This report investigates a scheduling problem where task duration is uncertain. The duration per task has a lower and upper bound, and is dependent on observed duration of other tasks. This tries to closer model real life. We reduce all possible different outcomes to a few extreme scenarios. The report compares two types of heuristcs: one which always chooses the longest duration task first, and one which tries to minimize the uncertainty by choosing tasks that reveal the most information. In the end, we find that the heuristic choosing the longest duration task first competes fairly well with the other type of heuristics.

It might surprise you that networks play a role in biology, but networks are ubiquitous. All living things have DNA within their cells. This DNA contains the building blocks of an organism. The process of creating such a building block requires mRNA and proteins. The concentration of a certain mRNA molecule plays a part in the making of these proteins. In biology these processes are called transcription and translation. These interacting processes can be transformed into a mathematical model, a network. This network is a collection of nodes and edges, which represent the interactions between different mRNA concentrations. Since these interactions are unknown, random matrix theory is used to model these interactions. It is interesting how the concentrations of mRNA molecules evolve over time. A system of differential equations can be used to model these changes of concentrations over time. This report aims to discover the properties of the model of interacting organisms. For this a linear model is found, which is a system of differential equations linearised around a given equilibrium. A system can be written as a matrix, to model multiple organisms this results in a block matrix. Each block can then be envisioned to model a certain gene pool that corresponds to an organism. Interactions between these gene pools can be modelled by adding an interaction block to the off diagonal blocks of the block matrix. Later on this linear model is improved with a non-­negative constraint, concentrations are after all non-­negative, which results in a new nonlinear model. Properties of both models are found by studying the distribution of the eigenvalues. Girko’s law and Wigner’s law are two important laws from random matrix theory, that help with the determination of the eigenvalues of a random interaction matrix. For the linear model it was found that the distribution of the eigenvalues is influenced by the entries of the block matrix and by the strength of the connection between the block matrices. Once the eigenvalues and the corresponding eigenvectors are found, the solution is deterministic. For the nonlinear model it was found that the distribution of the eigenvalues are influenced in a similar way as the linear model. But due the non­negative constraint, the stability of the system is not deterministic. The system can be partly asymptotically stable, partly stable and partly unstable for different time windows. It is living on the ‘edge of chaos’.

When is a deck of cards shuffled good enough? We have to perform seven Riffle Shuffles to randomize a deck of 52 cards. The mathematics used to calculate this, has some strong connections with permutations, rising sequences and the L1 metric: the variation distance. If we combine these factors, we can get an expression of how good a way of shuffling is in randomizing a deck. We say a deck is randomized, when every possible order of the cards is equally likely. This gives us the cut-off result of seven shuffles. Furthermore, this gives us a window to look at other ways of shuffling, some even used in casinos. It turns out that some of these methods are not randomizing a deck enough. We can also use Markov chains in order to see how we randomize cards by ”washing” them over a table.

There are several ways to write the number 5 as a sum of positive integers, disregarding order. A quick calculation shows that this can be done in 7 ways: 5 = 5, 5 = 4 + 1, 5 = 3 + 2, 5 = 3 + 1 + 1, 5 = 2 + 2 + 1, 5 = 2 + 1 + 1 + 1 and 5 = 1 + 1 + 1 + 1 + 1. In the same way, we can count the number of ways for each integer n. Although this calculation is trivial, a closed form for this function is not as easily obtained as one for combinations, for example. This work formulates and proves Rademacher's formula for these partition numbers. It also tries to uncover some key ideas behind the proof, the supporting theory and other inspirations.

Een BacheloreindProject over het verbeteren van de powerfactor in elektrische netwerken. Door een groter aantal frequenties in de spanning óf door bepaalde componenten in de belasting is er een compenserend gedeelte in het netwerk nodig en / of mogelijk. Met welke schakelingen, componenten en karakteristieke waarden dit gedaan kan worden, zal per situatie verschillen.