Examensarbeten för kandidatexamen

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https://odr.chalmers.se/handle/20.500.12380/45

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Kan en hopprocess förklara hur man tittar på tavlor?Aila Särkkä
(2022) Ahl, Linnéa; Ahlman, Filip; Alsaberi, Rani; Kindberg, Carl; Chalmers tekniska högskola / Institutionen för matematiska vetenskaper; Dinger, Ulla; Särkkä, Aila
Ögonrörelser växlar mellan fixeringar och hopp och ögonrörelser kan användas inom många olika tillämpningsområden. Arbetets syfte är att utveckla en modell, baserad på en hopprocess, som förklarar hur man tittar på en tavla genom att använda ögonrörelser från den som tittar på tavlan. Dessutom ingår en jämförelse mellan ögonrörelser hos konstexperter, individer som är vana vid att titta på tavlor, och konstnoviser, individer som inte har en sådan vana. Tre olika modeller har jämförts, en Poissonprocess och två olika varianter av en Markovprocess. Datan består av ögonrörelser som registrerades från 20 deltagare, tio konstexperter och tio konstnoviser, när de studerade tavlan. Vidare studerade deltagarna tavlan under tre minuter men endast den första minuten har undersökts. Enligt de erhållna resultaten kan en fungerande modell konstrueras med hjälp av en Markovkedja för att simulera ögonens hopp och en gammafördelning för att simulera fixeringslängder. Vissa skillnader mellan konstexperter och konstnoviser kan noteras. En skillnad är att konstexperternas blick stannar på ett område på tavlan under kortare tid, i jämförelse med konstnovisernas. Vidare tenderar konstexperter att göra fler hopp än konstnoviser.
Losing the sensation of touch: Mathematical modeling of diabetic neuropathy using spatial point processes
(2022) Kaewchino, Sirada; Nylander, Maximilian; Ujam, Jadd; Chalmers tekniska högskola / Institutionen för matematiska vetenskaper; Dinger, Ulla; Konstantinou, Konstantinos
Diabetes has led to an epidemic of complications associated with this disease. Diabetic neuropathy, which causes pain and loss of sensation due to degeneration of nerve fibers, is one of the most common complications of diabetes. Confocal microscopy made it possible to observe that the nerve endings in the outer skin of sick patients tend to be more clustered than in healthy subjects. Therefore, it is imperative to understand the process of degeneration and the spatial thinning of nerve fibers to detect diabetic neuropathy at an early stage. The two main hypotheses were tested are whether the nerve thinning occurs randomly and independently of other points and whether the nerve thinning is conditional on the other points. Three mathematical models were developed based on spatial thinning. The first is an independent random thinning model, the second is a deterministic thinning model and the third is a Gaussian thinning model. The second and third models are conditional on the location of the base point and the distance from it. When evaluating the spatial statistics, we used the centered L-function as a summary function when conducting a global envelope test with N = 500 simulations, where we tested the hypothesis under the empirical mild data. We also evaluated the different models based on non-spatial statistics which were compared to the mild data. The spatial results from the first model showed that nerve thinning does not occur randomly and independently (p = 0.01), thus rejected the null-hypothesis for significance level α = 0.05. The second model could not be rejected under the null-hypothesis (p = 0.624) as well as the third model (p = 0.056) for significance level α = 0.05. The non-spatial results showed that the first model sufficed if the desired outcome is to observe just non-spatial characteristics of the data whilst the second and third model lacked in this area. A likely explanation as to why the second and third models performed worse in the non-spatial regard, may be that spatial thinning isn’t a sufficient explanation behind the underlying mechanisms.
Optimaltransport för styrning av en svärm av agenter
(2022) Holmberg, Linnéa; Lemann, Emelie; Sörstrand, Elias; Wärnsäter, Alfred; Chalmers tekniska högskola / Institutionen för matematiska vetenskaper; Dinger, Ulla; Ringh, Axel
The purpose of this report is to derive and implement a solver for a multimarginal optimal transport problem. This type of multimarginal optimal transport problem can be used to model and calculate how a swarm of agents should be controlled in an optimal way. Interpolation, entropic regularization and Sinkhorn iterations are used in order to do this. We applied the solver to two different cases. In the first case, the agents started according to a certain distribution inside a 100 × 100 grid and their goal was to evenly spread out. Furthermore, an obstacle was placed in the model that moved through the grid for each time step. In the last case, the algorithm was required to find an optimal way out through a maze.
Dynamisk modellering av signalvägar som reglerar kol- och kvävemetabolismen i bagerijäst
(2022) Andrekson, Leo; Hellberg, Rakel; Johansson, Emma; Olsson, Jesper; Chalmers tekniska högskola / Institutionen för matematiska vetenskaper; Dinger, Ulla; Persson, Sebastian
Understanding how signaling functions within and between cells is relevant to get a better grasp of several diseases. A central signaling process in organisms ranging from everyday baker’s yeast to humans is the nutrient signaling network, which is responsible for sensing availability of nutrients, such as carbon and nitrogen. The TORC1, SNF1 and cAMP-PKA signaling pathways are central components of the nutrient signaling network in baker’s yeast, which regulates the metabolism based on nutrient levels. In this project, we investigated these pathways by implementing and analyzing an ODE model of said pathways from Jalihal et al. [1]. As a first step, the model was implemented and some results from Jalihal et al. were recreated in order to confirm the implementation. Furthermore, the model consists of several unknown parameters that have to be estimated. This was done using a quasi-Newton algorithm. Two parameter vectors were obtained which give a better description of data than the parameter vector reported by Jalihal et al. The conclusion drawn from the identifiability and sensitivity analysis is that the majority of the parameters have low sensitivity. Qualitative data was used to examine the model’s validity for scenarios where no quantitative data is available. This was done for the parameter vectors obtained from the parameter estimation and Jalihal et al. [1]. Most of the scenarios analyzed were consistent with qualitative data, except for a few cases. For these cases, experiments were proposed that could improve and expand the model. Despite the shortcomings of the model, it is still useful as a basis for further development. Measures such as extending the model, combined with new parameter estimations, may in the long run lead to a well-describing model of the nutrient signaling pathways in baker’s yeast.
Galton-Watson-processen och åldrande celler
(2022) Eriksson, Lotta; Hildingsson, Joel; Ibahim, Taha; Varghaei, Laleh; Chalmers tekniska högskola / Institutionen för matematiska vetenskaper; Dinger, Ulla; Sagitov, Serik
The theory of the multitype Galton-Watson-process has been applied to build a mathematical model for describing biological aging in cells. We also study the phenomenon of rejuvenation. The theoretical results of the model have been applied and populations have been simulated in order to study how biological aging affects cell populations and individual cells. This project investigates how different parameters of the model affect the properties of cell populations. We study what the distribution of different biological ages looks like in the population, guaranteed extinction of populations and the distribution of life lengths in cells. We also introduce a rejuvenation index to provide a formal measure of rejuvenation in the populations. The rate of accumulation of damage and how frequently the cells divide is essential in whether or not the population becomes extinct. In the surviving populations, asymmetric distribution of damage between the mother and daughter cells results in smaller populations but with a higher proportion of biologically young cells. The individual cells live for a shorter period of time but rejuvenation occurs to a higher extent. In populations where the damage is distributed symmetrically, populations become larger and the individual cells live longer but rejuvenation does not occur. Aging in cells has been modeled in a simple but insightful way but there is still some sensitivity in the choice of parameters. The mathematical model that has been developed has also compared the aging and rejuvenation with the yeast cell. We suggest that future research aims at adjusting the parameters of the model in order to be able to describe the mechanisms behind the aging of the yeast and gain an insight into human aging.

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