Department of Mathematical Sciences / Institutionen för matematiska vetenskaperhttps://hdl.handle.net/2077/176062024-04-16T20:23:04Z2024-04-16T20:23:04ZSharp bounds on the height of some arithmetic Fano varietiesAndreasson, Rolfhttps://hdl.handle.net/2077/793412023-12-01T21:06:20Z2023-01-01T00:00:00ZSharp bounds on the height of some arithmetic Fano varieties
Andreasson, Rolf
In 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.
2023-01-01T00:00:00ZNew AI-based methods for studying antibiotic-resistant bacteriaInda Díaz, Juan Salvadorhttps://hdl.handle.net/2077/786752023-11-03T21:00:17Z2023-11-03T00:00:00ZNew AI-based methods for studying antibiotic-resistant bacteria
Inda Díaz, Juan Salvador
Antibiotic resistance is a growing challenge for human health, causing millions of deaths worldwide annually. Antibiotic resistance genes (ARGs), acquired through mutations in existing genes or horizontal gene transfer, are the primary cause of bacterial resistance. In clinical settings, the increased prevalence of multidrug-resistant bacteria has severely compromised the effectiveness of antibiotic treatments. The rapid development of artificial intelligence (AI) has facilitated the analysis and interpretation of complex data and provided new possibilities to face this problem. This is demonstrated in this thesis, where new AI methods for the surveillance and diagnostics of antibiotic-resistant bacteria are presented in the form of three scientific papers.
Paper I presents a comprehensive characterization of the resistome in various microbial communities, covering both well-studied established ARGs and latent ARGs not currently found in existing repositories. A widespread presence of latent ARGs was observed in all examined environments, signifying a potential reservoir for recruitment to pathogens. Moreover, some latent ARGs exhibited high mobile potential and were located in human pathogens. Hence, they could constitute emerging threats to human health. Paper II introduces a new AI-based method for identifying novel ARGs from metagenomic data. This method demonstrated high performance in identifying short fragments associated with 20 distinct ARG classes with an average accuracy of 96. The method, based on transformers, significantly surpassed established alignment-based techniques. Paper III presents a novel AI-based method to predict complete antibiotic susceptibility profiles using patient data and incomplete diagnostic information. The method incorporates conformal prediction and accurately predicts, while controlling the error rates, susceptibility profiles for the 16 included antibiotics even when diagnostic information was limited.
The results presented in this thesis conclude that recent AI methodologies have the potential to improve the diagnostics and surveillance of antibiotic-resistant bacteria.
2023-11-03T00:00:00ZEfficient training of interpretable, non-linear regression modelsAllerbo, Oskarhttps://hdl.handle.net/2077/773672023-10-31T13:08:57Z2023-06-30T00:00:00ZEfficient training of interpretable, non-linear regression models
Allerbo, Oskar
Regression, the process of estimating functions from data, comes in many flavors. One of the most commonly used regression models is linear regression, which is computationally efficient and easy to interpret, but lacks in flexibility. Non-linear regression methods, such as kernel regression and artificial neural networks, tend to be much more flexible, but also harder to interpret and more difficult, and computationally heavy, to train.
In the five papers of this thesis, different techniques for constructing regression models that combine flexibility with interpretability and computational efficiency, are investigated. In Papers I and II, sparsely regularized neural networks are used to obtain flexible, yet interpretable, models for additive modeling (Paper I) and dimensionality reduction (Paper II). Sparse regression, in the form of the elastic net, is also covered in Paper III, where the focus is on increased computational efficiency by replacing explicit regularization with iterative optimization and early stopping. In Paper IV, inspired by Jacobian regularization, we propose a computationally efficient method for bandwidth selection for kernel regression with the Gaussian kernel. Kernel regression is also the topic of Paper V, where we revisit efficient regularization through early stopping, by solving kernel regression iteratively. Using an iterative algorithm for kernel regression also enables changing the kernel during training, which we use to obtain a more flexible method, resembling the behavior of neural networks.
In all five papers, the results are obtained by carefully selecting either the regularization strength or the bandwidth. Thus, in summary, this work contributes with new statistical methods for combining flexibility with interpretability and computational efficiency based on intelligent hyperparameter selection.
2023-06-30T00:00:00ZThe influence of numbers when students solve equationsHolmlund, Annahttps://hdl.handle.net/2077/763182023-10-06T12:09:29Z2023-01-01T00:00:00ZThe influence of numbers when students solve equations
Holmlund, Anna
Is 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.
2023-01-01T00:00:00Z