Faculty of Science
Recent Submissions
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(2024)In this study, we will construct approximation scheme for the orbits within Chua's system. Chua's system is a chaotic dynamical system modeling currents and voltages of a specific circuit, called Chua's circuit. This circuit displays remarkable chaotic nature while being very simple and having only one non-linear component. The corresponding system of differential equations is also relatively simple, and its solutions are always explicitly given within certain regions of $\mathbb R^3$. These solutions give rise to complicated phenomena, when they cross from one region to another. We define novel concepts to understand the structure of these orbits and show how their changes can be controlled. We find that in a geometrical sense, the orbits' changes over time are relatively predictable and this allows us to approximate future positions of orbits to some degree. In the end we also give a general outlook of the system, its stages, nature and potential for applications.
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(2024)Electrical impedance tomography (EIT) is an imaging method where an image of the interior of an object is obtained from electrical measurements at the surface. Electrodes are placed on the surface of the imaged object and low-frequency current is applied on the electrodes. The current induces a voltage distribution that can be measured through the same electrodes, resulting in a current-to-voltage map. EIT is an inverse problem, meaning that we use the result, the current-to-voltage map, and some prior information about the object to find the cause: the conductivity distribution of the imaged object. Recent findings show that the Beltrami equation based method for reconstructing the conductivity distribution from the measured data can be deconstructed into steps that reveal a link to parallel-beam X-ray tomography. The complex geometrical optics solutions to the Beltrami equation are treated as a scattering series, and the scattering terms are studied numerically in a simplified setting in this thesis. It is known from previous publications that the conductivity distribution can be reconstructed from the leading order term, but the leading order term cannot be extracted from the measured current-to-voltage map. Numerical results illustrate the effect of the conductivity, radius, and location of the inclusion on the multiple scattering terms. The effects are noticeable but small, and difficult to generalize due to the highly ill-posed nature of EIT. However, the results clearly show that while the scattering terms decrease, convergence is not fast enough to disregard the effect of the higher order terms on the full scattering series that corresponds to measured data. To make use of the link between EIT and X-ray tomography, some method of extracting the leading order term from the current-to-voltage map is necessary.
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(2024)In model theory, a theory $\Sigma$ is called categorical in a cardinal $\kappa$, if all the models of $\Sigma$ of cardinality $\kappa$ are isomorphic. A breakthrough result in the study of categoricity was introduced in $1965$, when Michael Morley proved that a theory in a countable language is categorical in every uncountable cardinal, if it is categorical in one of them. This result was named after him as Morley's Categoricity Theorem. The aim of this thesis is to prove Morley's Theorem by using the contraposition of the claim in the theorem using the original methods of Morley rank, indiscernible sequences and prime models. But taking a modern approach on the way the proofs presented in the thesis are constructed. The required concepts for the proof are split into five steps, forming the composition of the thesis. The thesis starts in the second chapter by going through the required prerequisites of set theory, first order logic and model theory. Following with a deeper look into model theory by covering types, saturation and stability; and formalizing the idea of a monster model, the workspace of the thesis. The middle three chapters of the thesis consist of developing the concepts required to prove Morley's Theorem. In the third chapter Morley rank, Morley degree and totally transcendental theory are defined. Focus is given on proving basic properties of Morley rank and establishing the link between total transcendence and stability: a countable theory is $\omega$-stable if and only if it is totally transcendental. In the fourth chapter Skolemization, indiscernible sequences and Ehrenfeucht-Mostowksi models are introduced, and used to prove that a theory which is categorical in some uncountable cardinal is $\omega$-stable. Afterwards in the fifth chapter totally transcendental theories are considered more. The chapter contains the main lemmas to prove Morley's Theorem. Concretely that totally transcendental theories have prime models over any set and how to build indiscernible sequences over sets in totally transcendental theories by selecting them with Morley rank and Morley degree. In the final chapter of the thesis the results are put together and the main theorem is proven. The method of the proof is by contradiction. The assumption is that a theory is categorical in some uncountable cardinal, and not categorical in some other. Then by $\omega$-stability two models of the same cardinality are constructed, such that one is $\omega_1$-saturated and the other is not. This leads to a contradiction since these two models are not isomorphic, proving the theorem.
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(2024)Solar electric energy production has significantly increased over the past decades to reduce reliance of non-renewable energy sources. Among photovoltaic technologies, silicon-heterojunction solar cells stand out for their efficiency and performance. Transparent conducting oxides, most prominently indium-tin-oxide, serve as front and rear electrodes facilitating light access to the silicon (Si) active layer and charge-carrier transport. This thesis explores ultra-thin silver (Ag) films as an alternative, highlighting Ag’s high conductivity and absence of inter-band transitions in the visible range as a potential platform for new transparent-conducting metals (TCMs). Thin films are layers ranging from a few nanometers to several micrometers thick, with properties distinct from bulk materials, closely tied to structure and morphology. Understanding the factors influencing film morphology is vital for improving their properties. Within this framework, thin Ag films are deposited by magnetron sputtering at three different working pressures (P = 5, 10, 15 mTorr), at three vapor arrival rates F ≈ 0.5, 1.0, 3.0 Å/s) and on four types of Si substrates, native Si(100); plasma etched Si(100); Si(100) covered with a 5 nm thick amorphous-Si (a-Si) layer; microcrystalline Si (µc-Si) with a textured pyramid-like surface topography. Growth on the first three substrates are studied with in situ and real-time spectroscopic ellipsometry to monitor the evolution of the opto-electronic properties, and ex-situ X-Ray Reflectivity and Four-Point-Probe. Growth on textured µc-Si substrates is analyzed ex-situ with Scanning Electron Microscopy. It is found that the type of substrate surface significantly affects film growth, resulting in notable variations in continuous formation thicknesses (hcont). For nearly every combination of pressure and substrate, hcont decreases as the growth rate increases. The analysis of growth on Si (100) substrates covered with a-Si is the most comprehensive and it demonstrates that hcont increases with pressure. Moreover, it is shown that when those substrates are contaminated post-growth by exposure to the atmosphere hcont increases considerably. Analysis of film morphologies grown on textured µc-Si substrates shows that for thicknesses of hf ≈ 150 Å films are not fully continuous. This indicates that shadowing effects and difference in surface energies between a-Si and µc-Si need to be considered when designing synthesis strategies for TCMs for photovoltaic devices.
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(2024)This thesis explores the uncertainty and variability in microbial dose-response relationships, focusing on the case of Salmonella infection. We employed three single-hit Dose-Response models: Exponential, Beta-Binomial(hierarchical and non-hierarchical), and Beta-Poisson. We used Bayesian inference to account for variability between salmonella strains and uncertainty in model parameters. The Beta-Binomial and Beta-Poisson models, which allow for individual variability, outperformed the simpler Exponential model in capturing the complexity of the Salmonella data. Bayesian methods provided posterior distributions for critical parameters, offering a deeper understanding of the uncertainties involved in microbial risk assessments. The results suggest that models accounting for strain-specific variability and parameter uncertainty provide a more realistic framework for assessing foodborne infection risks. While data availability and model assumptions limited the scope of this study, the thesis highlights the utility of flexible Bayesian models in microbial risk modeling. Future work should focus on expanding these models to other microbes than Salmonella. We should also work on improving the data, considering a more comprehensive range of dose levels and hierarchical structures with more than two levels to account for variability.