige rijtjes die convergeren. De ruimte is separabel, maar voldoet niet aan het tweede aftelbaarheidsaxioma.
Het is bekend dat elke aftelbare reguliere ruimte normaal is en dus `veel' continue functies naar het eenheidsinterval heeft. Daarom is een aftelbare reguliere ruimte niet samenhangend. Dit geldt echter niet voor aftelbare Hausdorff ruimten. We bestuderen een ruimte die aftelbaar Hausdorff is én samenhangend. Deze ruimte heet Gustin's rijruimte.
Daarna bekijken we een andere samenhangende ruimte: de topologische sinus-kromme. Deze is niet compact, maar we kunnen de ruimte uitbreiden en er een compacte ruimte van maken. We zien in dit hoofdstuk dat niet elke kromme padsamenhangend is. Dus samenhang impliceert niet altijd padsamenhang.
De vierde ruimte die we bestuderen is Van Douwen's ruimte. Deze is regulier en $T_1$. Elke continue reëelwaardige functie in deze ruimte is constant.
Verder bekijken we een overaftelbaar product van normale ruimten, waarvan de productruimte zelf niet normaal is. Of het product separabel is of niet, hangt af van de `grootte' van het product.
Tot slot nog een productruimte: de ruimte van Helly. Dit product is wel normaal, maar bevat een niet-normale deelruimte. We zullen namelijk zien dat de ruimte compact Hausdorff is en een deelruimte heeft die homeomorf is met de Sorgenfreylijn. De ruimte voldoet aan het eerste maar niet aan tweede aftelbaarheidsaxioma. De ruimte is wel separabel.

The automation of replanting seedlings into bigger trays (transplanting) has been a major industrialisation step in the horticultural sector. Modern machines are abundant in large companies and are quite effective, but they are very expensive both in purchase and maintenance. With major clients taking these costs for granted, designers have not stopped to consider ways to alleviate them. This work introduces an affordable electronic system that makes transplanters wireless, allowing for hot-swappable actuators and thus greatly reducing the cost of technicians and downtime as well as procurement. The expensive servo motor drives are replaced by dedicated microcontrollers, allowing the actuators to make their own decisions based on the constraints imposed by various standards of tray sizes and other circumstances. A method is presented to derive the most favourable trajectories, which are then enforced on the system through a closed-loop feedback system. The resulting performance approaches that of the system's predecessor, but at a much more reasonable price. This makes transplanters more affordable for small companies, allowing a broader market to reap the harvest of technological development and increasing worldwide horticultural yield as a result.

In dit verslag worden drie noties van eindigheid van een verzameling gedefinieerd.
Een verzameling is eindig wanneer er een bijectie bestaat tussen de verzameling en een natuurlijk getal. Uiteraard worden eerst de natuurlijke getallen gedefinieerd.
Een verzameling is surjectie-eindig als iedere surjectieve functie met als domein en codomein deze verzameling ook injectief is.
Een verzameling is Dedekind-eindig als iedere injectieve functie met als domein en codomein deze verzameling ook surjectief is.
Er wordt bewezen dat deze drie noties in ZFC equivalent zijn.
Ook wordt bewezen dat in ZF iedere eindige verzameling surjectie-eindig is, en iedere surjectie-eindige verzameling Dedekind-eindig is.
Ten slotte wordt aangetoond dat de uitspraken `iedere Dedekind-eindige verzameling is surjectie-eindig' en `iedere surjectie-eindige verzameling is eindig', niet te bewijzen zijn in ZFA.
De conclusie is dat de hiërarchie eindig impliceert surjectie-eindig impliceert Dedekind-eindig strikt is.

This thesis concerns the cap set problem in affine geometry. The problem is illustrated by the card game SET and its geometrical interpretation in ternary affine space. The maximal cardinality of a cap is known for the dimension one to six. For the four lowest dimensions, a maximal cap is constructed and the optimality of its size proven. From there, two recursive methods are described and applied to obtain upper bounds for the maximal size of caps in dimensions seven to ten. The best found upper bounds are 291, 771, 2070 and 5619, respectively.