This report aims to study the phenomenon of neutrino oscillations through derivation of the formulas for their transition probabilities in two flavors and expansions thereof in three flavors. By studying the series expansions of the transition probabilities in three flavors, one could get a clearer understanding of the effect of matter and the CP-violating phase, $\delta_{CP}$ on neutrino oscillations. The purpose of investigating this is to be able to investigate the impact of matter and $\delta_{CP}$ at the potential experiment ESSnuSB. The ESSnuSB experiment would be an extension of the linear accelerator project European Spallation Source, ESS, that is currently under construction in Lund, Sweden. The ESSnuSB aims to study neutrino oscillations at the second oscillation maximum which, through the derivations, was found at 540 km from the ESS which is consistent with other studies. The mine in Garpenberg is located 540~km from the ESS, which makes it a prime candidate for a detector to be built within.

A promising new technology in medical imaging is photon-counting detectors (PCD). Itcould allow for images with higher resolution, less noise, improved material decomposi-tion while possibly reducing radiation exposure for patients. Recently, the possibility touse deep-learning denoising in tandem with PCD to increase image quality is starting tobe investigated. In this report we use a variety of standard image quality metrics suchas MSE, SSIM and MTF, on different image phantoms, to evaluate two ways of imple-menting neural networks in the reconstruction process: in the image domain and in thesinogram domain. We show that implementing the network in the image domain seemsto be the most promising choice to increase image quality, observing higher contrast,reduced noise and smaller errors than for the sinogram domain network. We also discusswhy this might be the case. Additionally, we study the effects of optimizing the networksand how well the neural networks generalize to types of phantoms other than the onesthey were trained on.

The primary subject matter of the report is the Hirota Direct Method, and the primary goal of the report is to describe and derive the method in detail, and then use it to produce analytic soliton solutions to the Boussinesq equation and the Korteweg-de Vries (KdV) equation. Our hope is that the report may also serve as an introduction to soliton theory at an undergraduate level. The report follows the structure of first introducing Hirota's bi-linear operator and giving an account of its relevant properties. The properties of the operator are then used to find soliton solutions for differential equations that can be expressed in a "bilinear" form. Thereafter, a set of methods for finding the bilinear form of a more general non-linear differential equation are presented. Finally, we apply the tools to the Boussinesq and KdV equations respectively to derive their soliton solutions.

The purpose of this paper is to implement and simulate Grover's algorithm on one and several qubits on a classical computer. The theory behind the algorithm and its components are described in detail. This paper provides a proof of concept for one of the most remarkable results in the theory of quantum computation. I have constructed a library in Python to simulate the gates used in the algorithm that can be used up to an arbitrary number of qubits. The results of the simulations are supposed to demonstrate the characteristics of the algorithm and advantages compared to classical search.

A few quantum systems on the line with weighted classical orthogonal polynomials as eigenstates are studied. Explicit expressions of the propagators,i.e. the integral kernels of the time evolution operators, are derived. In the case of Hermite polynomials, the system is the harmonic oscillator, while forgeneralized Laguerre and Gegenbauer polynomials, the corresponding quanutum system are equivalent to two-particle Calogero-Sutherland systems.

X-ray computed tomography (CT) has since its introduction in the early 1970s become one of the most important tools used for medical imaging. In CT, a large number of x-ray attenuation measurements are combined and reconstructed to form a three-dimensional image of the targeted area. In the recent years, a new type of detector called photon counting detector (PCD) has attracted considerable interest. This new type of detector acquires spectral information is associated with several benefits and has shown to be very valuable.  Furthermore, the use of deep learning to reconstruct images produced by CT has attracted significant attention in the last couple of years. However, the best way of incorporating deep learning into the reconstruction chain into the reconstruction chain is still incompletely understood. Additionally, the use of deep learning has mainly been investigated for the case of conventional CT and not for CT performed with PCDs. It these two points that this work aims to address.  Multiple deep learning architectures were implemented and evaluated on material images acquired by simulating a PCD. The deep-learning part of the reconstruction took the form of image-domain denoising after the material images had been obtained from the material sinograms through filtered back projection. Then, a comparison between the different deep learning architectures was made to find out which architecture is the most suited for denoising images produced by PCDs in the image domain.

This thesis investigates the nonlinear partial differential equation known as sine-Gordon and its special soliton solutions.Simpler analytical results are derived and more advanced methods and their results are discussed.Further, a finite difference scheme is derived, implemented and compared against a known energy conserving scheme of sine-Gordon in terms of stability, accuracy, convergence and computation time.The complete solvability of the equation enables comparison between numerical solutions and their analytical counterparts.No unified answer to which numerical scheme is best was determined as they both were shown to have pros and cons.

New EU regulations are currently taking effect, which means there will be stricter requirements on lighting units and more necessity for labs to perform photometric quality control. In the project a flickermeter setup was constructed at the Swedish Energy Agency's lab for performing quality assurance of luminaries by measuring the relevant metrics PST, SVM, and ITHD. By accomplishing this in accordance with the relevant IEC Standards, it creates an example setup which can be reproduced in other labs without the need for expensive nontransparent commercial devices. The project also examines the viability of creating such a setup from scratch without prior expertise, using the IEC Standards as a guide. To ensure the reliability of our setup multiple verifications were performed in accordance with the IEC Standards. Most gave positive results, but some issues were also discovered, mainly in regards to the optical filter needed for the setup. A suitable replacement for the missing optical filter was introduced, with which good results were acquired for both SVM and PST when compared to commercial flickermeters. Good results were also acquired in regards to the effects of dimming and the ITHD measured. Aside from the results, some issues regarding the procedures used for quantifying flicker were raised and discussed, along with discussing inconsistencies and discrepancies in the standards used.

In this project a laser setup was constructed, quantum entangled photons were generated and two different methods were used to show said entanglement. The entangled photons were generated through a process called spontaneous parametric down conversion (SPDC). The process includes pumping a non-linear crystal with photons, such that an incoming photon is converted into a pair of photons. A quantum mechanical model was developed which predicted polarization entanglement of two photons through correlation or anti-correlation depending on the choice of basis. The first of the two methods involved directly measuring the state of both photons and consequently confirming entanglement. The second method involved directly measuring one of the photons while the second entered a quantum eraser which consequently led to an interference pattern verifying entanglement. Several improvements were made during the course of the project, all aimed to increase the contrast ratio, a rate of quantum entanglement. The contrast was improved by increasing indistinguishability between the generated photons. At the end of the project a contrast ratio of 78 % was achieved.

Understanding biological systems oftentimes requires mapping the behaviors of biomolecules on small scales such as single proteins or nucleic acids. Fluorescence Correlation Spectroscopy (FCS) is a group of particularly noninvasive and sensitive fluorescence-­based techniques that can be used for this purpose. In this project an FCS-­apparatus has been built and consequently used to test the boundaries of a newly developed variant of FCS called FRET­FCS (Förster Resonance Energy Transfer). Unlike regular FCS, this technique employs fluorescence resonance energy transfer (FRET) in addition to fluorescence. In practice, testing the boundaries means adding increasing amounts of FRET­ DNA to a mix of red­ respectively green marked DNA and determine the lower limit for which FRET­-DNA can be detected. A functioning FCS-­setup was built. When FCS­ and FCCS (Fluorescence Cross­ Correlation Spectroscopy)­ measurements were conducted, the desired curves for the intensity functions were generated. The result of the sensitivity analysis was that the lower limit of detectable FRET­-DNA was approximately 1.3% of a sample containing FRET­-DNA and single­ marked DNA.