One of the main focuses of this thesis is a special kind of symmetrical arrangement called a "Steiner system." Imagine trying to organize a group of objects in such a way that every possible subset of them fits together in a very specific and symmetrical pattern. These patterns are not just mathematical coincidences; they have real-world applications, such as in coding theory.

More precisely, among these Steiner systems, there are some particularly interesting ones known as "Witt designs." These designs are important because their symmetrical properties are used in constructing certain types of error-correcting codes. These codes can correct a certain amount of errors within information that is sent or stored, making it more reliable.

To better understand these Witt designs, the thesis explores two main ways to create them. One method uses something called the Golay code, which is a specific type of error-correcting code. The other method involves a mathematical group called the projective special linear group. Both of these constructions are explained in detail, with proofs provided to show that they indeed form the desired symmetrical patterns.

In short, the Gibbs phenomenon describes the oscillating behaviour followed by an overshoot or undershoot of a Fourier partial sum expansion compared to the original function near jump discontinuities. In particular, the fact that this overshoot or undershoot will never decrease below 9% of the height of the “jump”.
First, some historical remarks about the discovery of the Gibbs phenomenon are given and several fundamental elements of Fourier analysis are introduced. By thoroughly analyzing the Saw-tooth wave function, an example of a function that exhibits the Gibbs phenomenon, we provide some figures in order to visualize this phenomenon. Furthermore, we address and fill in the gaps of the existing mathematical proof to advance the theoretical understanding of the Gibbs phenomenon. This analysis is then generalized to apply to a broader class of functions with jump discontinuities, by first considering a function with a jump discontinuity at 0 and then proving the occurrence of the phenomenon at a general point of jump discontinuity.
Furthermore, a literature research is conducted to evaluate different methods for resolving the Gibbs phenomenon. This thesis includes an in-depth analysis of filtering methods and spectral reprojection methods. Specifically, the Gegenbauer reconstruction method is described and its theoretical foundations in removing the Gibbs phenomenon is examined.
Finally, a real-life application of the Gibbs phenomenon is given by looking at its occurrence in Magnetic Resonance Imaging.
Overall, this thesis not only advances the theoretical understanding of the Gibbs phenomenon but also provides an insight into its resolution and application in Magnetic Resonance Imaging.

This thesis explores and proves the theorem of De Rham using sheaf cohomology. The theorem of De Rham states that for a smooth manifold, the singular cohomology is isomorphic to the De Rham cohomology. Intuitively, this theorem states that the number of holes in a smooth manifold coincides with the failure of the fundamental theorem of calculus for closed differential forms defined on the manifold. This theorem allows us to make analytical conclusions based on the geometry of a surface.

Initially, the homological algebra of abelian groups is studied to quantify the holes in a smooth manifold and the failure of the fundamental theorem of calculus. An isomorphism of these two cohomologies is locally established. Subsequently, sheaves and their corresponding cohomology theory are introduced to lift this local isomorphism to a global one. Ultimately, using a double diagram of sheaves, the singular and De Rham cohomologies are shown to be isomorphic, thereby proving the theorem of De Rham.

Staphylococcus aureus and Pseudomonas aeruginosa are two species of bacteria that are involved in numerous conditions, including lung infections and chronic wound infections. The aim of this project was to study the short-term interactions that occur when P. aeruginosa first encounters an established S. aureus colony, which it then seeks to break apart whilst mixing with S. aureus. Limoli et al. have studied these interactions using experiments, and have thus identified several key aspects involved in these interactions, such as the mechanisms that P. aeruginosa employs to approach the S. aureus colony. The means by which we intended to study interactions between S. aureus and P. aeruginosa is a model that was made by previous members of the Idema group and that was based on the experiments by Limoli et al. In this report, we discuss this model and the biological background relevant to it. We also document the problems that we encountered while trying to run simulations using an existing implementation of this model.

Let (S, Σ, μ) be a divisible measure space. Let F be a collection of subsets of S that are in Σ. For some applications it can be useful to describe the overlap between the sets in F. The sparse and Carleson constant both describe this overlap in a different way. The closer both of these constants are to 1, the closer the sets in F are to being pairwise disjoint. It has been shown that the sparse and Carleson condition are actually equivalent: we always have that F is Λ-Carleson if and only if F is Λ^-1-sparse. Proving that a Λ^-1-sparse collection is Λ-Carleson is quite simple, but proving that every Λ-Carleson collection also is Λ^-1-sparse turns out to be much harder. Previous proofs of the fact that Λ-Carleson are Λ^-1-sparse, such as the one by Hänninen [14] and Rey [25], have all relied on difficult theory. There is also no known method to exactly find the sets E_Q for each Q ∈ F that we need to satisfy the sparse condition. Rey is able to approximate these sets, but his algorithm has a logarithmic loss that can only be removed when imposing geometric restrictions.

In this paper I will give a proof of the equivalence of the sparse and Carleson condition for any finite collection F that relies only on basic set and optimisation theory. This proof can be extended to infinite collections F with only a minimal restriction. Asides from proving the equivalence, I also describe an algorithm that can find the sets E_Q for each Q ∈ F that we need to satisfy the sparse condition if F is finite and Carleson with respect to a divisible measure μ. Finally, I will describe an algorithm that we can use to find the Carleson constant of a finite collection F if it is unknown.

Within this report, an analytical research will be performed on the effectivity of Tuned Mass Dampers (TMDs) and Viscoelastic Dampers (VEDs), and a conclusion will be drawn as to which is most effective in damping the oscillations of an earthquake. First of all, a simple model of second-order differential equations describing the oscillations of each floor within a building of n floors is derived, in which no frictional forces are taken into account. The solution to this model is derived for a building of two floors, which demonstrates the characteristics of oscillations in this model and which shows how resonance is represented mathematically, and what patterns of oscillations look like. Then, a TMD and VED are added to the model of one floor to expose what their working principles are. Subsequently, the methods of adding these dampers to the models for two, five and ten floors are given. For varying earthquake-frequencies, the maximum displacement within 5 seconds is determined, which shows the effect of TMDs and VEDs for different positions within the building. The conclusion from these results is that the effect of the TMDs is always that at the eigenfrequencies, the oscillations of the building are no longer resonating. However, the building will start resonating for other eigenfrequencies due to the TMD. The VED is more consistent, though its damping-effect on resonating frequencies is smaller than for the TMDs. The conclusion is that VEDs are most effective, though more dampers have to be added to improve its effectiveness. After this, a new model is introduced, which does contain a friction force: air resistance. Based on the new model, it is concluded that even oscillations for resonating frequencies are eventually damped out, though they do still grow large. Again, TMDs and VEDs are added to the model, and similar conclusions about the effectivity of the dampers as for the first model were drawn.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes.

First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes.

Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode.

Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide.

Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.

Context: In astronomy, collision detection is the computational problem of finding when planets, asteroids or satellites collide. Computer simulations make it possible to show very precisely how bodies move through space, which makes it possible to detect collisions. However, these simulations require a lot of computing power if there are many bodies, which is of course undesirable.
Aims: The aim of this report is to provide more insight into a collision detection method that does not use the computationally expensive simulations, but rather fast mathematical techniques.
Methods: It can be shown that the collision detection problem between two planets is equivalent to finding the integer point (𝑘, 𝑙) between two parallel lines, closest to the origin. The solution uses the continued fraction of the ratio of the orbital periods of the planets, as the slopes of the lines are this ratio. The convergents of the continued fraction form basis vectors with which the point (𝑘, 𝑙) is searched for. This is called lattice basis reduction.
Results: Where brute force always takes 𝑘 steps to find the integer point (𝑘, 𝑙), the method with continued fractions and lattice basis reduction often finds the point in about √𝑘 steps. The worst case however is still 𝑘 steps, the same as brute force. This only happens if there are no basis transformations, so luckily 𝑘 is then often small.
Conclusions: When there are many bodies between which collisions need to be detected, lattice basis reduction provides a fast method. For the basis vectors, the convergents of the continued fraction of the ratio of the orbital periods of the planets must be used. If the solution cannot be reached exactly, (𝑘, 𝑙) can also be estimated.

In this thesis, we use a variation of a commutator technique to prove that l^p-stability is independent of p, for p greater than or equal to one, and for convolution-dominated matrices indexed by relatively separated sets in groups of polynomial growth. Moreover, from the inverse-closedness of the Schur matrices we deduce a Wiener type Lemma for the matrices in the intersection of the weighted convolution-dominated matrices, over all polynomial weights.
Finally, applications of the convolution-dominated matrices are presented. We prove the inverse-closedness of a non-commutative space generated by a discrete series representation restricted to a lattice in a nilpotent Lie group. In addition, we apply the aforementioned result on l^p-stability to show that if \pi(\Lambda)g is a p-frame for the coorbit space Co(L^p) for some p in [1,\infty], then \pi(\Lambda)g is a q-frame for the coorbit space Co(L^q) for each q in [1,\infty], where (\pi, H) is a discrete series representation of a group G of polynomial growth, \Lambda is a relatively separated set in G, and g is a vector in H such that the matrix coefficient V_g g is in the Amalgam space W_{w_a}(G). Moreover, we prove that the frame operator of the frame \pi(\Lambda)g is invertible on the coorbit spaces Co(L^p) for each p in [1,\infty], under the assumption that g is a vector in H, such that V_g g belongs in W_{w_a}(G) for each polynomial weight w_a.

In this thesis the Conjugate Gradient method used in the forecasting of the number of students in the Netherlands is investigated. The addition of international students to the forecasting led to mathematical problem of which a solution could not be computed. While the numerical method is working as intended, the work presented in this thesis shows that the unsolvability lies with a linear system that has no solution. The cause of this is attributed to the use of a selection process in the linear system. Leading to a hugely singular matrix and a forecasting problem without a solution. Several methods were tried to change the use of the selection process but without any results .

Implementing a correction to vesicle fluctuation analysis that accomodates non-zero exposure times, analysing the effects of the changes.

In this thesis, we give a primal-dual interior point method specialized to clustered low-rank semidefinite programs. We introduce multivariate polynomial matrix programs, and we reduce these to clustered low-rank semidefinite programs. This extends the work of Simmons-Duffin [J. High Energ. Phys. 1506, no. 174 (2015)] from univariate to multivariate polynomial matrix programs, and to more general clustered low-rank semidefinite programs.  Clustered low-rank semidefinite programs are programs with low-rank constraint matrices where the positive semidefinite variables are only used within clusters of constraints. The free variables can be used in any constraint, and can be used to connect clusters. The solver uses this structure to speed up the computations in two ways. First, the low rank structure is used to reduce matrix products to products of the form uT M v, where M is a matrix and u and v are vectors, as already suggested by Löfberg and Parrilo in [43rd IEEE CDC (2004)]. Second, an additional block-diagonal structure is introduced due to the clusters. This gives the possibility to do operations such as the Cholesky decomposition block-wise.   We apply this to sphere packing and spherical cap packing. For sphere packing, the speed of the solver is compared to the often used arbitrary precision solver SDPA-GMP, showing the theoretical speedup in time complexity. We give a new three-point bound for the maximum density when packing spherical caps of $N$ sizes on the unit sphere.    

In this thesis, we introduce the quantum groups Uq(SL(2,C)) and Aq(SL(2,C)) as Hopf algebras. We study their representations, including their similarities and differences with the classical theory. We show that the eigenvectors of Koorwinder's twisted primitive elements of Uq(SU(2)) are dual q-Krawtchouk polynomials. We use this explicit expression to define generalised matrix elements and spherical functions in Aq(SL(2,C)). Then we use the Haar functional to show that these generalised matrix elements are Askey-Wilson polynomials with two continuous and two discrete parameters. Next, we show a new result. Namely, two twisted primitive elements of Uq(SL(2,C)) generate Zhedanov's Askey-Wilson algebra AW(3). Consequently, AW(3) is embedded as a subalgebra into Uq(SL(2,C)). We use this to show that overlap functions of twisted primitive elements in Uq(SU(2)) are q-Racah polynomials. With that, we derive a summation formula connecting q-Racah and dual q-Krawtchouk polynomials.

The concept of entanglement is one of the distinguishing features in quantum mechanics. Information about one particle can determine the state of another particle. This information and entanglement in subsystems is quantified by the entanglement entropy. The entanglement entropy has become an important research topic in recent years. Entanglement entropy is expected to grow with the boundary size for systems described by Hamiltonians with local interactions, described by the area law. For one dimension there exists a bound quantifying this area law S = O (2^1/ΔE) in terms of the energy gap ΔE between the ground-state energy and the first excited-state energy. It is an open problem whether the area law holds in higher spatial dimensions. Therefore it is of great interest to study the entanglement entropy in systems that violate the area law. In this thesis spin chain models with local interactions are studied for arbitrary spin. Hamiltonians with a unique ground state are constructed by mapping spin chains onto (coloured) Dyck and (coloured) Motzkin paths. We show that these models express logarithmic or power law violations of the area law for the bipartite entanglement entropy. These results are presented in the table below. The known bound of the area law in one dimensional systems is dependent on the energy gap of these models. The energy gap of the Motzkin-path model is investigated by constructing the an orthogonal excited state with small energy. In doing so we associate the Hamiltonian with the Laplacian of a graph. We present an alternative proof for the bound on the energy gap of the colourless Motzkin-path model than S. Bravyi et al and R. Movassagh et al. We show that the energy gap ΔE scales in terms of the chain length n as ΔE = O(n^-2) in the limit n →∞. Therefore ΔE → 0 in this limit, resulting in area-law violations of the entanglement entropy while maintaining the known bound on the entanglement entropy.

Dynamic Time Slot Management (DTSM) is a system often used in online retail to manage the delivery of goods to customers. With DTSM customers arrive over time and place orders. They get presented with a set of time slots and the customer picks the time slot in which he wants the goods to be delivered to his home. A DTSM system creates a time slot offer for the customers based on the current delivery schedule, which is a solution to a Vehicle Routing Problem (VRP). Each accepted customer is added to this delivery schedule. This delivery schedule gets periodically optimized by an optimization algorithm. During this optimization, new customers may arrive and receive time slots based on the non-optimized schedule, which causes a discontinuity in the system. In this thesis, five different procedures have been created that deal with this problem. The impact of the different procedures is analyzed by simulating the DTSM system and comparing the final delivery schedules. Solving the problem by leaving out the optimization or insertion step of the DTSM system is outperformed by all procedures. Procedures that barely make use of the optimized schedule accept the least number of customers. The procedure that delays customers, which makes it similar to the theoretical setting, accepts the most customers, but this comes at the cost of poor customer service. The most promising results are found by the procedure that inserts new customers into the optimized schedule. Enhancing this procedure with a merge algorithm improves the performance slightly.

In line with the growing trend of using machine learning to improve solving of combinatorial optimisation problems, one promising idea is to improve node selection within a mixed integer programming branch-and-bound tree by using a learned policy. In contrast to previous work using imitation learning, our policy is focused on learning which of a node's children to select. We present an offline method to learn such a policy in two settings: one that is approximate by committing to pruning of nodes; one that is exact and backtracks from a leaf to use a different strategy. We apply the policy within the popular open-source solver SCIP. Empirical results on four MIP datasets indicate that our node selection policy leads to solutions more quickly than the state-of-the-art in the literature, but not as quickly as the state-of-practice SCIP node selector. While we do not beat the highly-optimised SCIP baseline in terms of solving time on exact solutions, our approximation-based policies have a consistently better optimality gap than all baselines if the accuracy of the predictive model adds value to prediction. Further, the results also indicate that, when a time limit is applied, our approximation method finds better solutions than all baselines in the majority of problems tested.

Online P2P file sharing networks rely on the cooperation of participants to function effectively. Agents upand download files to one another without the need for any central authority. If agents all contribute to the network and share roughly the same amounts of data as they contribute the network will operate, however if some agents decide to defect and consume far more resources than they contribute the file sharing will stagnate. In online networks with some kind of central authority, such as Ebay, Airbnb, etc. cooperation is achieved through a review system, which is maintained and secured by the central authority. P2P networks are however distributed and cooperation must be achieved without this central mitigator. One way of approaching this problem is by observing cooperative biological communities in nature. One finds that cooperation among biological organisms is achieved through a mechanism called indirect reciprocity. Indirect reciprocity is based on a reputation scheme in which agents share information about each other’s cooperativeness aiding one another in deciding who to interact with and who to shun. In this work we analyse properties a reputation mechanism must satisfy in order to achieve cooperation in P2P networks, incentivising contributions and penalising excessive comsumption of data. In particular, we determine under what conditions reputation mechanisms are resistant to attacks on the P2P network. We focus on one attack above all, namely that of a sybil attack in which a malicious agent creates multiple fake identities who report high levels of cooperativeness about one another. We determine properties accounting mechanisms must satisfy in order to prevent attackers from obtaining arbitrarily high reputation and to consequently be able to consume arbitrarily large amounts of data. This thesis offers a theoretical framework for evaluating the effectiveness of reputation mechanisms on the basis of their ability to induce cooperation and their resistance to sybil attacks.

Optimizing supply chain management in hospitals can lead to a significant reduction in costs. One of the supply chain processes that can be optimized is the sterilization of reusable instruments used during surgical procedures. This process takes place at the Central Sterile Supply Department (CSSD) and consists of two main steps, `washing and disinfection' and `sterilization'. These steps are executed by batch processing machines, with a setup time prior to each step. The setup at each stage is executed manually. The machines of a CSSD are renewed every 10 years, which provides a opportunity to assess all capacity planning decisions. In this thesis, the aim is to determine the required resources of the CSSD, while minimizing the total costs and guarantying the availability of instruments for scheduled surgeries. Costs include: acquiring machines, machine batch costs, machine maintenance costs, and staff costs depending on the number of opening hours. This thesis contributes to current research by proposing a framework for the capacity planning decisions at a CSSD, by extending existing models by taking specific characteristics of the CSSD into account, and considering a new objective function, namely minimizing the total costs. Capacity planning decisions are considered on three hierarchical levels. First, the strategic level involves long-term decisions, where the number and type of required machines have to be determined. Second, on a tactical level, the amount of opening time has to be determined. Third, on an operational level, the instrument sets have to be scheduled within batches and machines. The sterilization process is formulated as a mixed integer linear problem (MILP). The problem is described as a multi-stage hybrid flow shop with additional constraints to represent the specific characteristics of the CSSD. This model takes into account capacity planning decisions on all three levels. The MILP formulation is proven to be NP-hard. The demand for sterilized instruments is determined from historical data from the Leiden University Medical Center. To evaluate the model, three instances with a timespan of a week are created. Preliminary results showed that the multi-stage flow shop is difficult to solve for real-life instances. Hence, a decomposition, based on the three hierarchical levels, is proposed. The decomposition leads to three levels of optimization models. The strategic model takes into account the scheduling of instruments per day, while the tactical model schedules per day parts, and the operational models determine a specific point in time. The proposed models are individually tested on their performance. The results from the strategic model are used as input for the tactical model. Results show that the strategic model underestimated the required amount of opening time, as it does not take into account the spread of release times. Furthermore, the computational results show that the tactical and operational models are difficult to solve for real-life instances. Hence, a heuristic approach is proposed by forming a chain of the models. The best results were obtained by setting a minimum amount of opening time for the strategic model and using the resulting machines as input for the tactical model. The results show that the instrument sets are equally spread over the week. To conclude, the results of this thesis contribute to quantifying the required resources for a sterilization process within a hospital. However, to obtain more practical results, future research is required. Suggestions for future research include: taking uncertainties within the process into account, the application of metaheuristics, and the required number of employees.

Waste-to-energy is the process of generating energy from municipal waste. The most conventional energy recovery method is incineration, which takes place at a waste-to-energy plant. The incoming waste is collected in a bunker, where an overhead crane stacks and mixes the material before feeding it to the furnaces. The more homogeneous the waste reaches the furnaces, the more stable the incineration process. In automated mode, crane movements are controlled by the generation and execution of crane orders.

The waste crane scheduling problem is formulated as a mixed integer linear program, describing the scheduling of crane orders based on order characteristics such as type, origin and destination. The objective of the problem is to optimise the crane performance, which is interpreted as the minimisation of the total crane driving time while performing as many mixing orders as possible.

In the base model, simulated order schedules are resequenced. Given the type, origin and destination of the orders, the starting times are reconsidered. The obtained reductions in total driving time compared to the simulation model vary from 2% to 7%, depending on the crane strategy, whether stacking or mixing is allowed, and the maximum time between the generation and completion of feeding orders. Since the order types are given as input, the number of mixing orders is fixed.
In the extended model, the determination of the order characteristics is included in the scheduling process. Unfortunately, the size of the model increases so fast over time that 15 minutes proved to be the maximum length of a scheduling period for which a solution could be obtained. With a rolling horizon approximation, a stacking period of 10 hours has been scheduled.

Decision rules on the generation of orders are derived from the output of the optimisation models. These decision rules are implemented in the simulation model. The obtained results are assessed based on key performance indicators mainly related to the mix quality. The rules resulted in up to 35% more mixing orders, which translates into material being mixed more often, not necessarily in more material being mixed.

A look at Wythoffs Nim, Wythoff rows and Akiyama's nim. We try to find a row and a game that match Wythoff's row and game. A row is found in the Lieshoutrow, but the game we chose: Akiyama's nim, wasn't close to the beauty of Wythoff's nim. We exolain why.