In this thesis, we analyze a roller - pinned supported beam of length L subjected to a point load at a distance `1 and a uniformly distributed load over the entire span of the beam. The aim of this study is to determine the slope and the vertical deflection curve of the beam. The analysis is conducted by applying the finite element method, which consists of dividing the whole domain into geometrically simple subdomains, namely elements. The analysis is then performed for each subdomain separately. The approximate solution is obtained by assembling all the elements. The problem is reduced into solving a system of linear algebraic equations. It can be expressed in the matrix form as A = where A is the stiffness matrix, the vector of nodal variables, and the load vector. The finite element method has produced a solution close to the analytical solution.The finite element solution is improved by increasing the number of elements with variablestep sized element. The error based on the L2-norm with respect to the displacement is presented as well as the condition number of the stiffness matrix. The finite element method is implemented in Matlab, and the diagram of the deflected beam is presented.

A big challenge for many companies, road haulers business among others, is how to make a schedule for all employees. Scheduling problems handles a big number of constraints which also increase the complexity of the problem.   It is important to SJ that all train departures are staffed. SJ offers both permanent and temporary employments but during recruitment in connection to the summer they only hire temporary employees. It often happens that SJ hires a larger number of workers than needed, and as a consequence of this some of the employees does not get to work the number of hours the wishes and they might turn to other employers. Temporary employments can result in uncertainty for SJ as they can choose when to work. Due to this, SJ now wants to examine the possibility to hire permanent employees during the summer and they also want to examine how the distribution of employments should look like for different stations and time periods.   Important constraints for scheduling and planning have been identified and formulated, and these have formed the basis to the optimization problem that is indented to be solved. A scheduling tool, EAPS, has been developed and with help from this minimal staff set and employment arrangements for boarders can be identified. EAPS ensure that each worker is free every other weekend, that no worker is assigned more shifts than allowed for the analyzed period and that they do not work more shifts in a row than allowed. EAPS ensure that all departures are covered by boarders at the same time as the number of workers are minimized.

Centrala gränsvärdessatsen (CGS) är en av grundpelarna inom statistik och sannolikhetsteori. CGS säger att ”summan av ett stort antal oberoende likafördelade slumpmässiga variabler är approximativt normalfördelad”. Det finns olika bevis för CGS. I denna uppsats kommer jag att bevisa centrala gränsvärdessatsen genom att utnyttja karaktäristiska funktioner, eftersom karaktäristiska funktioner har bra egenskaper vilket vi kommer att få se när vi definerar dem. I detta arbete har jag kopplat samman olika pusselbitar som behövs för att kunna bevisa CGS. Genom att göra det så blir det enklare för läsaren att förstå CGS mer grundläggande. / The Central Limit Theorem (CLT) is one of the pillars of statistics and probability theory. CLT states that ”the sum of a large number of independent equally distributed random variables is approximately normally distributed”. There are different proofs for CLT. In this essay I will prove the central limit theorem by utilizing characteristic functions, since characteristic functions have good features, which we will see when defining them. In this paper I have linked various pieces of the puzzle needed to prove CLT. By doing so it becomes easier for readers to gain a deeper understanding of CLT.

This is a survey of decoupling results in harmonic analysis and their use in combinatorial number theory. The main results are an expository proof of a decoupling inequality for exponential sums in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BR%7D%5E%7B2%7D" /> and a new lower bound on the cardinality of the sumsets of certain convex sequences. There is also some discussion regarding the multilinear Kakeya inequality, the uncertainty principle as well as the use of decoupling in number theory in general. / Denna uppsats fördjupar sig i tekniken decoupling inom harmonisk analys och dess användning inom kombinatorisk talteori. Huvudresultaten är ett förklarande bevis av en decoupling-olikhet for exponentiella summor i <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BR%7D%5E%7B2%7D" /> samt en ny undre begränsning på storleken av summamängder av vissa konvexa sekvenser. Utöver detta diskuteras den multilinjära Kakeyaolikheten, osäkerhetsprincipen samt användningen av decoupling inom talteori från ett mer generellt perspektiv.

The topic of this thesis is the connectivity of the k-nearest neighbor random geometric graph model. The main result is an expository proof of the fact that there is a critical constant for connectivity. In addition to this, other results related to the connectivity of the k-nearest neighbor model, as well as the closely related Gilbert disc model, are discussed. / Denna uppsats fördjupar sig i "the k-nearest neighbor graph", en geometrisk slumpgraf, och när den är sammanhängande. Huvudresultatet är ett förklarande bevis av att det finns en kritisk konstant för egenskapen att vara sammanhängande med stor sannolikhet. Utöver detta diskuteras andra resultat rörande när k-nearest neighbor-grafen och den närstående Gilberts diskmodell blir sammanhängande.

Lärande genom problemlösning utvecklar elevernas matematiska förståelse. Idag har problemlösning en liten roll i matematikundervisningen. Istället lär sig elever att imitera procedurer. Forskning visar på att problemlösningsstrategier inte går att lära sig utan att få ett sammanhang och tid till att träna. För att kunna lära ut problemlösningsförmågan krävs kunskap. Syftet med denna studie är att undersöka om hur eleverna i två femteklasser visar sin problemlösningsförmåga. För att konkretisera syftet har vi formulerat en forskningsfråga: Hur hanterar eleverna problemlösningsprocessen i matematiska diskussioner. Metoden för att undersöka detta har varit att samla in empiri i form av ljudinspelningar, genom en fältstudie i två grundskoleklasser i södra Sverige. Resultatet visar att eleverna går igenom de olika faserna (ingång, attack och granskning) som forskare i problemlösning skriver om. Däremot uppmärksammades också att eleverna inte relaterar till liknande uppgifter och att de reflekterar i låg utsträckning vilket är viktiga delar av faserna. Vidare forskning hade behövts om i vilken utsträckning lärarnas identifiering och reflektion i helklass över olika strategier påverkade elevernas val av strategier.

De modellering van een wet of mortaliteit heeft een consequente interesse van een grote meerderheid van onderzoekers en vele modellen door de jaren is voorgesteld. The first aim of this thesis is to systematically evaluate a selection of models --- Modified Perks, Heligman-Pollard and Power-exponential --- to determine their relative strengths and weaknesses with regard to forecasting the mortality rate using the Lee-Carter model. Den andre målsætningen er at tilpasse dødelighedsdata ved de selektive modeller fra USA, Sverige og Grækenland ved hjælp af numeriske teknikker til kurvefitting med den ikke-lineære mindst kvadratmetode. The results indicate that the Heligman-Pollard model performs better especially when the phenomenon of the `` accident hump '' occurs during adulthood.

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This thesis uses modern portfolio theory together with machine learning techniques to generate stable portfolio returns over eleven currency pairs with spreads included. The backtests show that support vector machine predicted future returns better than neural network and linear regression. Principal component analysis and data smoothing combined with the local outlier factor further improved the performance of the trading algorithm. However, the ensemble of the top performed predictor performed below the individual predictors. Also, the use of different error estimates showed the criticality of mean arctangent absolute percentage error over mean absolute error and over mean squared error for profitability. For obtaining sensible results in a transaction costless setting, adopting risk adjusted leverage proved necessary. Otherwise, the profit-maximizing leverage surpassed the risk adjusted in a spread setting.