Licentiate Thesis / Licentiatuppsatser Institutionen för matematiska vetenskaper
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Item Sharp bounds on the height of some arithmetic Fano varieties(2023) Andreasson, Rolf; Department of Mathematical Sciences, Division of Algebra and Geometry, Chalmers University of Technology and the University of GothenburgIn the framework of Arakelov geometry one can define the height of a polarized arithmetic variety equipped with an hermitian metric over its complexification. When the arithmetic variety is Fano, the complexification is K-semistable and the metrics are normalized in a natural way, we find in this thesis a universal upper bound on the height in a number of cases. For example for the canonical integral model of toric varieties of low dimension (in paper 1) and for general diagonal hypersurfaces (in paper 2). The bound is sharp with equality for the projective space over the integers equipped with a Fubini-Study metric. These results provide positive cases of a conjectural general bound that we introduce, which can be seen as an arithmetic analog of Fujita’s sharp upper bound on the anti-canonical degree of an n-dimensional K-semistable Fano variety in [11]. An extension of the toric result to arbitrary dimension hinges on a conjectural sharp bound for the second largest anti-canonical degree of a toric K-semistable Fano variety in a given dimension. A version of the conjecture for log-Fano pairs is also introduced (in paper 2), which is settled in low dimensions for toric log-pairs and for simple normal crossings hyperplane divisors in projective space. Along the way we define a canonical height of a K-semistable arithmetic (log) Fano variety, making a connection with positively curved (log) Kähler-Einstein metrics.Item The influence of numbers when students solve equations(2023) Holmlund, AnnaIs it possible that some students’ primary difficulty with equation-solving is neither handling the literal symbols nor the equality, but the numbers used as coefficients? It is well known that many students find algebra a difficult topic, and there is much research on how students experience this strand of mathematics, with indications of how it can be taught. Still, a perspective not often fronted in this research – that has been suggested as an area potentially important – is how numbers, other than natural numbers, in algebra, are perceived by students. Such kinds of numbers (negative numbers and decimal fractions) have been used in this thesis to explore how the numbers influence students’ equation-solving. Two studies with a phenomenographic approach have explored how students (n1=5, n2=23) perceive linear equations of similar structure but with different kinds of numbers as coefficients, e.g., 819 = 39 ∙ 𝑥 and 0.12 = 0.4 ∙ 𝑥. In the second study, a test was also used to investigate the magnitude of the influence of a change of coefficients for 110 students while solving equations with a calculator. The findings show that equations with decimal fractions and negative numbers are less likely to be solved by these students, and decimal fractions as coefficients can even make a student unable to recognize a kind of equation they just solved with natural numbers. The interviews display that, depending on the number in a linear equation, some students focus on different aspects of the equation, and that the numbers influence what meaning the students see in the equation and how they can justify their solution. Following the phenomenographic approach, differences in the way that students experience the equations were specified, and critical aspects were formulated. This implies a wider use of different kinds of numbers in teaching algebra, as different kinds of numbers hold different challenges, thereby also varying learning potential, for students.Item Numerical approximation of mixed dimensional partial differential equations(2023) Mosquera, MalinIn this thesis, we explore numerical approximation of elliptic partial differential equations posed on domains with a high number of interfaces running through. The finite element method is a well-studied numerical method to solve partial differential equations, but requires alterations to handle interfaces. This can result in either unfitted or fitted methods. In this thesis, our focus lies on fitted methods. From finite element methods, one obtains large linear systems that need to be solved, either directly or via an iterative method. We discuss an iterative method, which converges faster when using a preconditioner on the linear system. The preconditioner that we utilise is based on domain decomposition. In Paper I, we consider this kind of partial differential equation posed on a domain with interfaces, and show existence and uniqueness of a solution. We state and prove a regularity result in two dimensions. Further, we propose a fitted finite element approximation and derive error estimates to show convergence. We also present a preconditioner based on domain decomposition that we use together with an iterative method, and analyse the convergence. Finally, we perform numerical experiments that confirm the theoretical findings.Item Matematisk och pedagogisk kunskap – Lärarstudenters uppfattningar av begreppen funktion och variabel(2017) Borke, Mikael; Matematiska Vetenskaper Chalmers Tekniska Högskola och Göteborgs UniversitetItem Resampling in network modeling of high-dimensional genomic data(University of Gothenburg and Chalmers University of Technology, 2017) Kallus, Jonatan; Department of Mathematical SciencesNetwork modeling is an effective approach for the interpretation of high-dimensional data sets for which a sparse dependence structure can be assumed. Genomic data is a challenging and important example. In genomics, network modeling aids the discovery of biological mechanistic relationships and therapeutic targets. The usefulness of methods for network modeling is improved when they produce networks that are accompanied by a reliability estimate. Furthermore, for methods to produce reliable networks they need to have a low sensitivity to occasional outlier observations. In this thesis, the problem of robust network modeling with error control in terms of the false discovery rate (FDR) of edges is studied. As a background, existing types of genomic data are described and the challenges of high-dimensional statistics and multiple hypothesis testing are explained. Methods for estimation of sparse dependency structures in single samples of genomic data are reviewed. Such methods have a regularization parameter that controls sparsity of estimates. Methods that are based on a single sample are highly sensitive to outlier observations and to the value of the regularization parameter. We introduce the method ROPE, resampling of penalized estimates, that makes robust network estimates by using many data subsamples and several levels of regularization. ROPE controls edge FDR at a specified level by modeling edge selection counts as coming from an overdispersed beta-binomial mixture distribution. Previously existing resampling based methods for network modeling are reviewed. ROPE was evaluated on simulated data and gene expression data from cancer patients. The evaluation shows that ROPE outperforms state-of-the-art methods in terms of accuracy of FDR control and robustness. Robust FDR control makes it possible to make a principled decision of how many network links to use in subsequent analysis steps.Item A two-stage numerical procedure for an inverse scattering problem(Chalmers University of Technology and University of Gothenburg, 2015) Bondestam Malmberg, John; Department of Mathematical Sciences, Chalmers University of Technology and University of GothenburgIn this thesis we study a numerical procedure for the solution of the inverse problem of reconstructing location, shape and material properties (in particular refractive indices) of scatterers located in a known background medium. The data consist of time-resolved backscattered radar signals from a single source position. This relatively small amount of data and the ill-posed nature of the inversion are the main challenges of the problem. Mathematically, the problem is formulated as a coefficient inverse problem for a system of partial differential equations derived from Maxwell’s equations. The numerical procedure is divided into two stages. In the first stage, a good initial approximation for the unknown coefficient is computed by an approximately globally convergent algorithm. This initial approximation is refined in the second stage, where an adaptive finite element method is employed to minimize a Tikhonov functional. An important tool for the second stage is a posteriori error estimates – estimates in terms of known (computed) quantities – for the difference between the computed coefficient and the true minimizing coefficient. This thesis includes four papers. In the first two, the a posteriori error analysis required for the adaptive finite element method in the second stage is extended from the previously existing indirect error estimators to direct ones. The last two papers concern verification of the two-stage numerical procedure on experimental data. We find that location and material properties of scatterers are obtained already in the first stage, while shapes are significantly improved in the second stage.