In this thesis we consider orbital stability of certain patterns in stochastic partial differential equations. We study two examples: a rotating wave in a two-dimensional reaction-diffusion equation and a soliton in a parametrically forced nonlinear Schrödinger equation. In both cases, we show that, for small noise, solutions to the stochastic equations remain close to a version of the pattern which is shifted according to some stochastic phase. We give explicit expressions for this phase, and show that it is optimal to first order in the strength of the noise.

To show stability, we construct a multiscale expansion of the solution around an arbitrarily shifted version of the pattern, and show that this expansion is accurate to second order. From this expansion an obvious candidate for the correct phase shift arises. For technical reasons we then construct a sequence of approximations to this phase shift, which is necessary to show the multiscale expansion around the correctly shifted pattern. We then combine this expansion with a deterministic stability result to get stochastic stability.

Finally, we take first steps towards formulating and proving the same results in a more general setting, where the pattern shift is represented by the action of a Lie group. We obtain some estimates necessary for the multiscale expansion, find the correct phase, and formulate necessary assumptions for the stability to hold.

In this thesis we consider the incompressible and stationary Stokes problem with Navier-slip boundary conditions on an infinite two-dimensional wedge with opening angle θ. As is common for differential equations on domains with corners, the problem is decomposed into a singular expansion near the corner (polynomial problem) and a regular remainder (smooth problem). We prove existence and uniqueness of solutions to the smooth problem related to the Stokes equation which is given by -PΔu = f, where P is the Helmholtz projection. By means of the Lax-Milgram theorem it is found that this problem has a unique strong solution in a certain class of weighted Sobolev spaces if the opening angle θ is small enough. Direct application of the Lax-Milgram theorem would normally only yield a weak solution. However, by introducing additional bilinear forms we gain control on all second order derivatives and therewith obtain a strong solution. Finally, we touch upon the time-dependent Stokes problem and the polynomial problem.

This thesis treats the thin-film equation which models the film height $h$ for a viscous film in the complete wetting regime. We show existence and uniqueness to the thin-film equation with mobility m(h) = hⁿ and mobility exponent n∈ (1,3/2)∪ (3/2,3). The thin-film equation is rewritten as an abstract Cauchy problem and usage of semi-group theory yields maximal L^p-regularity for the linearized problem. With a fixed point argument, analogous to the one used by Giacomelli, Gnann, Knüpfer and Otto in their 2014 paper, the nonlinear problem is treated. Under a smallness condition on the initial value to a suitably transformed version of the thin-film equation, we obtain a solution in L^p(0,∞;H_{k-2,α-1/2})∩ \dot{W}^1,p(0,∞;H_{k+2,α+1\2}), where the H-spaces denote weighted Sobolev spaces. The novelty of this work lies in the usage of L^p-spaces in time, where the existing literature only deals with L²-spaces. It is found that the L^p setting allows for treatment of all n∈ (1,3/2)∪ (3/2,3).

This thesis is about the wave equation. The wave equation describes waves that propagate through a certain medium. The solution of the wave equation is a mathematical description of what that wave looks like. There are many fields of study where the wave equation is used to model processes that behave like travelling waves. For instance, a vibrating string or membrane can be modelled by the wave equation very well. Furthermore, the wave equation can be used to model light or sound waves and how they reflect and refract when travelling
through different materials.
Because the wave equation is such an important tool to model these phenomena, there are many people working on solving the wave equation in different contexts, i.e. finding a solution. However, it is difficult (often impossible) to find an exact solution. Therefore, people usually calculate a ’solution’ that is approximately correct.
An important question to ask is: ’Does the wave equation always have a solution?And is there only one solution?’. Physically, the answer is clear. If a string is vibrating, it clearly cannot vibrate in two ways at the same time and it will always vibrate in some particular way. Even though the answer is clear physically, answering this question mathematically is more difficult. It is good to remember that the wave equation is just a mathematical model of propagating waves, and it could very well be that the wave equation has more than one solution or no solution at all in some particular context.
The answer of this question is important for the people calculating approximate solutions to the wave equation. If there does not exist a unique solution, the process of calculating the approximate solution may break down.
Answering that question is the subject of this thesis. We will first introduce the necessary mathematical tools, after which we will use existing research to find out under which conditions the wave equation has one, and only one solution. Finally, we will extend our results to more general equations than just the wave equation.

In various scientific disciplines, measurement data is collected across space and over time with the aim of inferring information regarding an underlying stochastic spatiotemporal phenomenon. The high computational costs of current methods render this task intractable for the large datasets typically encountered in spatiotemporal statistics.

We propose a model based on Gaussian random fields, defined as the solutions to a class of space-time fractional stochastic partial differential equations (SPDEs) driven by Gaussian noise. The tunable parameters characterizing this class are expected to admit intuitive interpretations, in which case an efficient numerical approximation scheme for solutions to these SPDEs yields a powerful spatiotemporal model which is usable in practice.

The study of this class of SPDEs is the focus of this work. We define weak and mild solution concepts and show their equivalence under lenient conditions on the differential operators involved. Subsequently we establish results linking the well-posedness, spatiotemporal regularity and asymptotic covariance structure of the SPDEs to conditions on the model parameters, confirming their interpretability. Lastly, we specialize the SPDEs to a class of fractional ordinary differential equations in time, and describe a method to compute numerical approximations of their solutions. Future work is needed to analyze this scheme and generalize it to the original class of SPDEs.

In this thesis, we analyse the spectrum of the generator of the one-dimensional Zig-Zag process defined on the torus $\mathbb{T}$. This is a piecewise deterministic Markov process (PDMP) used in Monte Carlo Markov chain methods (MCMC) for sampling from a probability distribution and calculating integrals \cite{Rejectionfree}, \cite{ZigZag}, \cite{Bouncy}. We show for Lipschitz potentials $U$ and bounded refreshment rates $\lambda_0 \in L^{\infty}(\mathbb{T})$ that the spectral gap $\kappa = \sup\{\operatorname{Re} \lambda : \lambda \in \sigma(\mathcal{L})\} \setminus \{0 \}$ of the associated $J$-self-adjoint generator $\mathcal{L}$ on $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ is positive. Moreover, we give two lower bounds for $\kappa$ by making use of one of the Schur complements associated with a block operator that is unitarily equivalent to $\mathcal{L}$. In addition we show that the spectrum of $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ are the same and that the generator defined on both spaces generates a contraction semigroup. Under the assumption of unimodality of the potential $U$ and a zero refreshment rate, we show that a vertical "asymptotic line" exists to which all of the eigenvalues converge. Furthermore, we show that a spectral mapping theorem exists where, due to the spectral line, the spectrum of the semigroup can become uncountable or countable depending on the time parameter of the semigroup $P(t)$ generated by $\mathcal{L}$. Lastly, we show that a discretisation of the spectrum generates a semigroup that converges uniformly on each bounded time interval to the semigroup of the Zig-Zag process and we use these discretisations to numerically analyse the behaviour of general potentials and refreshment rates.

This thesis considers the thin-film equation in partial wetting. The mobility in this equation is given by h³+λ^3-nhⁿ, where h is the film height, λ is the slip length and n is the mobility exponent. The partial wetting regime implies the boundary condition dh/dz>0 at the triple junction. The asymptotics as h↓0 are investigated. This is done by using a dynamical system for the error between the solution and the microscopic contact angle. Using the linearized version of the dynamical system, values for n when resonances occur are found. These resonances lead to a different behaviour for the solution as h↓0, so the asymptotics are found to be different for different values of n. Together with the asymptotics for h→∞ as found in [Giacomelli et al., 2016], the solution to the thin-film equation in partial wetting can be characterized. Also, via this solution, the relation between the microscopic and macroscopic contact angles can be analyzed. From the main result of this thesis, it can be seen that the macroscopic Tanner law for the contact angle depends smoothly on the microscopic contact angle.

This thesis covers the power load flow problem and three numerical methods which can be used as a solution. We firstly discuss the concepts from electrical engineering required to discribe the problem. This includes alternating current and voltage and admittances of various electrical elements. We then use these concepts to derive a set of equations that govern how power flows through a network. To solve these equations, we consider the Newton-Raphson, line-search and trust-region algorithms. For these methods, we also look at its convergence and the amount of computational power required. We conclude in the cases where the parameters of the electrical network are within reasonable bounds, the Newton-Raphson algorithm is sufficient in precision and requires the least computational power. Otherwise, one of the other two methods may be tried.

In this thesis we study the boundedness of a generalization of the Hardy-Littlewood maximal operator, involving rearrangement invariant Banach function space and indices of the spaces.
We first consider a classical proof of boundedness of the Hardy-Littlewood maximal operator on rearrangement invariant Banach function spaces. After establishing necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, we consider a generalization of the Hardy-Littlewood maximal operator introduced by C. Pérez. We investigate and slightly improve the known sufficient conditions under which this more general maximal operator is bounded on a rearrangement invariant Banach function space. After which we search and find necessary conditions for boundedness in a general setting. In the final section we study Boyd indices and fundamental indices, especially how they are related to boundedness of the more general maximal operator. We also introduce weak fundamental indices and investigate some of their properties and uses. Finally we show how under certain assumptions we can state equivalent necessary and sufficient conditions for boundedness on Lorentz spaces and Orlicz spaces.

In this master thesis the proof of Lotz that Weak Lp spaces have the Grothendieck property is studied. The proof is slightly modified to be more explicit and easier to comprehend by introducing lemma’s to better separate different parts of the proof that more clearly reveal its structure. Furthermore, the more general Marcinkiewicz spaces are shown to sometimes have the Grothendieck property, using the sufficient conditions for a Banach lattice to have the Grothendieck property that Lotz derived to prove the Grothendieck property of Weak Lp spaces. For most of these conditions that together are sufficient, proving that Marcinkiewicz spaces satisfy them is done in a way very similar to the case of Weak Lp spaces. However, the proof of the (necessary) condition that the dual sometimes has order continuous norm does not allow for such a simple generalization and requires more work. Finally, by using some more recent results about the existence of symmetric functionals in the dual, the conditions that are given for the Grothendieck property of Marcinkiewicz spaces are shown to be necessary, and we thereby obtain a characterization of the Marcinkiewicz spaces that have the Grothendieck property

In this thesis, we investigate whether maximizing the coverage of taxis can be beneficial when the goal is to minimize the waiting times of the clients. When dispatching taxis, often only current requests are taken into account and not future ones. We examined how beneficial it can be to take coverage into account. For taxi companies it is important to keep their customers satisfied, by serving them as quickly as possible. Often companies assign the taxis to the nearest requests. We investigate whether there are better dispatch methods. We developed three dispatch policies which use ILP models, and discussed what the best option is. The first policy does not consider the coverage and minimizes the waiting times of the clients among the requests that are taking place at that moment. The second one aims to maintain a good coverage and short waiting times when assigning taxis to requests. The third one also aims to maintain a good coverage and short waiting times when assigning taxis to requests and also relocates taxis to gain a better covered area. Those policies can be a contribution for taxi companies because they help to serve clients more quickly. To determine what the best way is of dispatching taxis, we run a simulation on the different policies on a real-data map and compared the results. The literature over taxi dispatch mostly conducts research on a small area like a city or a village. In this thesis, taxis dispatch takes place on a larger area where taxis commute between cities.