Graph burning is a process that models the spread of information through a network. This process is divided into rounds and is visualized as a spreading fire. Each round, a source of fire may be chosen from where the fire spreads and burns surrounding points. From every burned point, the fire propagates through adjacent points, eventually covering the entire network. The burning number of a graph, b(G), was introduced to measure the time required to burn the entire network.

It was proven that path graphs on n vertices have burning number ⌈√n⌉. It was then conjectured that all connected graphs have at most this burning number. The burning number has been estimated or identified for several classes of graphs. We provide a lower and upper bound for the burning number of a new class of graphs with a path-like structure, namely necklace graphs. Necklace graphs are constructed by concatenating smaller graphs, called pearls. To obtain bounds, we define lower-bound paths and upper-bound paths, which are paths connecting two designated vertices. We show that a lower-bound path always exists in the form of a shortest path, while finding an upper-bound path for an arbitrary necklace is NP-complete. For necklace graphs where a shortest path is an upper-bound path, we show that the burning number can take only two values.

Finally, we consider cycle-forests, which are disconnected graphs whose components are cycle graphs. We prove that burning cycle-forests is NP-complete.

Compound symmetry groups are defined as groups generated by a set of isometries
of subspaces of a metric space. Here the main focus is on groups generated by discrete
rotations of overlapping disks in the plane. The simplest case occurs when only two disks
are involved, which are called two-disk systems. When the disks are rotated, they
are partitioned into pieces. Below a certain radius of the disks, we can label the pieces
and express the rotations as permutations, from which we can identify the group structure.
However, after a certain radius, the group may become infinite, and we will call this radius
the critical radius.

Two-disk systems can be extended to three-disk systems, and generalised to k-disk systems. Estimates of critical radii for three-disk systems are found using a brute-force approach.

In this paper, we provide a method to find all equable triangles on a given grid. Equable triangles are triangles that have the same perimeter and area. The amount of equable triangles was already found for the integer and the Eisenstein lattice. We adapt the proof by Aebi and Cairns for the Eisenstein lattice to work on general lattices. This is done in three steps:
1. Find a constraint on the side lengths of equable triangles on a given grid.
2. Find all equable triangles subject to this constraint.
3. Check for each triangle found if it can be placed on the grid.
Using this method, we can find all equable triangles on a given grid. On most grids, no equable triangles can be placed. When it is possible, they often had integer side lengths.

The problem of finding even cycles efficiently in a directed graph can be rewritten to a problem of finding the permanent and determinant of the graph’s adjacency matrix. Although this has been known for a long time, it did not solve the shortest even cycle problem, as mathematicians lacked an efficient method to compute the permanent.
In 1979 Valiant found out that in some specific cases, when one works modulo a power of 2, the permanent could be computed efficiently. Although this did not directly lead to a breakthrough in finding even cycles, it laid the groundwork for Björklund, Husfeldt and Kaski to solve the problem of finding the shortest even cycle. Instead of working with the integers modulo 2, Björklund, Husfeldt and Kaski used a polynomial quotient ring E4d , which is an extension of some polynomial quotient ring F2d . The polynomial quotient ring E4d , it turns out, benefits from having a characteristic of four and relies on F2d being a field. Surprisingly, these two properties are exactly what is needed to efficiently compute both the permanent and determinant.
The aim of this thesis is to give the reader a better understanding of the polynomial quotient rings F2d and E4d that are used by Björklund, Husfeldt and Kaski in. These structures are then used to explain the part of their algorithm that computes the determinant of matrices with entries in E4d . The reader is given a deeper insight into the algorithm by comparing it to the standard algorithm for computing the determinant learned in most linear algebra classes. By using the polynomial rings and algorithm for the determinant a Python script is written. This script can perform calculations within E4d and compute the determinant for a matrix with entries in E4d . Finally, the difference between computing the permanent and the determinant will be briefly explained.

Algebras are vector spaces with a bilinear product. When we fix a finite dimension n and a finite field K with q elements, there are a finite number of non-isomorphic algebras. Seeing as vector spaces are completely determined by the dimension and scalar field, the number of non-isomorphic algebras is the number of ways we can define a bilinear product of vectors. Exploiting the bilinearity of the product we can construct a surjection from vectors of n n by n matrices over K to non-isomorphic n-dimensional algebras over K. This function can be made injective by reducing to orbits under a group action on the set of vectors of matrices, thus providing a bijection. So the number of non-isomorphic algebras is equal to the number of orbits of this group action. In 2020 Verhulst used Burnside's Lemma to count the orbits and thus the number of non-isomorphic algebras. However, the formula he derived still requires a number of complicated calculations.

The aim of this thesis is to implement Verhulst's formula in python. In Table 4.1 you can find the output of the python script. The results for n=1 merely show some trivial cases. The results for n=2 fit the formulas obtained, using completely different methods, by Petersson and Scherer. The results for n=3 and n=4 are, to my knowledge, new. Currently, the limiting factor is computation time. A more powerful computer could squeeze out a few more results, but a more efficient formula is necessary. Based on the results, it seems that the contribution of the identity matrix in Verhulst's formula provides a decently tight lower bound.

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 1977 the RSA system was designed by Ron Rivest, Adi Shamir and Len Adleman. The RSA system is a part of modern cryptography. Modern cryptography makes sure that one can secure important information - like data, communication or banking accounts - on the computer. There are several attacks on the RSA system where the private key can be cracked. In this report two of such attacks are discussed.