| 1. |
- Xia, Jiacheng, 1991
(författare)
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Vector-valued Eisenstein series of congruence types and their products
- 2019
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Historically, Kohnen and Zagier connected modular forms with period polynomials, and as a consequence of this association concluded that the products of at most two Eisenstein series span all spaces of classical modular forms of level 1. Later Borisov and Gunnells among other authors extended the result to higher levels. We consider this problem for vector-valued modular forms, establish the framework of congruence types and obtain the structure of the space of vector-valued Eisenstein series using tools from representation theory. Based on this development and historic results, we show that the space of vector-valued modular forms of certain weights and any congruence type can be spanned by the invariant vectors of that type tensor at most two Eisenstein series.
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| 2. |
- Karlsson, Carl-Joar, 1995
(författare)
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Game theory and applications
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Individual-level interactions and decisions spread though populations and change the collective-level dynamics in an intricate way. Nevertheless, game theory is well suited for the unification of these viewpoints. This thesis introduces the works The strength of diversity and Decisions and disease: a mechanism for the evolution of cooperation, which show that games have broad applications in population dynamics modeling. In The strength of diversity we calculate the equilibrium strategies of the so-called game of teams. The game of teams is an individual-level competition between teams, and a team's strategy in this context is a distribution of strength over the team members. It turns out that the equilibrium strategies are flat distributions or 'alternating' flat distributions whenever there exists equilibrium strategies. In Decisions and disease: a mechanism for the evolution of cooperation we combine the classic SIR and SIS models from epidemiology with the prisoner's dilemma game. The transmission rate is computed as the average over defecting and cooperating individuals, and the individuals are subjected to a replicator equation that takes into account the portion of infectious members of the population. We compute the steady state solutions and interpret the results.
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| 3. |
- Lundengård, Karl, 1987-
(författare)
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Generalized Vandermonde matrices and determinants in electromagnetic compatibility
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility.In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.
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| 4. |
- Jansson, Erik, 1996
(författare)
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Geometric Discretization in Shape analysis
- 2022
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Discretizations in shape analysis is the main theme of this licentiate thesis, which comprises two papers. The first paper considers the problem of finding a parameterized time-dependent vector field that warps an initial set of points to a target set of points. The parametrization introduces a restriction on the number of available vector fields. It is shown that this changes the geometric setting of the matching problem and equations of motion in this new setting are derived. Computational algorithms are provided, together with numerical examples that emphasize the practical importance of regularization. Further, the modified problem is shown to have connections with residual neural networks, meaning that it is possible to study neural networks in terms of shape analysis. The second paper concerns a class of spherical partial differential equations, commonly found in mathematical physics, that describe the evolution of a time-dependent vector field. The flow of the vector field generates a diffeomorphism, for which a discretization method based on quantization theory is derived. The discretization method is geometric in the sense that it preserves the underlying Lie--Poisson structure of the original equations. Numerical examples are provided and potential use cases of the discretization method are discussed, ranging from compressible flows to shape matching.
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| 5. |
- Gyllingberg, Linnéa
(författare)
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Mathematical models of biological interactions
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Mathematical models are used to describe and analyse different types of biological interactions. From self-propelled particle models capturing the collective motion of fish schools to models in mathematical neuroscience describing the interactions between neurons to individual-based models of ecological interactions. A question that arises for all such models is how we scale from one level to another. How do we scale from fish interactions to the movement of the school of fish? How do we scale from neuronal interactions to the functioning of the brain? How do we scale from animal competition to population dynamics? It is approaches to this question that we study in this thesis for two different systems. In paper I, we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. The first is based on local interactions of the individuals and approximates the individual-based model well for small dispersal distances. The second approximates the long-range interactions of the individual-based model. Both approximations incorporate the demographic stochasticity from the individual-based model and show that dispersal stabilizes the population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the classical mean field approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Taken together, our results deepen the understanding of spatial population dynamics and introduces new approximate analytical descriptions.In paper II, we propose a model of social burst and glide motion in pairs of fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and pendulum-like tit-for-tat turn taking. Unlike self-propelled particle models, which assume that fish move at a constant speed, the model produces realistic motion of individual fish. Moreover, unlike previous studies where a random component is used for leadership switching to occur, we show that leadership switching, both periodic and chaotic, can be the result of from a deterministic interaction. We give several empirically testable predictions on how fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.
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| 6. |
- Maximov, Stepan, 1994
(författare)
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Classification of classical twists of the standard Lie bialgebra structure on a loop algebra
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This licentiate thesis is based on the work "Classification of classical twists of the standard Lie bialgebra structure on a loop algebra" by R. Abedin and the author of this thesis. The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. We study classical twists of the induced Lie bialgebra structures and obtain their full classification in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced Lie bialgebra structures are determined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of the CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by A. Belavin and V. Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of the CYBE.
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| 7. |
- Zheng, Yu, 1992
(författare)
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Gaussian conversion protocols forcubic phase state generation
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This licentiate thesis will introduce the background of the appended paper. The scope of this thesis is to find a Gaussian conversion protocol that allows one to prepare cubic phase states, which are resourceful states for achieving universality in continuous-variable quantum computation. The starting point of the protocol is a trisqueezed state, which has been realized experimentally with superconducting microwave circuits. With the help of machine learning, large fidelity can be achieved in these protocols. Furthermore, these protocols could also be helpful for other applications in continuous-variable systems as well.
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| 8. |
- Mannaa, Bassel, 1982
(författare)
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Constructive Newton–Puiseux Theorem, Sheaf Model of the Separable Closure and Dynamic Evaluation
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Computing the Puiseux expansions of a plane algebraic curve defined by an affine equation over an algebraically closed field is a an important algorithm in algebraic geometry. This is the so-called Newton–Puiseux Theorem. The termination of this algorithm, however, is usually justified by non-constructive means. By adding a separability condition we obtain a variant of the algorithm, the termination of which is justified constructively in characteristic 0. Furthermore, we present a possible constructive interpretation of the existence of the separable closure of a field by building, in a constructive metatheory, a suitable site model where there is such separable closure. Consequently, one can dispense with the assumption of separable closure and extract computational content from proofs involving this assumption. The theorem of Newton-Puiseux is one example where we use the sheaf model to extract computational content. We then can find Puiseux expansions of an algebraic curve defined over a non-algebraically closed field K of characteristic 0. The expansions are given as a fractional power series over a finite dimensional K-algebra.
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| 9. |
- Gustafsson, Henrik, 1988
(författare)
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Automorphic string amplitudes
- 2015
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis explores the non-perturbative properties of higher derivative interactions appearing in the low-energy expansion of four-graviton scattering amplitudes in toroidal compactifications of type IIB string theory. We summarise the arguments for finding such higher derivative corrections in terms of automorphic forms using U-duality, supersymmetry and string perturbation theory. The perturbative and non-perturbative parts can then be studied from their Fourier expansions. To be able to compute such Fourier coefficients we use the adelic framework as an intermediate step which also gives a new perspective on the arithmetic content of the scattering amplitudes.We give a review of known methods for computing certain classes of Fourier coefficients from the mathematical literature as presented in Paper I of the appended papers, and of our own work in Paper II towards computing some of the remaining coefficients of interest in string theory.
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| 10. |
- Johansson, Helena, 1971-
(författare)
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Mathematical Reasoning in Physics Tests : Requirements, Relations, Dependence
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- By analysing and expanding upon mathematical reasoning requirements in physics tests, this licentiate thesis aims to contribute to the research studying how students’ knowledge in mathematics influence their learning of physics. A sample of physics tests from the Swedish National Test Bank in Physics was used as data, together with information of upper secondary students’ scores and grades on the tests. First it was decided whether the tasks in the tests required mathematical reasoning at all and if they did, that reasoning was characterised. Further, the relation between students’ grades and mathematical reasoning requirements was examined. Another aim in this thesis is to try out if the Mantel-Haenszel procedure is an appropriate statistical method to answer questions about if there is a dependence between students’ success on different physics tasks requiring different kinds of mathematical reasoning. The results show that 75% of the tasks in the physics tests require mathematical reasoning and that it is impossible to pass six out of eight tests without mathematical reasoning. It is also revealed that it is uncommon that a student gets a higher grade than Pass without solving tasks that require the student to come up with not already familiar solutions. It is concluded that the Mantel-Haenszel procedure is sensitive to the number of students each teacher accounts for. If there are not too few students, the procedure can be used. The outcome indicates that there is a dependence between success on tasks requiring different kinds of reasoning. It is more likely that a student manages to solve a task that requires the produce of new reasoning if the student has solved a task that is familiar from before.
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