This thesis investigates the conserved currents of spinor fields. These can be computed through Noether’s theorem, which links symmetries to conserved quantities. This requires the transformation behaviour of spinors, which is given using the Clifford algebra. Furthermore, other relevant concepts for describing spinors, such as vector bundles and connections, are introduced. Finally, the concept of a universal spinor bundle is introduced, which can be used to describe spin-1 2 particles under the effect of and as a cause of gravity.

The analysis of equilibria of complex systems is challenging due to the high dimensionality and non-linear interactions. There are different kinds of complex systems such as cooperative systems in which all interactions help growth and mixed-weight systems, in which some interactions help growth, while others hinder it. In this thesis, we focus on the correlation between the equilibria of cooperative and mixed-weight systems. We simplified the equilibrium equations by combining the attributes of a system into a one-dimensional equation. This reduced equation is easy to compute and provides an upper bound on the equilibrium value of each node. Although this bound may exceed many actual equilibrium values, it still defines the subspace in which all equilibria must lie. For cooperative systems, we presented a theorem that provides constraints on two vectors. If these vectors satisfy the given conditions, then there exists an equilibrium between the components. We also discussed methods to find such vector pairs. We applied this theorem to relate the equilibria of mixed-weight and cooperative systems. The equilibria of the mixed-weight system are always less than or equal to some equilibrium in the cooperative system. We introduced a framework for classifying cooperative equilibria. On any subset of nodes, an equilibrium may have entries that are maximal compared to all other equilibria on that subset. This leads to a single equilibrium that is the largest at every entry, called the principal equilibrium, which is component-wise maximal. The principal equilibrium upper bounds all equilibria of the mixed-weight system. Finally, we discussed the inherent difficulty of translating cooperative equilibria into the mixed-weight system, which stems from high dimensionality and non-linearity. We stated the conditions that mixedweight equilibria must satisfy and provided constraints determining if a cooperative-system equilibrium remains valid when competitive interactions are added. This concludes the comparison by showing that the principal equilibrium provides a component-wise upper bound for all equilibria of the mixed-weight system.

From the identification of the pathogens causing major human diseases, through the understanding of the most basic forms of life, and in the struggle to develop new antibiotics to counter the increasing bacterial resistance to these widely used substances, research on microbial growth and survival has rightfully seen a booming interest across the last two centuries. The guanosine tetraphosphate (ppGpp) signaling system is of particular significance for these fields of research. It is a bacterial response to stress and starvation which allows these organisms to activate the necessary genes to survive in these conditions. It also plays a key role in modulating the abundance of various machineries necessary for growth in order to maximize the rate at which bacteria are growing in different nutrient conditions. In poorer nutrient conditions requiring more enzymes to import and digest these nutrients into the same amount of essential building blocks for growth, production of ppGpp is triggered in response to the lack of these building blocks. Higher ppGpp concentrations lead to downregulation of the abundance of ribosomes, the machineries operating growth, allowing larger abundance of enzymes producing the building blocks from nutrients. Recent advances, by genetically modifying bacteria, allowed to finely tune ppGpp concentrations independently of growth conditions. Using this approach, we investigate the scope of the regulation operated by ppGpp: which proteins does it influence in changing growth conditions and to what extent? We present novel quantifications of different proteins and other relevant biomolecules in E. coli strains artificially modified to have higher or lower ppGpp concentration than naturally found. These results allow us to identify which proteins are regulated by ppGpp and why the right concentration of ppGpp is necessary for bacteria to achieve optimal growth. We confirm the role of ppGpp in upregulating the proteins related to the synthesis of new proteins: translation. We also show that, apart from translation-related proteins, ppGpp is not responsible for consistently regulating other groups of proteins, including some that do vary with growth rate variations caused by change in nutrient source, which were suspected to be under ppGpp’s control. With a simplified mathematical model of the ppGpp regulation mathematical model, we attempt to understand why slowing down bacterial growth by artificially increasing ppGpp requires a lot more ppGpp than naturally found. By doing so, we identify characteristics that a model seeking to explain ppGpp perturbations should have. These conditions strengthened our understanding of the ppGpp system by refuting some of our intuitions and laying a foundation for future models elucidating the missing pieces. We also present another story regarding how yeast can survive one of the harshest stress: being completely desiccated. With these different studies, we extend our knowledge of two different systems relevant to growth and survival of microorganisms, which leads to potential new directions for those who seek to investigate such systems and answer some of the questions that our findings raise.

This thesis investigates the world of optics at the sub-wavelength scale, with a focus on understanding the behaviour of singularities that can arise in the electromagnetic field when confined to two dimensions. Predominantly we have investigated random waves, which are the most generic form of wave patterns. By treating these singularities as if they are real particles, we make use of methods from statistical physics to investigate their behaviour, both their spatial correlations, as well as their temporal dynamics through time-resolved measurements. Finally we show that it is possible to measure both infrared and visible light simultaneously, while still retaining sensitivity to amplitude, phase and polarization for both colours.

Models governed by systems of ordinary differential equations (ODEs) often produce complex and unpredictable behaviors. To address this, we can use dimension reduction techniques, which simplify these models, allowing for the retention of specific behaviors while greatly decreasing the cost of numerical solutions and, in some cases, enabling analytical derivations of sufficient conditions for the existence of nonzero fixed points of the model’s ODEs. This thesis reviews the state-of-the-art reduction theories and extends the established proofs by Wu et al., by providing necessary assumptions, lemmas, and a novel proof of an important proposition used in their work. We additionally verify and confirm Wu et al.’s findings and predictions for a cooperative version of the Cowan-Wilson model, which describes a population of neurons’ firing activity. We derived a one-dimensional reduction, inspired by Laurence et al., for a generalized Cowan-Wilson model, which we introduced in this thesis. Unlike the original, this generalized model can produce oscillatory behavior without external stimulus. A valuable finding is that the method of reduction does not depend on the specific form of the Cowan-Wilson function, allowing it to be applied to a broader class of nonlinear dynamics. Our reduction is able to predict system behavior effectively, given the network yields a unique reduction. However, the reduction parameter was not unique in approximately half of the networks studied, which saw a 66% increase in average error, suggesting these networks are inherently multi-dimensional. This opens the door for future research into the existence of a multi-dimensional reduction framework that could mitigate this discrepancy.

This thesis investigates the peeling-off property of zero rest-mass fields in asymptotically flat space-times, as described by Penrose. Unlike other literature, this work is designed to be accessible to undergraduate physics or mathematics students. It provides a more detailed derivation including numerous explicit calculations, which is appropriate for the assumed background knowledge. The thesis is structured to build from fundamental mathematical concepts to advanced applications in general relativity. The mathematical concepts introduced are topology, compactifications and differential geometry. The main body of the work applies these mathematical tools to the Minkowski space-time and extends the analysis to more general asymptotically flat space-times.

A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r^-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds.

Cables are fundamental components in numerous technical implementations, such as cable-stayed bridges. As cables are prone to vibration due to e.g. wind, it is necessary to find ways to reduce these oscillations.
This thesis aims to build upon the work conducted by Su et al. [1]; their paper studies the vibration of an inclined cable with an attached Tuned Mass Damper (TMD). In particular, as Su et al. assume that the cable takes the shape of a parabola in equilibrium, the goal is to find a better estimate of the equilibrium configuration of the cable. To this end, this thesis will utilise a modified version of the method used by Caswita [2]; Caswita derives the equations of motion of a cable without any attached mass by applying Lagrangian mechanics.
The results show that the equilibrium position differs meaningfully from a parabola. The ordinary differential equations that Su et al. obtain by using Galerkin’s method are also considerably different when using the alternative equilibrium position. These differences are mainly caused by the fact that the cable hangs on an incline, rather than by the addition of the TMD.

Quantum algorithms have shown much potential in solving complex problems more efficiently than our current classical algorithms, particularly in search problems with a large and complex search space. When leveraging the principles of superposition and interference, quantum walk based algorithms can lead to a faster convergence on a target state. The thesis begins with a theoretical framework for the study of classical and quantum walks on graphs. The usage in search algorithms is discussed together with the effect of a modified Hamiltonian that introduces a bias toward the target state. This is analyzed using perturbation theory showing a decrease of the ground state energy. Ultimately, this thesis focuses on the optimization of quantum random walk search algorithms using resetting techniques. Simulations of quantum random walks on simple graph structures, such as path and cycle graphs are used to provide concrete examples. Results show that resetting not only improves the probability of finding the target state, but also has a faster convergence. Furthermore, decoherence effects are shown to be reduced when using resetting techniques.

There has been interesting research on the superconducting diode recently. A possible structure for this is the Superconductor/Ferroelectric/Superconductor (S/FE/S) Josephson junction. Our goal is to simulate a S/FE/S junction with a magnetic field and see if this leads to a superconducting diode effect. First this junction and its properties are studied through a numerical simulation. Then a magnetic field is applied and we look at how this affects the junction. For our simulation we use the Velocity Verlet method. We start by simulating a normal RCSJ. The results agree with what we expect from the theory, however there is small but noticeable error which we attribute to the numerical method. Next a S/FE/S junction is simulated. The polarisation of the ferroelectric only affects the RCSJ when it oscillates at a frequency which is close to the plasma frequency. If we apply a magnetic field we see small gaps for the retrapping current however these are close to the numerical error thus they are to small too be detected by our simulation.

In the last decades there has been an increasing interest in computing the local strain at the atomic scale of materials. By knowing aspects of the local strain in a lattice, one has information about measurements of distortions of lattice parameters concerning shifts, deformations and defects computed with respect to a smooth, defect-free reference region. Multiple methods have been implemented so far in order to map the strain of two-dimensional lattice patterns, which are obtained through means of a High Resolution Electron Microscope (HRTEM). The functioning of a HRTEM is based on the same principles as an optical microscope, but it uses a beam of electrons instead of visible light. One of the computational methods which then processes the obtained two-dimensional images is called the Geometrical Phase Analysis (GPA) and makes use of a very important mathematical tool, the Fourier transform. The GPA method lies at the center of this project and consists of several steps. First, the Fourier transform of the lattice image is plotted and two Bragg peaks corresponding to two linearly independent frequency vectors in the power spectrum are chosen. Next, a mask is applied around these peaks, separately. In my project I have chosen to apply the Hann smoothing filter. Then, the inverse Fourier transform is applied to the masked image and the phase (also called the raw phase) is plotted. The next step is to compute the reduced phase, which is defined at a local pixel as being the raw phase from which the following product is subtracted: 2𝜋 ⃗𝑔 ⋅ ⃗𝑟, where ⃗𝑟 is the vector corresponding to a pixel in the real space and ⃗𝑔 the frequency vector corresponding to the Bragg peak around which one has applied the mask. At this point the reference region is computed by choosing a smooth, homogeneous area in the reduced phase image. In order to obtain the strain, one needs an optimal frequency vector ⃗𝑔 defined at every pixel of lattice. In order to do so, a minimization process defined in the context of a computer algorithm in the programming language Python has been implemented. These computations should lead to obtaining the lattice strain, which is calculated by taking the symmetric part of the derivation of the displacement obtained from the two linearly independent Fourier components, which is in turn called the distortion. The antisymmetric part of the distortion is the rotation component and serves as a check for the correctness of the computational method. The goal of my project is not only to provide a solid theoretical background for the GPA method and to discuss the strain at atomic level in several lattice patterns, but to also provide a rigorous computer algorithm that makes these computations reality. This algorithm, opposed to pre-existing software, facilitates the reader’s process immensely by the large amount of detail which is given at every step, detail which easily motivates and supports the reader in potential side-steps they would want to take in order to make the method their own.

Chiral induced spin selectivity (CISS) is an unexplained phenomenon in physics, in which the interaction within chiral molecules brings about a selectivity in the spin of the electrons.
The Breit interaction has been suggested as a possible explanation, but has not yet been researched in the context of CISS.
In this thesis, first a derivation of the Breit equation is given, which in the second part is being used to construct a simulation of electron transport through a simple chiral molecule.
The spin-polarization for the transmission including the Breit interaction was compared to the case excluding the Breit interaction, and the two were found to be substantially different. From that it followed that the influence of the Breit interaction on the transport of an electron through a chiral molecule is significant and it should therefore not be disregarded in calculations on CISS.

Synchronisation refers to the tendency of coupled oscillators to move in phase with one another over time, departing from their own natural frequency. This phenomenon occurs in almost all branches of science, engineering, and social life. Synchronisation is a well understood concept in classical mechanics. Quantum synchronisation involves attempting to extend this concept to quantum mechanics, where there are number of phenomena such as damping that are more difficult to realise. This thesis focuses on two things. Firstly, it explores the bridge between classical and quantum synchronisation. Both the equivalencies as well as the difficulties that must be overcome. This is initially done by looking at a simple classical model for two coupled oscillators and extending this to a quantum model, and subsequently by exploring the derivation of a master equation for a damped quantum harmonic oscillator with non-linear damping. Secondly, it explore the synchronisation of four coupled quantum van der Pol oscillators. To this purpose, the six different systems (tree, chain, loop, tower, spade, all-to-all) with four coupled classical van der Pol oscillators first analysed and the Arnold tongues are determined, showing for what parameters synchronisation is expected to occur. The equivalent systems with four quantum van der Pol oscillators are explored to determine whether the classical Arnold tongues give a good indication of when quantum synchronisation is expected to occur. An alternative measure for synchronisation is also investigated for these systems based on the relative entropy of coherence. The Arnold tongues for each of the four classical oscillator systems were determined. For some systems (chain and loop) the Arnold tongues were much smaller than for the three oscillator system while for others (tree, all-to-all) they were only slightly smaller. The quantum Arnold tongues showed very similar behaviour to the classical Arnold tongues. Furthermore, when investigating the correlations between different pairs of oscillators, it was found that the correlation decreased as the number of connections separating two oscillators increased. In the chain system, an expected symmetry was found. The relative entropy of coherence is found to be a good measure of synchronisation, better than the more frequently used complex correlator. This is because the relative entropy can take into account the entire system, while the correlator describes the synchronisation between two individual oscillators in the system.

Quantum technology is evolving faster than ever. Currently, all eyes are on the quantum computer, the promising computer that can solve problems which are unsolvable for regular computers. In order to understand this new technology, it is necessary to understand the qubit, the basic unit of quantum information. This can be done by means of the density matrix: a mathematical representation of the state of a system. The aim of this thesis is to find the density matrix of a single qubit in closed (isolated) and open quantum systems. In the case of a closed system, an alternating sequence of two processes with different Hamiltonians is considered, which both last a fixed amount of time after one another. These systems have been solved using a direct formula for a 2 × 2 matrix with distinct eigenvalues raised to the power n for any n ∈ N. In the case of an open system, dissipation is taken into account compared to a single process in a closed system. These open systems have been solved numerically using the Lindblad master equation and the spin-boson model to model the environment as a bath of bosons. However, the use of multiple approximations and assumptions questions the validity of the results for ’strong’ interactions. Suggestions for further research include investigating quantitatively when the Lindblad equation is valid to use and solve open quantum systems using different models for the environment.

Quantum communication has been shown to be vastly superior to classical communication in many problems. However no general statements exist which tells us how much better quantum communication is to its classical counterpart. In this thesis it was studied the minimum amount of classical bits required to exactly simulate a quantum communication process. The quantum communication process specifically studied was a quantum prepare and measurement communication problem. It has been shown that the calculation of the amount of classical bits of communication required for simulation reduces to a minimization-maximization optimization problem. Several results have been presented for for solving this optimization problem and in addition a link was made between classical simulations of quantum communication and a recent debate on the reality of the quantum state.

Open questions are whether life can be enabled in uninhabitable environments, and whether there is a limit to howmuch one can tune the speed of proliferation. Answering such questions has broad implications. It may reveal whether we can live in unforeseen habitats, whether we can slow down aging, and whether there are limits to lifespan. In this dissertation, we explore such questions for the budding yeast by changing the temperature. We will use temperature, a physical parameter, as a knob to tune the speed of cellular life. Temperature affects all organisms and habitats, and is of contemporary interest in light of climate change. Indeed, cells of microbes, plants and cold-blooded animals often endure temperatures that can be considered extreme. For reference, budding yeast lives comfortably at 30 ◦C and has a doubling time of roughly 1.5 hours – the time a cell needs to grow and divide into two cells. During our studies, we will elucidate the common principles that govern the life of yeast at extreme temperatures – how a cell survives, grows, replicates, ages, and dies. We combine models, experiments and measurements of single cells and at amolecular level, and integrate these into a systems-level view of the life of yeast at extreme temperatures..

Quantum Markov Semigroups (QMS) describe the evolution of a quantum system by evolving a projection or density operator in time. QMS are generated by a generator obeying the well-known Lindblad equation. However, this is a difficult equation. Therefore, the result that the Lindblad form greatly simplifies in the case of the generator commuting with the modular automorphisms group, is useful. Unfortunately, the proof only works for finite dimensional Hilbert spaces, which is why the aim of this thesis is to generalise this result to countably infinite dimensional Hilbert spaces. To this end, the Lindblad equation is derived from both a mathematical and physical perspective. Where the former relies on rigorous proof and the latter relies on approximations.   In the rigorous case the theory of unital completely positive maps is used. Furthermore, multiple topologies are considered which put less stringent conditions on the operators of interest than the norm topology. Additionally, the Haar measure is used on the unitaries of the bounded linear operators to construct the explicit Lindblad form. To derive the result by employing physical assumptions the interaction picture is used. The physical derivation starts from the Von Neumann equation and uses multiple assumptions to obtain the final Lindblad form. The most important physical assumptions are: the Born approximation, the Markov approximation and the rotating wave approximation.   Furthermore, the main result is the generalisation of the simplified Lindblad form. This simplified form holds for generators commuting with the modular automorphisms group in case the Hilbert spaces are countably infinite dimensional. However, this requires the domain of the generator to be restricted to trace class operators with the identity operator artificially added. Additionally, the generator needs to map strongly convergent sequences to weakly convergent sequences. It also needs to be self-adjoint with respect to the Hilbert-Schmidt inner product. Lastly, the generator is assumed to be self-adjoint with respect to the Gelfand-Naimark-Segal (GNS) inner product <X, Y>=Tr(σ X*Y) for σ a density operator. This last assumption implies that the generator commutes with the modular automorphisms group, which is the symmetry we are considering. Hence, the two previous assumptions are the additional requirements needed to generalise the result, besides the restriction of the domain. Therefore, it is recommended for further research to generalise the result for the domain extended to the bounded operators B(H). It should be noted that the proof heavily relies on the Hilbert space structure induced by the Hilbert-Schmidt inner product. Consequently, the generalisation for the bounded operators would probably require a different approach. Another recommendation is to try and lift the sequence and self-adjoint requirements on the generator. In addition, it is interesting to investigate which physical systems actually have the symmetry of generators commuting with the modular automorphisms group.

Magnons are quanta of spin waves, i.e. modes of collectively precessing spins. Thermally excited magnons in thin magnetic films generate stray fields at the film surface which can be detected using nitrogen-vacancy (NV) centers. NVs are lattice defects in diamond and are able to couple with magnon stray fields. Assuming a thermal occupancy of magnon modes, we study the magnetization dynamics of magnons propagating through thin magnetic insulators using the Landau-Lifshitz-Gilbert equation. We implement a numerical model to predict and understand the response of the NV center to proximal magnons in thin films. We investigate how the NV relaxation rate changes for different NV orientations by extending and generalizing the existing theory on chiral magnetic noise, and simulate an experimental setup for an NV placed just above the surface of a thin magnetic insulator. The simulation includes a static bias field in an arbitrary orientation with respect to the quantization axis of the NV center using the diamond's tetrahedral symmetry. This extended model is in demand due to limitations in present-day measurement techniques to align the bias field with an NV center. We use it to detect magnons that contribute to the relaxation rate of the NV, and determine an NV-to-film distance of 0.28(3) μm, from measured relaxation rates of an NV center placed above an yttrium-iron-garnet film with a thickness of 235(10) nm. Our model is available as an open-source Python module.

Synchronisation is the remarkable phenomenon of in-phase movement of coupled oscillators during a prolonged period of time, which even occurs when these have different natural frequencies. This property is used in many fields like physics, biology and chemistry, to model various system behaviours, for instance: circadian rhythms in the chemistry of the eyes \cite{Rompala2007}.
\newline\noindent This thesis focuses on synchronisation and entanglement in systems consisting of quantum Van der Pol oscillators. After looking at the exemplary properties of the classical VdP oscillator, it follows the methods of \cite{EshaqiSani2020} to explore the behaviour of two coupled QVdPOs, quantum Van der Pol oscillators, using Monte Carlo simulations for trajectories of this system. The system is then expanded to three-oscillator systems with all-to-all and chain coupling. The validity of the extension of the properties found in two QVdPOs in \cite{EshaqiSani2020} with regard to synchronisation and entanglement is tested. Does an Arnold tongue, the region of parameters for which a system shows synchronisation, still exist and if so has its shape changed? Do synchronisation and strong entanglement still show a positive correlation? \newline\noindent Simulations of the 2-oscillator system gave results which were just slightly different from those obtained in\cite{EshaqiSani2020}, validating the occurrence of synchronisation within the Arnold tongue and showing a positive correlation between synchronisation and entanglement of the system. Both the 3-oscillator systems showed synchronisation as well in their respective Arnold tongues, which were increasingly smaller for the all-to-all coupled and the chain coupled system when compared to the 2-oscillator system. Overall, the amount of synchronisation, shown in three oscillators was very close to the amount shown in two oscillators when strongly coupled and just a little less for weak coupling. The correlation between synchronisation and strong entanglement was also found for three oscillators, though for weaker coupling the chain coupled system showed a little more entanglement than before.

Tales of a Fountain of Youth and the invention of medicine illustrate our age-long obsession with two themes: life and death. What it takes to stay alive, and not to be dead, is a basic question in science that is easy to state, and yet difficult to address at a profound level. One striking feature of many living organisms is the ability of individuals to behave in unison by communicating with each other. At life’s microscopic level, living cells can also send and receive chemical signals to communicate with each other in their habitat but for a population of many thousands of cells it remains enigmatic who is communicating with whom, what are the signals, and how the signals work over space and time. We used quantitative experiments and mathematical modelling to systematically explore how mouse Embryonic Stem (ES) cells might cooperate by communicating when differentiating into the first two lineages. We discovered that differentiating mouse ES cells scattered across many centimeters on a dish form one macroscopic entity that either survives or dies in unison if and only if its population-density is above a threshold value. This switch-like behavior is determined by cells that secrete and sense FGF4 that diffuses over many millimeters to activate YAP1-induced survival mechanisms. Our work shows that living cells (in vitro) can rely on macroscopic cooperation to stay alive.