In this thesis we introduce the notion of elliptic functions, and in particular Weierstrass' Elliptic Function, in order to define the modular function $\lambda$ on the upper half plane. We establish a group ismorphism between the unimodular transformations and $\Aut{\C\setminus\{0,1\}}$. In particular this gives two functional equations involving $\lambda$, and with these equations we show th

We provide an introduction to Banach Algebras and C*-algebras. The discussion of C*-algebras leads to the two central theorems of this thesis: Gelfand Duality & the Gelfand-Naimark-Segal (GNS) Theorem. The latter tells us that every C*-algebra may be treated as some (special) subalgebra of bounded endomorphisms of a C-Hilbert space, while the former tells us that every commutative C*-algebra c

I syfte att förstå lösningar för nutida optimeringsproblem, ska vi se på det matematiska ramverket för den moderna teorin om variationskalkyl. Variationskalkyl är en gren av matematiken som utvecklades under 1700-talet för att lösa problem gällande fastställning av optimala trajektoria, former och strukturer, givet vissa begränsningar. Dess utveckling pågick parallellt med framsteg inom matematiskCalculus of variations is a branch of mathematics that was developed in the 18th century to solve problems of determining optimal trajectories, shapes and structures subject to constraints. It deals with the minimization of a certain type of integral functional. In the early days of the theory, the smoothness of the functions involved was assumed but, during the following centuries, advances in an

I den digitala tidsåldern spelar invecklad matematik ofta en undanskymd, men ändå avgörande roll. När vi löser bildproblem, som att avpixla foton och förbättra upplösningen, använder vi ofta komplex matematik. I den här avhandlingen vill vi lyfta fram den genom att beskriva tre viktiga algoritmer - ADMM, Plug and Play och Consensus Equilibrium. Denna avhandling utforskar hur dessa algoritmer byggeIn this thesis, we explore the Alternating Direction Method of Multipliers (ADMM) algorithm, the Plug & Play (PnP) algorithm and consensus equilbrium in order to understand their importance for solving complex imaging problems. We begin by motivating the invention of ADMM which combines the strengths of dual ascent and the method of multipliers. A proof of convergence for ADMM is then provided

A solution to an ordinary differential equation may be stable or unstable. Let x = φ(t) be a solution of an autonomous differential equation x′ = Ax. The solution x = φ(t) with φ(0) = t0 is stable if every other solution ψ(t) with initial condition sufficiently close to φ(t) will remain close to φ(t) for all t > 0. For an autonomous system of linear differential equations, it is possible to determ

Tre styck icke-triviella satser om kontinuerliga funktioner presenteras för läsaren. Kontinuerliga funkioner har ett flertal trevliga egenskaper, de över kompakta rum speciellt så. I och med att de saknar "stup," där funktionen hoppar från ett värde till ett annat, följer det att de kan lättare uppskattas. Den första satsen som presenteras handlar om just detta: Stone-Weierstraß sats, viThree main theorems are presented, all of them concerning different non-trivial properties of continuous functions on compact metric spaces. The first is the Stone-Weierstrass theorem, which states that a continuous functions on a compact metric space can always be uniformly approximated by functions in a smaller algebra -- for instance, on compact intervals of the real line then any continuous fu

Vi behandlar i denna uppsats några generaliseringar av resultat i grundläggande komplex analys, såsom existens av logaritmiskt konjugat som generalisering av harmoniskt konjugat. Detta använder vi sedan bland annat för att visa en sats om konforma avbildningar (d.v.s. avbildningar som bevarar vinklar), som utgör en analogi med Riemanns avbildningssats för områden i planet med ett hål. Detta resultThe existence of a primitive of analytic functions, of an analytic logarithm of zero-free analytic functions, and of a harmonic conjugate to any harmonic function are fundamental propositions of elementary complex analysis, holding on simply-connected domains. We find that with suitable corrections, analogous statements can be made on multiply connected domains, foremost of these is these being th

Klangen av ett piano och en gitarr, ett glas som börjar vibrera för att till slut gå i bitar på grund av att en viss ljudfrekvens spelats och svajningen av en hängbro under marscherande soldaters fötter, allt detta är exempel på resonans. Vilket kan förklaras som att en viss frekvens får en mycket starkare påverkan. Även om i fal- let med piano så är det effekten av resonans man söker för dess klaThis thesis presents a method, from an article by Gouveia and Stone, for solving resonant ODE:s as well as ODE:s with multiple roots, based on perturbation theory. It also gives a mathematical justification for the method as well as background on resonance and repeated roots. Later in the thesis we use the method to solve a number of ODE’s, illustrating its usefulness.

Under åren 1951–2002 släpptes en mängd på 2600–5000 ton PFAS ut i miljön. Det låter kanske inte så farligt. Men vad händer med miljön när denna mängd ökar? PFAS har konstaterats ha negativa effekter på daggmaskar, jordlevande bakterier och markhälsan. Dessa organismer är avgörande för att våra ekosystem ska fungera och för att vi ska kunna odla och bruka marken. Ändå är kunskapsläget kring PFAS påDuring the years 1951–2002, an amount of 2600–5000 tons of PFAS was released into the environment. The widespread occurrence of PFAS substances can have negative consequences for microbial species and communities within ecosystems, which is ex-pected to have negative effects also for higher trophic levels. Therefore, the purpose of this literature study was to compile information on which identifi

Denna studie undersöker möjligheten att använda geotermisk energi till fjärrvärmesystemet i Jönköping. Med ökande krav på lägre utsläpp från fjärrvärmeanläggningar kan geotermi som värmekälla till fjärrvärmenätet vara ett intressant alternativ till avfallsförbränning eller biobränslen. De lokala förutsättningarna för geotermisk energi har kartlagts med fokus på det sandstenslager som finns i centrThis study investigates the potential of using geothermal energy for the district heating system in Jönköping, Sweden. With an increasing demand for lower emissions from district heating plants, geothermal energy as a heat source for district heating networks may be an interesting alternative to waste incineration or biofuels. The local conditions for geothermal energy have been investigated focus

Tidewater glaciers on Svalbard are sensitive indicators of climate change and play a crucial role for Arctic ecosystems. The widespread retreat of tidewater glaciers across Svalbard highlights the need to quantify retreat rates and project when these glaciers will become land-terminating. This study investigates tidewater glacier retreat during historical (1936–1990) and present (2012–2024) period

The coupled climate and nature crisis requires rapid mitigation that does not compromise biodiversity. Forests have the potential to improve environmental objectives in synergies depending on management. To better understand the motivations behind forest management practices among non-industrial private forest (NIPF) owners in Sweden, this study conducted a cross-sectional comparative analysis wit

In the summer of 2023, the international World Scout Jamboree (WSJ) will take place in South Korea. A Swedish scout delegation will participate in the camp and provide a Swedish food house. In an international context such as the WSJ, the Swedish scouts will express their own nationality, through both the Swedish Food House and other national symbols. This expression of nationality is essentially

With the 25-year anniversary of the Öresund bridge, questions can be posed: how has the region benefited from the increased integration? What opportunities and problems can the individuals of this region tell of? This thesis contributes to the discussion regarding identity creation in border cultures. The results presented in this study show how new identities are shaped out of necessity through t

On not sharing ones home with ones partner: The gendered experience of women in LAT relationships in Sweden and Austria. This thesis explores the gendered experiences of women in Living Apart Together (LAT) relationships. It focuses on how LAT as a relational form is negotiated within the contested space of intimacy and coupledom, that is shaped by cultural norms and societal expectations around c

In China, young people face intense academic competition, a competitive job market, and changing cultural narratives about success and stability. Travel is often seen by young people as a means of escaping these pressures, but it is also influenced by economic conditions, social class, and cultural capital. This paper seeks to explore how Chinese young people aged 18-35 view depressive experiences

Background: The purpose of this analysis was to revisit the open US Food and Drug Administration (FDA) inspection list results from clinical trials and show the possibilities of evaluating inspection results between different world regions and explore what aspects of clinical trial deficiencies are most often reported.Aims: The aim of this study was to evaluate inspection results from different re

Aim of the review: To provide an overview of studies that have published dataregarding region and population size, procedure location, team composition, inclusionand exclusion criteria, outcome parameters, and cost–benefit analyses onextracorporeal membrane oxygenation use for refractory out-of-hospital cardiac arrest.Data sources: A structured systematic literature search of articles published pr

Aim of the review: This review aimed to evaluate the impact of Point-of-care Ultrasound during cardio- pulmonary resuscitation for adult non-traumatic CA on clinical outcomes and survival.Data sources: This review was conducted in accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines. The PubMed and Embase databases were searched on 14 November 2022. The