This study aims at investigating what opportunities mathematics textbooks used in Swedish and German upper secondary schools offer to practice modeling within the topic of mathematical analysis. The main interest of the study lies in how the textbook content relates to the national curriculum goals regarding modeling, but the Swedish and German textbooks are also compared to each other. In order to achieve this objective, two textbooks, one from each country, were analyzed with respect to the amount and to the quality of available modeling tasks. In the content analysis, theory on modeling competence and realistic tasks were used. The results indicate that the two analyzed textbooks contain modeling tasks to approximately same extent. Both textbooks support exclusively intra-mathematical aspects of modeling, which is in line with the findings of previous studies. While the scope of mathematical modeling in the textbooks does not correspond to the general curriculum goals of realistic modeling, it covers partly (Sweden) or completely (Germany) the curriculum requirements for at least the lowest proficiency level of the modeling competence. / Denna studie syftar till att undersöka vilka möjligheter matematikläroböcker som används i den svenska och den tyska gymnasieskolan erbjuder för att öva modellering inom matematisk analys. Studiens intresse ligger framför allt i hur läroböckernas innehåll relaterar till de nationella styrdokumentens lärandemål om modellering, men dessutom jämförs de svenska och tyska läroböckerna även med varandra. För att uppnå detta syfte analyserades två läroböcker, en från vardera land, med hänsyn till mängden och kvaliteten av befintliga modelleringsuppgifter. I innehållsanalysen användes teori om modelleringsförmåga och realistiska uppgifter. Resultaten indikerar att de två läroböcker som analyserats innehåller modelleringsuppgifter i ungefär samma utsträckning. Båda läroböckerna stödjer endast inommatematiska aspekter av modellering, vilket är i linje med tidigare forskningsrön. Även om omfattningen av matematisk modellering i läroböckerna inte överensstämmer med läroplansmål om realistisk modellering, motsvarar de befintliga modelleringsuppgifter delvis (Sverige) eller helt (Tyskland) åtminstone de lägsta kunskapskraven som styrdokumenten har angående modelleringsförmågan.

Vissa elever ser matematik som sitt favoritämne, andra känner ängslan för att de inte känner sig lika duktiga som övriga klasskamrater. Det vore därför användbart att dämpa elevers ångest och oro genom att arbeta med elevers matematiska självkänsla i klassrummet. Frågeställningarna för föreliggande examensarbete är följande: 1. Hur definieras begreppet matematisk självkänsla inom forskningslitteraturen? 2. Hur mäts matematisk självkänsla hos elever inom forskningslitteraturen? Det framkom i resultatet att det ej finns en entydig definition av matematisk självkänsla. En övervägande majoritet av de analyserade studierna definierade matematisk självkänsla som en elevs självvärdering av sin egna kompetens i matematik. Ett fåtal studier presenterade även andra komponenter till matematisk självkänsla, såsom en elevs uppfattning om sin kompetens i jämförelse med andra samt hur elevens tidigare erfarenheter såsom elevens framgångar, motgångar och hur stor emotionell vikt eleven sätter vid dessa. Samtliga analyserade studier som mätte matematisk självkänsla använde sig av ett påståendeformulär med tillhörande Likertskala. Påståendeformulären användes för att mäta bland annat elevernas kompetens inom matematik, elevernas kompetens inom matematik i relation till andra elever samt erfarenhet av matematik. / Some students see mathematics as their favourite subject, while some feel anxiety because they don’t experience themselves as good as other students. To reduce student’s anxiety and worry by working with student’s mathematical self-esteem in the classroom might, therefore, be productive. The questions for the present thesis were as follows: 1. How is mathematical self-esteem defined in research literature? 2. How are students’ mathematical self-esteem measured in research literature? It became apparent in the result that there was no unambiguous definition of mathematical self-esteem. A clear majority of the analysed studies defined mathematical self-esteem as a student’s self-evaluation of their own skills in mathematics. A few studies also presented other components of mathematical self-esteem, such as a student’s perception of their competence in comparison with others, and how the student’s previous experiences such as the student’s successes, setbacks and how great the emotional weight the student attach to them. All analysed research studies that measured self-esteem used a questionnaire with an associated Likert scale. The questionnaires were used to measure, amongst other things, students’ competence in mathematics, students’ competence in mathematics in relation to other students and also experience of mathematics

År 2045 ska Sverige uppnå territoriella nettonollutsläpp och till år 2030 ska utsläppen från transportsektorn ha minskat med 70 % jämfört med år 2010. Sveriges vägtrafik står för en tredjedel av de totala växthusgasutsläppen. För att uppnå klimatmålen bör de mest lämpade styrmedlen och åtgärderna prioriteras. En systematisk undersökning av de faktorer som påverkat utsläppsutvecklingen kan vägleda beslutsfattare att fördela resurserna där de gör mest nytta. Dekomponeringsanalys är en potentiell metod för detta syfte då flera olika faktorers effekter kan särskiljs och mätas. Fem additiva LMDI-I dekomponeringsanalyser genomfördes på utsläppsutvecklingen av fossilt CO2 inom personbilstrafiken mellan åren 1990–2015. De faktorer som undersöktes var befolkning, bil per capita, bränsleteknologier, motorstorlekar, trafikarbete per bil, emissioner och biobränsle. Data från emissionsmodellen HBEFA, Trafikverket och SCB användes i analyserna. Under hela perioden 1990–2015 minskade CO2-utsläppen och dekomponeringsanalyserna visade att alla de ingående faktorerna påverkat utvecklingen. Sett över hela tidsperioden 1990–2015 hade faktorerna påverkat utvecklingen mest i storleksordningen trafikarbete per bil (35 %), bränsleteknologier (15 %), befolkning (15 %), bil per capita (13 %), emissioner (11 %), biobränsle (7 %) samt motorstorlekar (5 %). Procenten anger andelen som faktorn utgjorde av effekternas absoluta summa. Trafikarbete per bil, emissioner, biobränsle och motorstorlekar minskade utsläppen. Bränsleteknologier, befolkning och bil per capita ökade utsläppen. Resultaten kan användas som en indikation för vilka faktorer som kan påverka den framtida utsläppsutvecklingen mest och för vilka åtgärder bör vidtas. Åtgärderförslag är incitament för att välja mer hållbara transportsätt, öka andelen av bilar med lägre utsläpp i fordonsflottan och använda mer biobränsle. / By year 2045 Sweden shall reach zero territorial net emissions and by year 2030 the emissions from the transport sector shall be reduced by 70% compared to year 2010. In Sweden the road traffic stands for a third of the total greenhouse gas emissions. In order to achieve the climate targets, the most suited policies and actions should be prioritized. A systematic investigation into the factors that affect the change in emissions can guide decision makers to distribute resources where they contribute the most. A decomposition analysis is a potential method for this purpose since the effect of different factors can be separated and measured. Five additive LMDI-I decomposition analyses were made on the change in fossil CO2 emission from passenger cars in Sweden between year 1990–2015. The factors that were investigated were: population, vehicle per capita, fuel technologies, engine sizes, distance travelled per vehicle, emissions and biofuel share. Data from the emissions model HBEFA, the Swedish Transport Administration and Statistics Sweden were used in the analyses. During the period of year 1990–2015 the CO2 emissions were reduced, and the decomposition analyses showed that all ingoing factors affected the change. Throughout the period the factors that contributed the most were in order of size: distance travelled per vehicle (35%), fuel technologies (15%), population (15%), car per capita (13%), emissions (11%), biofuel (7%) and engine size (5%). The percentage is the share of the factor’s effect of the absolute sum of all the different effects. Distance travelled per vehicle, emissions, bio fuels and engine size reduced the emissions. Fuel technologies, population and car per capita increased the emissions. The suggestions of actions are incentive for people to use more sustainable means for transportation, increase the share of cars with lower emissions in the fleet and use more biofuel.

<p>In this thesis we present a new space-time model of interacting agents in the financial market. It is a combination of the Curie-Weiss model and a model introduced by Järpe. We investigate properties such as the critical temperature and magnetization of the system. The distribution of the Hamiltonian function is obtained and a hypothesis test of independence is derived. The results are illustrated in an example based on real data.</p>

<p>This study is based on three models, Markov model, Hidden Markov model and the Radial basis function neural network. A number of work has been done before about application of these three models to the stock market. Though, individual researchers have developed their own techniques to design and test the Radial basis function neural network. This paper aims to show the different ways and precision of applying these three models to predict price processes of the stock market. By comparing the same group of data, authors get different results. Based on Markov model, authors find a tendency of stock market in future and, the Hidden Markov model behaves better in the financial market. When the fluctuation of the stock price index is not drastic, the Radial basis function neural network has a nice prediction.</p>

<p>Detta arbete är en textanalys av hur några svenska läroböcker i matematik introducerar algebra speglat i van Hieles teorier om tankenivåer vid inlärning. Van Hieles teorier poängterar språket som kunskapsbärare i matematik vilket går som en röd tråd genom analysen. Generellt börjar läroböckerna på van Hieles tankenivå 3. Enligt van Hieles teorier borde undervisningen i algebra börja på nivå 1, vilket då blir lärarens uppgift att göra utan stöd av matematikboken. Förslag på arbetssätt för nivå 1 och 2 ingår.</p> / <p>This paper is a text analysis of how some Swedish textbooks in mathematics introduce algebra filtered by van Hiele theories of levels of thinking. The van Hiele theories emphasize that the language constitutes the knowledge objects in mathematics, wich is used all through the analysis. Generally the textbooks start at the third van Hiele’s level of thinking. According to the van Hiele theories the teaching of algebra should start at level 1. This then becomes the teachers’ task to do without the support of the textbook in mathematics. Ideas on teaching level 1 and 2 are included.</p>

<p>In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D).</p><p>In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively.</p><p>In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels.</p><p>Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity.</p><p>These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.</p>

<p>For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.</p>

<p>This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C¹-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.</p>

<p>In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers.</p><p>Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f.</p><p>In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions.</p><p>In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.</p>