The Poincare Inequality on Smooth and Bounded Domains in R^d

In this thesis we study for which domain types the Poincare inequality holds for all functions having continuous first derivative. We first consider the classical Poincare inequality, which we prove holds for a very large class of open sets in R^d. We then constructively prove that bounded, open, and connected domains in R^d,...

Estimating normalizing constants using stochastic simulation

The computation of normalizing constants often brings (higher-dimensional) integrals, which could not be computed analytically or are computationally expensive. Stochastic methods will be explored to find an estimate for normalizing constants. These stochastic methods will be used to estimate the Bayes factor. With this Bayes factor can be...

Image Reconstruction for Dynamic Multi-Pinhole SPECT-imaging using Kernelised Expectation Maximisation

<i><b>Background </b></i>Dynamic SPECT scanning provides a non-invasive way to image the time-dependent distribution of radio-labelled tracers inside living tissue. Beside human medicine, dynamic SPECT also finds its applications in pre-clinical research on small animals. In pre-clinical research, multi-pinhole collimators are used to enable...

Sequence optimisation for Magnetic Resonance Fingerprinting

Magnetic Resonance Fingerprinting (MRF) is a relatively new approach for simultaneously estimating multiple quantitative maps in one acquisition. Sequence optimisation for MRF can be a powerful tool in increasing the accuracy an precision of the quantitative results. Multi-component analysis in the MRF framework can distinguish multiple...

Super Resolution Techniques Applied to Low-Field MRI

This work is part of the low-fieldMRI project, which aims to bring portable, affordable, low-fieldMRI scanners to low-income countries. Replacing the superconducting magnets of conventional scanners with standard ones can significantly reduce the costs, but it also has a negative impact on the Signal-to-Noise Ratio (SNR). In order to circumvent...

On Euler and Fibonacci Numbers: Why Pi is Bounded by Twice Phi

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...

Limiting behavior of a random variable conditional on a rare event

As an insurer you want identify the risks you take to prevent bankruptcy. The<br/>theory of large deviations formalizes the study of such rare events. We will use the<br/>theorem of Cramér, which is a main theorem in large deviation theory, to investigate<br/>the rate at which the probability of large deviations of the sums of random variables...

Het ontbinden van een matrix m.b.v. een getraind neuraal netwerk

Before making matrix calculations, it can be useful to decompose these matrices in some way, such as according to an LU-decomposition. This can save a lot of computing power and thus time. Several computer programs exist that can decompose matrices in multiple ways. However, this report answers the question of whether it is also possible to make...

An efficient metaheuristic to solve the project portfolio selection and scheduling problem for industrial projects

Industrial companies aim for optimizing profit from delivering project outcomes. Maximizing profit relies on optimization of using resources, production capacity and available time. To reach this goal, companies are typically reliant on planning and production schedules. This problem is known as the project portfolio selection and scheduling...

Codes for Noisy Channels with unknown offset

Storage systems, such as Flash memories, suffer, apart from the always present<br/>noise, also from offset. The presence of this noise can decrease the performance<br/>of a decoder using the Euclidean distance significantly. To negate the effects of<br/>offset, a new distance, the modied Euclidean distance, was introduced, which<br/>offers...