Majoriteten av elever saknar idag förståelsen för hur de ska addera bråktal med hjälp av varierande metoder trots flera års undervisning i bråkaritmetik. Studien syftar till att generera en fördjupad förståelse av de tidigare identifierade kritiska aspekter som årskurs 4 elever möter vid additionsberäkningar av tal i bråkform. Studien har två huvudsakliga frågeställningar: 1) Hur synliggörs tidigare identifierade kritiska aspekter inom additionsberäkningar med tal i bråkform i årskurs 4 elevers lösningar och resonemang? 2) Vilka kritiska aspekter framkommer i elevers lösningar med numeriska och bildliga representationsformer?  Studien omfattar en kombination av kvantitativa och kvalitativa metoder där inslag av variationsteorin integrerades. De tillämpade metoderna innefattar användning av arbetsblad samt genomförande av semi-strukturerade intervjuer. Totalt deltog 35 elever i undersökningen, varav 6 elever valdes ut för intervju. Resultaten av studien identifierade tre kritiska aspekter. Kritisk aspekt 1: Att urskilja hur additionsberäkning med tal i bråkform fungerar i jämförelse med hur heltal beräknas. Kritisk aspekt 2: Att urskilja hur additionsberäkning, med tal i bråkform, utförs med olika nämnare. Kritisk aspekt 3: Att urskilja hur additionsberäkning utförs med hjälp av representation av tal i bråkform som en del av en helhet. Dessa kritiska aspekter som framkommit i tidigare forskning bekräftar och stödjer resultaten från studien. / Most pupils today lack understanding of how to add fractions of how to add fractions using various methods, despite several years of instruction in fraction arithmetic. The study aims to generate a deeper understanding of the previously identified critical aspects encountered by fourth-grade pupils in addition calculations involving fractions. The study has two main research questions: 1) How are the preciously identified critical aspects manifested in fourth-grade pupils’ solutions and reasoning regarding addition calculations with fraction? 2) Which critical aspects emerge in pupils’ solutions using numerical and visual representations?  The study employs a combination of quantitative and qualitative methods, integrating elements of variation theory. The applied methods include the use of worksheet and conducting semi-structured interviews. A total of 35 pupils participated in the investigation, with 6 pupils selected for interviews. The results of the study identified three critical aspects. Critical aspect 1: Discerning how addition calculations with fractions differ from those involving whole numbers. Critical aspect 2:  Discerning how addition calculations with fractions are preformed when the denominators are different. Critical aspect 3: Discerning how addition calculations are carried out using representations of fractions as parts of a whole. These critical aspects, which have been highlighted in previous studies, confirm, and support the results of the study.

The muffle furnace main purpose is to anneal the rough sponge iron powder transported through it, which is done by burning natural gas. Heat is absorbed by the muffle and is transferred to the bed of sponge iron powder. In order to reduce the consumptions of fossil fuel, some companies of the industry aims to exchange the natural gas in their muffle furnace’s burners to syngas, produced from biomass. This will however affect the performance of the furnace in the heating aspect. For this work, it is assumed that the effect will be negative. Thus the aim of this study is to investigate how to compensate the loss of effect from the burners, by examining which other parameters have influence on the furnace heating performance of the sponge iron powder transported through the furnace. The investigation is executed by simulating a 1 meter of the furnace in COMSOL multiphysics for 10 min, not including the combustion chambers. The investigated parameters are the packing degree of the powder, surface emissivity of the muffle, process gas velocity, conveyor belt velocity and the heat transfer rate coefficient to muffle from the combustion chambers. Alas, the process gas velocity and conveyor belt velocity only have minor influence on the final result, according to this simulation. However, the simulation exhibited that the surface emissivity of the muffle and the packing degree of the powder has great impact on the heating of the powder and could compensate some of the lost effect from the burners. This could be obtained by using an unpolished and oxidized muffle surface, and use densely packed powder sample.

In this thesis three different approaches for solving partial differential equa-tions with neural networks will be explored; namely Physics-Informed NeuralNetworks, Fourier Neural Operators and the Deep Ritz method. Physics-Informed Neural Networks and the Deep Ritz Method are unsupervised machine learning methods, while the Fourier Neural Operator is a supervised method. The Physics-Informed Neural Network is implemented on Burger’s equation,while the Fourier Neural Operator is implemented on Poisson’s equation and Darcy’s law and the Deep Ritz method is applied to several variational problems. The Physics-Informed Neural Network is also used for the inverse problem; given some data on a solution, the neural network is trained to determine what the underlying partial differential equation is whose solution is given by the data. Apart from this, importance sampling is also implemented to accelerate the training of physics-informed neural networks. The contributions of this thesis are to implement a slightly different form of importance sampling on the physics-informed neural network, to show that the Deep Ritz method can be used for a larger class of variational problems than the original publication suggests and to apply the Fourier Neural Operator on an application in geophyiscs involving Darcy’s law where the coefficient factor is given by exponentiated two-dimensional pink noise.

In today’s pulp and paper industry the Kraft process is the most common method for pulp production. This method uses sodium based chemicals (white liquor) in the cooking process to remove lignin from the wood chips and create pulp. The remains from this process is called black liquor and is being sent to a recycling system for the purpose of recovering the cooking chemicals. Evaporation of black liquor is a big part of this recycling, and the evaporation plant consists of many different tanks that stores black liquor. At Smurfit Kappa Piteå a model has previously been created for the purpose of production planning. In this work the opportunity to add a part that simulates how the liquor stock in the chemical recovery system will change based on the planned production was investigated. This was done by estimating the amount of dry black liquor in the tanks through inflows and outflows. A formula for the produced black liquor was also developed. The results showed that simulating tank levels separately was difficult as data was lacking in some key areas. The final model is therefore a simplified version that estimates the total amount of dry black liquor in the evaporation plant. It simulates the black liquor buffer based on the planed production and how it will change over five days. This could be done with an error smaller than 6%, compared to measurements from sensors in the black liquor tanks. Attempts were also made to create similar models for the rest of the chemical recovery system. It was concluded that information about the inflow of green and white liquor has to be further investigated in order to implement these in the production planning model.

This research overview is an evaluation of knowledge and explores the impact of language linked to mathematics. This particularly in the context of conversations and discussions, on the mathematical understanding of students in the ages between ten to twelve years old. The research overview attempts to examine the role a conversation plays in a concept of learning, specifically in mathematics teaching. With this study we want to create an understanding between the relationship of a rewarding conversation and the understanding of mathematical concepts and systems. Through systematic search and analysis of scientific texts a total of 10 articles were used to gain a proper overview of the subject found through systematic searches of sources such as ERIC, EBSCO and LibSearch. The limitations set for the chosen articles were a timeframe between the years 2011 until 2023, as well as being peer-reviewed. All chosen articles and studies are written in English but originate from different places which provides a wider perspective on the subject. The results show that conversation has an essential role in students' understanding of mathematics. Through correct language that develops in mathematical conversations, a deeper understanding of the subject is created. The research overview also shows the importance of language in shaping students' mathematical knowledge, presenting it as a crucial component of their proficiency in the subject. The research overview further describes the importance of classroom conversations, the teacher's role in the classroom, and the impact of home environments on learning. An aspect related to the subject is how sociocultural thoughts about active learning in a social context reflect the present of Swedish school and its design. By delving deep into the functionality of conversation and the connection between discussion and learning, the school will be able to develop the learning offered in the classroom to better meet the students and their needs. Furthermore, our research concludes that for future educators it is a necessity to balance teacher guidance and student participation in classroom discussions.

This thesis will focus on generating the probability mass function using Fibonacci sequenceas the coefficient of the power series. The discrete probability, named Fibonacci distribution,was formed by taking into consideration the recursive property of the Fibonacci sequence,the radius of convergence of the power series, and additive property of mutually exclusiveevents. This distribution satisfies the requisites of a legitimate probability mass function. It's cumulative distribution function and the moment generating function are then derived and the latter are used to generate moments of the distribution, specifically, the mean and the variance. The characteristics of some convergent sequences generated from the Fibonacci sequenceare found useful in showing that the limiting form of the Fibonacci distribution is a geometricdistribution. Lastly, the paper showcases applications and simulations of the Fibonacci distribution using MATLAB. / <p></p><p></p><p></p>

In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach functionsetting. The thesis consists of three papers (A, B and C) and an introduction, which put these papers into a more general frame. This introduction has also independent interest since it shortly describe the dramatic more than 100 years of development of Hardy-type inequalities. It contains both well-known and very new ideas and results. In paper A we prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator, which is introduced in this paper (this operator generalizes the usualHardy kernel operator). These results generalize and unify several classical Hardy-type inequalities. In paper B we prove some new refined Hardy-type inequalities again in Banach function space settings. The used (super quadraticity) technique is also illustrated by making refinements of some generalized forms of the Jensen, Minkowski and Beckenbach-Dresher inequalities. These results both generalize and unify several results of this type. In paper C for the case 0&lt;p≤q&lt;∞ we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant. / In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit, Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach function setting.  We prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator. These results generalize and unify several classical Hardy-type inequalities. Next, we prove some new refined Hardy-type inequalities again in Banach function space settings. We used superquadraticity technique to prove refinements of some classical inequalities. Finally, for the case 0&lt;p≤q&lt;∞, we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant.

Syftet med den här kunskapsöversikten är att undersöka vikten av kreativt tänkande, dess relation till problemlösningsförmågan och den relationella förståelsen. Problemlösningsförmågan är en förmåga som understryks i läroplanen för matematik årskurs 4-6, därmed undersöker vi relationen mellan det kreativa tänkandet och problemlösningsförmågan. Undersökningen har gjorts genom att initialt skapa tre frågeställningar och därefter söka data från olika peer-reviewed forskningsartiklar på engelska mellan årtalen 2019–2023 från databaser som ERC, JSTOR, ERIC från olika delar av världen. Forskningen har granskats och sammanställts för att ge svar på frågeställningarna. Under den systematiska sökningsprocessen har vi stött på nya begrepp som open ended och heuristik som visat sig ha en nära relation till problemlösningsförmågan samt det kreativa tänkandet. Utöver det har Skemps lärandeteorier om instrumentell och relationell förståelse satt grunden för vår systematiska sökprocess. Resultatet påvisar att det kreativa tänkandet utvecklas efter arbete med problemlösningsförmågan samt öppna frågor. Det kreativa tänkandet kan i sin tur relateras till en relationell förståelse. De empiriska studierna påvisar att lärare brister i sin kunskap om hur man kan utforma matematiska problem som utvecklar ett kreativt tänkande hos eleverna. De valda empiriska studierna är genomförda i utlandet och Sverige skulle dra nytta av liknande forskning för att stötta pedagogerna i elevernas utveckling mot ett kreativt tänkande som står emot den snabbt utvecklande värld som väntar efter avslutad skolgång.

In this thesis we start off by ensuring that the reader is up to speed when it comes to some well known definitions and theorems from real analysis. We then introduce the reader to metric spaces and provide them with some examples such as the real numbers with the Euclidean distance, and compact sets with the Hausdorff distance. Then, we go on to define important concepts such as inner points, limit points, open sets, boundary and much more. We also show, whenever we can, how these concepts are connected. With these tools in place we move on to explain how limits and continuity are defined in metric spaces as well as providing the reader with several examples. We then introduce the reader to the concepts of compactness and uniform convergence, for which we show some interesting results such as how uniform convergence and the supremum norm are related. We finish off by covering curves and connectedness (including pathconnectedness) in metric spaces, before we briefly touch on topological spaces as to give the reader a hint of what further mathematics studies might hold. / I detta examensarbete börjar vi med att försäkra oss om att läsaren har de förkunskaper som behövs för att kunna ta del av arbetet. Detta görs genom att påminna läsaren om viktiga definitioner och satser från reell analys. Därefter introducerar vi läsaren till metriska rum och ger en mängd olika exempel på dessa som läsaren förhoppningsvis redan stött på. Detta inkluderar bland annat de reella talen med euklidiskt avstånd och slutna och begränsade mängder med Hausdorff-avstånd. När vi väl förklarat distanskonceptet introducerar vi inre punkter, hopningspunkter, öppna mängder, randpunkter och mycket mer. Vi visar dessutom, närhelst vi kan, hur dessa koncept hänger samman. När alla dessa grundbegrepp är etablerade kan vi fortsätta med att förklara gränsvärden och kontinuitet i metriska rum. Vi ger även läsaren flera exempel på detta. I arbetets andra hälft tar vi upp kompakthet och likformig konvergens, för vilka vi presenterar en del intressanta resultat, såsom hur likformig konvergens och supremumnormen är relaterade. Vi avslutar examensarbetet genom att gå igenom kurvor och sammanhängande mängder (inklusive bågvis sammanhängande mängder) i metriska rum, innan vi kort tar upp topologiska rum för att ge läsaren en föraning om vad vidare matematikstudier kan innehålla.

In this thesis, the credit worthiness of a company is modelled using a stochastic process. Two credit models are considered; Merton's model, which models the value of a firm's assets using geometric Brownian motion, and the distance to default model, which is driven by a two factor jump diffusion process. The probability of default and the default time are simulated using Monte Carlo and the number of scenarios needed to obtain convergence in the simulations is investigated. The simulations are performed using the probability matrix method (PMM), which means that a transition probability matrix describing the process is created and used for the simulations. Besides this, two variance reduction techniques are investigated; importance sampling and antithetic variates. / I den här uppsatsen modelleras kreditvärdigheten hos ett företag med hjälp av en stokastisk process. Två kreditmodeller betraktas; Merton's modell, som modellerar värdet av ett företags tillgångar med geometrisk Brownsk rörelse, och "distance to default", som drivs av en två-dimensionell stokastisk process med både diffusion och hopp. Sannolikheten för konkurs och den förväntade tidpunkten för konkurs simuleras med hjälp av Monte Carlo och antalet scenarion som behövs för konvergens i simuleringarna undersöks. Vid simuleringen används metoden "probability matrix method", där en övergångssannolikhetsmatris som beskriver processen används. Dessutom undersöks två metoder för variansreducering; viktad simulering (importance sampling) och antitetiska variabler (antithetic variates).