Syftet med denna litteraturstudie är att utifrån forskning undersöka vilka faktorer som är centrala vid bedömning av problemlösning i matematiken och koppla detta till kursplaner och betygskriterier.Utifrån tidigare forskning, rapporter och utvärderingar, kursplaner och betygskriterier från Skolverket beskrivs problemlösning, bedömning, kunskap och lärteorier med huvudfokus på bedömning av elevers problemlösningsförmåga.Analysen har visat att det finns vissa frågor som läraren bör tänka på vid bedömning av elevens problemlösningsförmåga; vilka förmågor som ska testas och hur detta ska göras, elevens medvetenhet om vad som ska bedömas, på vilket sätt eleven har förstått uppgiften, elevens möjlighet att lösa problemet, hur eleven har försökt lösa problemet, vilka kunskaper eleven har använt sig av, vilka slutsatser eleven kommit fram till, vilken tillförlitlighet bedömningen har samt vilka kriterier eleven har uppfyllt.

A study has been made of a few different freely available Quantum Computer simulators.All the simulators tested are available online on their respective websites. A number oftests have been performed to compare the different simulators against each other. Someuntested simulators of various programming languages are included to show the diversityof the quantum computer simulator applications.

The conclusion of the review is that LibQuantum is the best of the simulatorstested because of ease of coding, a great amount of pre-defined functionimplementations and decoherence simulation support among other reasons.

A study has been made of a few different freely available Quantum Computer simulators. All the simulators tested are available online on their respective websites. A number of tests have been performed to compare the different simulators against each other. Some untested simulators of various programming languages are included to show the diversity of the quantum computer simulator applications.

The conclusion of the review is that LibQuantum is the best of the simulators tested because of ease of coding, a great amount of pre-defined function implementations and decoherence simulation support among other reasons.

I detta kandidatarbete undersöker och jämför vi två olika metoder för att approximera gauss- och medelkrökningen hos en yta i rummet som är given som en mängd av punkter. Det är viktigt att försöka få en bra analogi mellan diskret krökning och analytisk krökning då man ofta startar med en mängd punkter i de praktiska fallen, som t ex i tillverkningsindustrin, igenkänning av objekt (inscannade bilder) och datorgrafik. Givet dessa punkter och en bra approximation av gauss- och medelkrökningen kan man få mer information om ytans geometri och beteende.

För att kunna förstå dessa begrepp och metoder/algoritmer så behandlas först den bakomliggande teorin och sedan metoderna.

Den första metoden är att återge ytan med hjälp av Bézierytor, vilka vi kan utföra geometriska operationer på utan problem och även få fram gauss- och medelkrökningen.

Den andra metoden kommer från artikeln ``Discrete Differential-Geometry Operators for Triangulated 2-Manifolds'' av Mark Meyer, Mathieu Desbrun, Peter Schröder och Alan H. Barr. Deras approximationer av krökningarna kräver en triangulering av ytan, vilket de inte ger någon algoritm för. De tittar på ett område runt varje punkt och approximerar krökningarna genom detta område, även Gauss-Bonnets sats används för approximering av gausskrökningen.

Mina simuleringar visar att Bézierytornas approximationer av gauss- och medelkrökningar är konvergenta och att alla värden ligger relativt nära varandra. Artikelns algoritm fungerar bra för gauss- och medelkrökning men deras algoritm beror väldigt mycket på trianguleringen vilket gör att man behöver ha krav på den triangulerade ytan, vilket i sig är ett svårt problem att lösa.

Using fnite difference methods for partial differential equations, this thesis focuses on the problem of connecting overlapping solution domains in the context of a frst order hyperbolic problem. Especially the stability properties of such constructions is studied, and a stable general implementation of the the test problem is proposed. However, no energy estimate could be found, and indeed proven not to exist in the natural norm. Finally, an example is also put forward where the interface conditions derived are, for stability considerations, incompatible with the boundary conditions in a coupled system of hyperbolic equations.

The focus in this thesis is on the calculations of an empirical null distributionfor likelihood ratio tests testing either separable or double separable covariancematrix structures versus an unstructured covariance matrix. These calculationshave been performed for various dimensions and sample sizes, and are comparedwith the asymptotic χ2-distribution that is commonly used as an approximative distribution. Tests of separable structures are of particular interest in cases when data iscollected such that more than one relation between the components of the observationis suspected. For instance, if there are both a spatial and a temporalaspect, a hypothesis of two covariance matrices, one for each aspect, is reasonable.

Fractal geometry is a new branch of mathematics. This report presents the tools, methods and theory required to describe this geometry. The power of Iterated Function Systems (IFS) is introduced and applied to produce fractal images or approximate complex estructures found in nature.

The focus of this thesis is on how fractal geometry can be used in applications to computer graphics or to model natural objects.

Inom vården utförs ofta schemaläggning av personal manuellt, vilket kräver mycket tid och resurser. Att planera arbetet för en grupp läkare, med dess ofta mycket komplexa sammansättning vad gäller exempelvis arbetsuppgifter och kompetenser, är ingen lätt uppgift. Detta examensarbete studerar huruvida en automatiserad taktisk bemanningsplanering med en tidshorisont på ett halvår till ett år, skulle kunna underlätta denna uppgift.

I rapporten presenteras en måloptimeringsmodell som implementerats i AMPL för att med CPLEX som lösare generera förslag till bemanningsplaner. För att utveckla en matematisk modell som väl representerar de förutsättningar som råder vid bemanningsplanering av läkare har alternativa formuleringar provats och utvärderats. Den mest lovande av modellerna, som baseras på måloptimering, har i en pilotstudie testats på data från Onkologiska kliniken vid Linköpings universitetssjukhus. Flexibiliteten i modellen gjorde att den enkelt kunde användas på de data som erhölls därifrån. Resultatet från pilotstudien indikerar att den utvecklade modellen har kapacitet att ge förslag till rimliga bemanningsplaner.

In the study of the Degasperis-Procesi dierential equation, an eigenvalue problem called the cubic string occurs. This is a third order generalization of the second order problem describing the eigenmodes of a vibrating string. In this thesis we study the eigenfunctions of the cubic string for discrete and continuous mass distributions, using the theory of total positivity, via a combinatorial approach with planar networks.

This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.

Fredriksons Verkstads AB i Vadstena är en modern verkstadsindustri med kompetens främst inom plåtbearbetning, svetsning, skärande bearbetning och systemmontage. Verksamheten kan delas in i de tre delarna kontraktstillverkning, konstruktion och produktion. Kontraktstillverkningen består av serier till livsmedels-, medicinteknisk och verkstadsindustrin. Fredriksons Conveyor Solutions konstruerar egna produkter i form av transportörsystem till livs- och läkemedelsindustrin.

Fredriksons kunder ställer i allt högre grad krav på reducerade ledtider och ökad leveransprecision. Syftet med examensarbetet är därför att försöka reducera Fredriksons interna ledtider för ett specifikt produktsortiment.

Ett första steg i arbetet med ledtidsreduceringen är att ta fram bra utarbetade prognoser. Framförallt har kvantitativa tidsseriemetoder analyserats. De enkla tidsseriemetoderna glidande medelvärde och exponentiell utjämning har undersökts i syfte att förbättra Fredriksons prognoser. Därutöver har en mer teoretisk fördjupning i avancerade tidsseriemetoder, främst ARMA-processer, genomförts.

Andra faktorer som påverkar produktionsledtiden är exempelvis partiformning, ställtider, säkerhetsmekanismer, kötider och sekvensering. Även dessa faktorer har analyserats och åtgärder har föreslagits i syfte att reducera Fredriksons interna ledtider.

Within telecommunication and routing of traffic in IP-networks a protocol named“Open Shortest Path First” (OSPF) is widely used. This means that a server dealswith the routing over a network with given weights by calculating shortest paths touse for routing. If we assume that a desired traffic pattern is given the problem isto find out if it is possible to set the weights so that the desired traffic pattern is apart of a shortest path graph. In this thesis we assume that it is a unique shortestpath. To search for weights that solve the problem leads to a complex LP-model. Analternative is to search in the LP-dual under certain restrictions. These solutions tothe LP-dual are called conflicts and a conflict means that there exists no weights sothat the desired traffic pattern is obtained. The goal of this thesis is to study, classifyand search for conflicts. An algorithm has been developed that finds some kind ofconflicts in polynomial time with respect to the size of the graph.

In this thesis we use the method of Semantic Mirrors to create a graph of words that are semantically related to a seed word. Spectral graph partitioning methods are then used to partition the graph into subgraphs, and thus dividing the words into different word senses. These methods are applied to a bilingual lexicon of English and Swedish adjectives. A panel of human evaluators have looked at a few examples, and evaluated consistency within the derived senses and synonymy with the seed word.

 The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, u_x]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0.

The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense.

The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.

The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors.

In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor.

These methods are also implemented in MATLAB.

We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.

In this paper the reflection-extension of one-dimensional quasiminimizers is studied.A brief introduction to quasiminimizers, focused on the one-dimensional ones, is given.The main result of the study concerns the size of the quasiminimizing constant of theextended function relative to the unextended one. Previous work by O. Martio gives anupper bound for this relation. This bound is lowered, and the new bound is proven to besharp.Sharp quasiminimizer constants are calculated for a few simple functions and theirreflection-extensions.

This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L¹ functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.

Image rendering is the process of creating realistic computer images from  geometric models and physical laws of light and reflection. This master thesis deals mainly with the numerical intricacies of implementing an image renderer using spherical harmonics. It investigates how to calculate the reflection of light in a surface using the Phong model, and employs ray tracing to create a realistic image of a geometric model. Further, it investigates different ways of calculating the spherical harmonic representation of a function defined on the sphere. The thesis also deals with the implementation of self-shadowing, and the effects of adding this component to the rendering equation.

Crew costs are one of the largest expenses for airlines and effective manpower planning is therefore important to maximize profit. The focus of research in the field of manpower planning for airlines has mainly been on the scheduling of crew, while other areas, surprisingly, have received very little attention. This thesis provides an overview of some of the other problems facing manpower planners, such as designing a career ladder, planning transitions and making course schedules.

Mathematical models are presented for some of theses problems, and for the problem of allocating training and vacation in time the mathematical model has been tested on data from SAS Scandinavian Airlines. When allocating training and vacation there are many aspects to consider, such as avoiding crew shortage, access to resources needed for training, and vacation laws. Comparisons between solutions obtained with the model and SAS Scandinavian Airlines manual plan show encouraging results with savings around 10%.

Many companies have traveling salesmen that market and sell their products.This results in much traveling by car due to the daily customer visits. Thiscauses costs for the company, in form of travel expenses compensation, and environmentaleffects, in form of carbon dioxide pollution. As many companies arecertified according to environmental management systems, such as ISO 14001,the environmental work becomes more and more important as the environmentalconsciousness increases every day for companies, authorities and public.The main task of this thesis is to compute reasonable limits on the mileage ofthe salesmen; these limits are based on specific conditions for each salesman’sdistrict. The objective is to implement a heuristic algorithm that optimizes thecustomer tours for an arbitrary chosen month, which will represent a “standard”month. The output of the algorithm, the computed distances, will constitute amileage limit for the salesman.The algorithm consists of a constructive heuristic that builds an initial solution,which is modified if infeasible. This solution is then improved by a local searchalgorithm preceding a genetic algorithm, which task is to improve the toursseparately.This method for computing mileage limits for traveling salesmen generates goodsolutions in form of realistic tours. The mileage limits could be improved if theinput data were more accurate and adjusted to each district, but the suggestedmethod does what it is supposed to do.

The expected equity premium is an important parameter in many financial models, especially within portfolio optimization. A good forecast of the future equity premium is therefore of great interest. In this thesis we seek to forecast the equity premium, use it in portfolio optimization and then give evidence on how sensitive the results are to estimation errors and how the impact of these can be minimized.

Linear prediction models are commonly used by practitioners to forecast the expected equity premium, this with mixed results. To only choose the model that performs the best in-sample for forecasting, does not take model uncertainty into account. Our approach is to still use linear prediction models, but also taking model uncertainty into consideration by applying Bayesian model averaging. The predictions are used in the optimization of a portfolio with risky assets to investigate how sensitive portfolio optimization is to estimation errors in the mean vector and covariance matrix. This is performed by using a Monte Carlo based heuristic called portfolio resampling.

The results show that the predictive ability of linear models is not substantially improved by taking model uncertainty into consideration. This could mean that the main problem with linear models is not model uncertainty, but rather too low predictive ability. However, we find that our approach gives better forecasts than just using the historical average as an estimate. Furthermore, we find some predictive ability in the the GDP, the short term spread and the volatility for the five years to come. Portfolio resampling proves to be useful when the input parameters in a portfolio optimization problem is suffering from vast uncertainty.

In this masters’ thesis report we present a graph theoretical method used for automatic bilingual lexicon acquisition with parallel texts. We analyze the concept of graph similarity and give an interpretation, of the parallel texts, connected to the vector space model. We represent the parallel texts by a directed, tripartite graph and from here use the corresponding adjacency matrix, A, to compute the similarity of the graph. By solving the eigenvalue problem ρS = ASAT + ATSA we obtain the self-similarity matrix S and the Perron root ρ. A rank k approximation of the self-similarity matrix is computed by implementations of the singular value decomposition and the non-negative matrix factorization algorithm GD-CLS. We construct an algorithm in order to extract the bilingual lexicon from the self-similarity matrix and apply a statistical model to estimate the precision, the correctness, of the translations in the bilingual lexicon. The best result is achieved with an application of the vector space model with a precision of about 80 %. This is a good result and can be compared with the precision of about 60 % found in the literature.

In this master's thesis, hidden Markov models (HMM) are evaluated as a tool for forecasting movements in a currency cross. With an ever increasing electronic market, making way for more automated trading, or so called algorithmic trading, there is constantly a need for new trading strategies trying to find alpha, the excess return, in the market.

HMMs are based on the well-known theories of Markov chains, but where the states are assumed hidden, governing some observable output. HMMs have mainly been used for speech recognition and communication systems, but have lately also been utilized on financial time series with encouraging results. Both discrete and continuous versions of the model will be tested, as well as single- and multivariate input data.

In addition to the basic framework, two extensions are implemented in the belief that they will further improve the prediction capabilities of the HMM. The first is a Gaussian mixture model (GMM), where one for each state assign a set of single Gaussians that are weighted together to replicate the density function of the stochastic process. This opens up for modeling non-normal distributions, which is often assumed for foreign exchange data. The second is an exponentially weighted expectation maximization (EWEM) algorithm, which takes time attenuation in consideration when re-estimating the parameters of the model. This allows for keeping old trends in mind while more recent patterns at the same time are given more attention.

Empirical results shows that the HMM using continuous emission probabilities can, for some model settings, generate acceptable returns with Sharpe ratios well over one, whilst the discrete in general performs poorly. The GMM therefore seems to be an highly needed complement to the HMM for functionality. The EWEM however does not improve results as one might have expected. Our general impression is that the predictor using HMMs that we have developed and tested is too unstable to be taken in as a trading tool on foreign exchange data, with too many factors influencing the results. More research and development is called for.

This thesis studies the allocation of stock in a two-level inventory system with stochastic demand. The system consists of one central warehouse which supplies N non-identical retailers with one single product. Customer demand occurs solely at the retailers and follows independent Poisson processes. The purpose is to investigate the value of using a more advanced allocation policy than First Come-First Serve at the central warehouse. The focus is on evaluating how well the simple First Come-First Serve assumption works in a system where the warehouse has access to real-time point-of-sale data, and where shipments are time based and consolidated for all retailers. The considered allocation policy is a myopic policy where the solution to a minimization problem, formulated as a constrained newsvendor problem, determines how the warehouse allocates its stock to the retailers. The minimization problem is solved using (a heuristic method based on) Lagrangian relaxation, and simulation is used to evaluate the average inventory holding costs and backorder costs per time unit when using the considered policy. The simulation study shows that cost savings around 1-4 percent can be expected for most system configurations. However, there were cases where savings were as high as 5 percent, as well as cases where the policy performed worse than First Come-First Serve. The study also shows that the highest cost savings are found in systems with relatively low demand, few retailers, short transportation times and a short time interval between shipments.

Penrose's inequality which relates the total mass of a space-time containing a black hole with the area of the event horizon, is a yet unproven condition that is required for the cosmic censorship hypothesis. It is believed that the inequality could be proved by using properties of the Hawking mass. This thesis gives analytical expressions for the Hawking mass in slowly rotating Kerr and Kerr-Newman space-times. It is also shown that the expressions are monotonically increasing, a result that does not contradict Penrose's inequality.

In this thesis we address the problem of how to construct an optimal algorithm for displaying banners (i.e advertisements shown on web sites). The optimization is based on the revenue each banner generates, with the aim of selecting those banners which maximize future total revenue. Banner optimality is of major importance in the e-commerce industry, in particular on web sites with heavy traffic. The 'micropayments' from showing banners add up to substantial profits due to the large volumes involved. We provide a broad, up-to-date and primarily theoretical treatment of this global optimization problem. Through a synthesis of mathematical modeling, statistical methodology and computer science we construct a stochastic 'planning algorithm'. The superiority of our algorithm is based on empirical analysis conducted by us on real internet-data at TradeDoubler AB, as well as test-results on a selection of stylized data-sets. The algorithm is flexible and adapts well to new environments.

A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.

For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.

Vid Onkologiska klinikens behandlingsmottagning ges cytostatika till de patienter som inte är inlagda vid kliniken. Arbetsförhållandena för sjuksköterskorna på mottagningen är ibland mycket påfrestande, genom att arbetsbelastningen kan vara mycket varierande, både mellan olika dagar och mellan olika sjuksköterskor. Vidare ger dagens bokningsrutiner inte någon samordning mellan sjuksköterskorna, vilket gör att mottagningen även har svårigheter att på ett bra sätt utnyttja sina behandlingsresurser.

Behandlingstiderna för patienterna bokas idag löpande. Examensarbetet syftar till att simulera förändringar i dagens verksamhet. Vi undersöker möjligheterna att genom samordnad bokning dels förbättra arbetsförhållandena för sjuksköterskorna och dels förbättra resursutnyttjandet vid mottagningen. Detta görs genom att konstruera dagsscheman för sjuksköterskorna. Arbetsförhållandena i dessa dagsschemana styrs av regler. Genom att utnyttja historisk statistik över utförda behandlingar genereras dagsscheman som uppfyller mottagningens behandlingsbehov på lång sikt. På så sätt kan även mottagningens maximala behandlingskapacitet undersökas. Dagsscheman skapas med hjälp av kolumngenerering, varvid kolumngenereringsproblemet löses med en girig heuristik.

Analyser visar att det finns stora möjligheter att med hjälp av samordnad bokning förbättra arbetsförhållandena för sjuksköterskorna, samtidigt som samma antal patienter, eller till och med fler, kan behandlas.

GPGPU is a collective term for research involving general computation on graphics cards. A modern graphics card typically provides more than ten times the computational power of an ordinary PC processor. This is a result of the high demands for speed and image quality in computer games.

This thesis investigates the possibility of exploiting this computational power for image processing purposes. Three well known methods where implemented on a graphics card: FFT (Fast Fourier Transform), KLT (Kanade Lucas Tomasi point tracking) and the generation of scale pyramids. All algorithms where successfully implemented and they are tree to ten times faster than correspondning optimized CPU implementation.

In this thesis a method for solving the Generalized Assignment Problem (GAP) is described. It is based on a reformulation of the original problem into a Set Partitioning Problem (SPP), in which the columns represent partial solutions to the original problem. For solving this problem, column generation, with systematic overgeneration of columns, is used. Conditions that guarantee that an optimal solution to a restricted SPP is optimal also in the original problem are given. In order to satisfy these conditions, not only columns with the most negative Lagrangian reduced costs need to be generated, but also others; this observation leads to the use of overgeneration of columns.

The Generalized Assignment Problem has shown to be NP-hard and therefore efficient algorithms are needed, especially for large problems. The application of the proposed method decomposes GAP into several knapsack problems via Lagrangian relaxation, and enumerates solutions to each of these problems. The solutions obtained from the knapsack problems form a Set Partitioning Problem, which consists of combining one solution from each knapsack problem to obtain a solution to the original problem. The algorithm has been tested on problems with 10 agents and 60 jobs. This leads to 10 knapsack problems, each with 60 variables.

In case of the PDE's the concept of solving by separation of variables

has a well defined meaning. One seeks a solution in a form of a

product or sum and tries to build the general solution out of these

particular solutions. There are also known systems of second order

ODE's describing potential motions and certain rigid bodies that are

considered to be separable. However, in those cases, the concept of

separation of variables is more elusive; no general definition is

given.

In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.

Syftet med det här examensarbetet var att vi ville undersöka om eleverna blev motiverade att arbeta med och lära sig matematik om den framställs på ett lustfyllt och verklighetshetsanknutet sätt. Att arbeta med matematik i projektform som ett komplement till den ordinarie matematikundervisningen skulle kunna vara ett sätt att motivera eleverna. För att få svar på våra frågeställningar planerade, utformade och genomförde vi en projektvecka i matematik på en gymnasieskola. Under veckans gång observerade vi eleverna och deras arbete. Utifrån våra observationer utvärderade vi själva projektveckan. Dessutom lät vi eleverna besvara en enkät.

När vi utformade projektveckan valde vi att utforma den som att det vore ett realityprogram som skulle sändas på TV. Vi lät eleverna arbeta med projekt i grupp och samtidigt med laborationer enskilt eller två och två. Vi valde att ge projekten och laborationerna nya namn, missions och sidospår, både med förhoppning att öka elevernas intresse och för att knyta ihop dessa med projektveckans upplägg, TV-programmet.

För att få innehållet i en projektvecka i matematik på gymnasiet verklighetsanknutet anser vi att det är viktigt att utgå från en verklig situation, istället för att utgå från ett matematiskt problem och utifrån detta arbeta fram ett projekt. Vi arbetade mycket med att få innehållet i projektveckan så verklighetsanknutet som möjligt och det anser vi att vi lyckades bra med. Vi kan även utläsa från enkäten eleverna besvarade att de upplevde matematiken i projektet verklighetsanknuten.

Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(u) ≡ u²_xu_xx + 2u_xu_yu_xy + u²_yu_yy = 0. For numerical reasons ∆q(u) = div (|▼u|q▼u) = 0, which (formally) approaches as ∆∞(u) = 0 as q → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(u) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.

Gas sensors are today used in many different application areas, and one growing future market is battery operated sensors. As many gas sensor components are heated, one major limit of the operation time is caused by the power dissipated as heat. AppliedSensor is a company that develops and produces gas sensor components, modules and solutions, among which battery operated gas sensors are one targeted market.

The aim of the diploma work has been to simulate the heat transfer on a hydrogen gas sensor component and its closest surroundings consisting of a carrier mounted on a printed circuit board. The component is heated in order to improve the performance of the gas sensing element.

Power dissipation occurs by all three modes of heat transfer; conduction from the component through bond wires and carrier to the printed circuit board as well as convection and radiation from all the surfaces. It is of interest to AppliedSensor to understand which factors influence the heat transfer. This knowledge will be used to improve different aspects of the gas sensor, such as the power consumption.

Modeling and simulation have been performed in FEMLAB, a tool for solving partial differential equations by the finite element method. The sensor system has been defined by the geometry and the material properties of the objects. The system of partial differential equations, consisting of the heat equation describing conduction and boundary conditions specifying convection and radiation, was solved and the solution was validated against experimental data.

The convection increases with the increase of hydrogen concentration. A great effort was made to finding a model for the convection. Two different approaches were taken, the first based on known theory from the area and the second on experimental data. When the first method was compared to experiments, it turned out that the theory was insufficient to describe this small system involving hydrogen, which was an unexpected but interesting result. The second method matched the experiments well. For the continuation of the project at the company, a better model of the convection would be a great improvement.

This thesis grew out of a problem encountered by a subsidiary of a Swedish multinational industrial corporation. This subsidiary is responsible for the corporation’s customer financing activities. In the thesis, we refer to these entities as the Division and the Corporation. The Division needed to find a new approach to finance its customer loan portfolio. Risk control and return maximization were important aspects of this need. The objective of this thesis is to devise and implement a method that allows the Division to make optimal funding decisions, given a certain risk limit.

We propose a funding approach based on stochastic programming. Our approach allows the Division’s portfolio manager to minimize the funding costs while hedging against market risk. We employ principal component analysis and Monte Carlo simulation to develop a multicurrency scenario generation model for interest and exchange rates. Market rate scenarios are used as input to three different optimization models. Each of the optimization models presents the optimal funding decision as positions in a unique set of financial instruments. By choosing between the optimization models, the portfolio manager can decide which financial instruments he wants to use to fund the loan portfolio.

To validate our models, we perform empirical tests on historical market data. Our results show that our optimization models have the potential to deliver sound and profitable funding decisions. In particular, we conclude that the utilization of one of our optimization models would have resulted in an increase in the Division’s net income over the past 3.5 years.

Traffic planning in a modern congested society is an important and time consuming procedure. Finding fast algorithms for solving traffic problems is therefore of great interest for traffic planners allover the world.

This thesis concerns solving the fixed demand traffic assignment problem (TAP) on a number of different transportation test networks. TAP is solved using the Frank-Wolfe algorithm and the shortest path problems that arise as subproblems to the Frank-Wolfe algorithm are solved using the network simplex algorithm. We evaluate how a number of existing pricing strategies to the network simplex algorithm performs with TAP. We also construct a new efficient pricing strategy, the Bucket Pricing Strategy, inspired by the heap implementation of Dijkstra's method for shortest path problems. This pricing strategy is, together with the actual use of the network simplex algorithm, the main result of the thesis and the pricing strategy is designed to take advantage of the special structure of TAP. In addition to performing tests on the conventional Frank-Wolfe algorithm, we also test how the different pricing strategies perform on Frank-Wolfe algorithms using conjugate and bi-conjugate search directions.

These test results show that the updating shortest path calculations obtained by using the network simplex outperforms the non-updating Frank-Wolfe algorithms. Comparisons with Bar-Gera's OBA show that our implementation, especially together with the bucket pricing strategy, also outperforms this algorithm for relative gaps down to 10E-6.

Sweden experiences heavy snowfall during the winter season and cost effective road maintenance is significantly affected by the routing of snowplows. The routing problem becomes more complex as the SwedishNational Road Administration (Vägverket) sets operational requirements such as satisfying a time window for each road segment.

This thesis focuses on route optimization for snowplows after snowfall; to develop and implement an algorithm for finding combinations of generated routes which minimize the total cost. The results are compared to those stated in the licentiate thesis by Doctoral student Nima Golbaharan (2001).

The algorithm calculates a lower bound to the problem using a Lagrangian master problem. A common subgradient approach is used to find near-optimal dual variables to be sent to a column-generation program which returns routes for the snowplows. A greedy heuristic chooses a feasible solution, which gives an upper bound to the problem. This entire process is repeated as needed.

This method for routing snowplows produces favorable results with a relatively small number of routes and are comparable to Golbaharan's results. An interesting observation involves the allocation of vehicles in which certain depots were regularly over- or under-utilized. This suggests that the quantity and/or distribution of available vehicles may not be optimal.

Detta arbete handlar om elevers kunskaper i att översätta en matematisk problemtext till en ekvation. Jag har dels studerat tidigare forskning inom området och dels gjort en egen studie. Huvudsyftet med arbetet är att ta reda på om det finns problem för elever i årskurs nio att finna en ekvation som kan lösa en bestämd uppgift och i så fall vilka är svårigheterna för eleverna. För att uppfylla syftet med detta arbete har jag valt att göra en litteraturstudie, en kvantitativ studie samt en mindre kvalitativ studie. I litteraturstudien tar jag bland annat upp vad algebra och ekvationer är för något, algebrans betydelse i skolan och vad tidigare undersökningar säger i det område jag undersöker. I min undersökning har 49 elever i årskurs nio deltagit i den skriftliga undersökningen och tre elever gjorde en gruppintervju.

Litteraturstudien och min studie avslutas med en diskussion. I diskussionen dras det slutsatser mellan min undersökning och litteraturstudien Några slutsatser som har framkommit av undersökningen och litteraturstudien är att elever i årskurs nio har svårt med att översätta ett problem till en ekvation. Det är framförallt förståelsen för bokstäver och likhetstecknets funktion i ekvationer som är svårt för eleverna. Avslutningsvis tar jag upp några didaktiska perspektiv på ekvationer som kan vara till hjälp för lärarna i undervisningen.

Intresset är lågt för matematik hos elever i grundskolans senare del visar färska undersökningar. Skolverket efterlyser bland annat variation av arbetsformer. Jag har kommit i kontakt med en lärare som arbetar med laborativ matematik som återkommande inslag i matematikundervisningen. Denna uppsats är en fallstudie av hans arbete.

Syftet med detta examensarbete är att undersöka och beskriva hur man kan utveckla och arbeta med laborativ matematik som återkommande inslag i matematikundervisningen. Frågeställningarna har varit: Vilken är lärarens definition av laborativ matematik? Hur arbetar läraren inför, under och efter en matematisk laboration och hur svarar eleverna på lärarens intentioner? Vad har läraren för erfarenhet av hur kollegor och skolledning påverkar när det gäller utveckling av laborativ matematik?

För att finna svar på dessa frågor har jag genomfört lektionsbesök och intervjuer.

Resultatet beskriver dels lärarens definition på laborativ matematik, men även hur rapportskrivning kan användas i matematikundervisning, elevers motstånd till att byta arbetsmetod och hur viktigt stöd från skolledning och kommun är i utvecklingsarbete.

Syftet med detta examensarbete var att undersöka om det är läromedlet som styr matematikundervisningen. Mina frågeställningar har varit: Hur påverkar läromedel matematikundervisningen? Hur använder olika lärare samma läromedel? För att söka svar på dessa frågor valde jag att genomföra litteraturstudier, genomgång av läromedlet Mattestegen samt observationer och intervjuer med fem olika lärare som alla jobbar med Mattestegen i åldersintegrerad verksamhet, år 4-6.

Uppsatsen bygger på en litteraturstudie, som visar på tidigare forskning om matematikundervisning och lärarens roll i matematikundervisningen. Uppsatsen bygger även på en empirisk del där läromedlet Mattestegen presenteras, observations- och intervjuresultaten redovisas, samt en analys av dessa.

Resultatet av min undersökning visar att lärarna uppfattar sig vara mycket styrda av läromedlet i sin undervisning. Det går att se en viss skillnad mellan de lärare som varit verksamma i några år, de vågar släppa läroboken och använda annat material, jämförelsevis med de lärare som är relativt nyexaminerade, de använder i stort sett inget annat än läromedlet i sin undervisning. Alla lärare i undersökningen har visioner om att de ska bli bättre på att använda annat material. Lärarna tror och litar på att läromedlet överensstämmer med kunskapsmålen, att läromedelsförfattaren har byggt upp boken på så sätt att eleverna klarar de nationella proven.

The transportation cost of logs from forest to customers is a large part of the overall cost for the Swedish forestry industry. Finding good routes from harvesting points to saw and pulp mills is a complex task, where the total number of feasible routes is extremely high. In this thesis we present two methods for log truck scheduling.

The first is to, from a given set of routes, find the most valuable subset that fulfils the customers demand. We use a model that is similar to the set partitioning problem and a method that is referred to as a composite pricing coupled with Branch and Bound. The composite pricing based method prices the routes (columns) and chooses the most valuable ones that are then added to the LP relaxation. Once an LP optimum is found, the Branch and Bound method is used to find an integer optimum solution. We have tested this on a case of realistic size.

The second method is a tabu search heuristic. Here, the purpose is to create efficient and qualitative routes from a given number of trips (referred to as predefined trips). From a start solution tabu search systematically generates new solutions. This method was tested on a small problem and on a five times larger problem to study how the size of the problem affected the result. It was also tested and compared on two cases in which the backhauling possibilities (i.e. instead of traveling empty the truck picks up another load on the return trip) had and had not been studied. The composite pricing based method and the tabu search method proved to be very useful for this kind of scheduling.

Uppsatsen ger en beskrivning av olika lärares resonemang kring hur man organiserar matematik- undervisningen under skolår 7-9. Ingår gör också lärares resonemang om vilka moment/förmågor i matematik som olika elever bör ha med sig efter avslutad grundskola. Den tredje frågan som behandlas i uppsatsen är vilka moment/förmågor inom matematiken som undervisningens organisation ger eleverna möjlighet att lära sig. Dessa frågor besvaras med hjälp av observationer på tre olika skolor och intervjuer med sex lärare på dessa tre skolor. I uppsatsen ingår en analys av de läromedel som skolorna använder sig av inom matematikundervisningen.

Ett gemensamt resultat, enligt den forskning som ingår i litteraturstudien och enligt de lärare som jag har intervjuat, är att frågan om hur man ska organisera matematikundervisningen är mycket svår att svara på. Enligt forskningen kan man hitta både för- och nackdelar med olika sätt att organisera undervisningen på. De lärare som jag har intervjuat pekar också på olika för- och nackdelar med olika sätt att organisera matematik-undervisningen. Det visar sig att läromedlet i matematik kan spela en ganska stor roll för hur man organiserar undervisningen.

The Swedish National Debt Office (SNDO) is the Swedish Government’s financial administration. It has several tasks and the main one is to manage the central government’s debt in a way that minimizes the cost with due regard to risk. The debt management problem is to choose currency composition and maturity profile - a problem made difficult because of the many stochastic factors involved.

The SNDO has created a simulation model to quantitatively analyze different aspects of this problem by evaluating a set of static strategies in a great number of simulated futures. This approach has a number of drawbacks, which might be handled by using a financial optimization approach based on Stochastic Programming.

The objective of this master’s thesis is thus to apply financial optimization on the Swedish government’s strategic debt management problem, using the SNDO’s simulation model to generate scenarios, and to evaluate this approach against a set of static strategies in fictitious future macroeconomic developments.

In this report we describe how the SNDO’s simulation model is used along with a clustering algorithm to form future scenarios, which are then used by an optimization model to find an optimal decision regarding the debt management problem.

Results of the evaluations show that our optimization approach is expected to have a lower average annual real cost, but with somewhat higher risk, than a set of static comparison strategies in a simulated future. These evaluation results are based on a risk preference set by ourselves, since the government has not expressed its risk preference quantitatively. We also conclude that financial optimization is applicable on the government debt management problem, although some work remains before the method can be incorporated into the strategic work of the SNDO.