The main component of this essay is the numerical analysis of stochastic differential equations driven by Wiener and Poisson processes. In order to do this, we focus on two model problems, the geometric Brownian motion and the linear stochastic oscillator, studied in the literature for stochastic differential equations only driven by a Wiener process. This essay covers theoretical as well as numerical investigations of jump - or more specifically, Poisson - processes and how they influence the above model problems. / Den huvudsakliga komponenten av uppsatsen är en numerisk analys av stokastiska differentialekvationer drivna av Wiener- och Poisson-processer. För att göra det så fokuserar vi på två modellproblem, den geometriska Brownska rörelsen samt den linjära stokastiska oscillatorn, studerade i litteratur för stokastiska differentialekvationer som bara drivs av en Wiener-process.Den här uppsatsen täcker teoretiska samt numeriska undersökningar av hopp - eller mer specifikt, Poisson - processer och hur de påverkar de ovan nämnda modellproblemen.

In this thesis Clarkson's inequalities and their generalizations are the main tools. The technique that can be used to prove Clarkson type inequalities in more dimensions is shown. We also establish Clarkson type inequalities in general Banach spaces and point out the connections between Clarkson's inequalities and the concept of type and cotype. The classical results on the von Neumann-Jordan constant, closely related to Clarkson's inequalities, are shortly presented. The concepts of moduli of convexity and smoothness, which are connected with the geometry of Banach spaces, are discussed. Some equivalent ways of describing modulus of convexity and some properties of this function are formulated. The estimation of the modulus of convexity for L(p)-spaces is presented as well. Finally, several examples of moduli of convexity and smoothness for different spaces are described. / <p>Godkänd; 1999; 20070320 (ysko)</p>

Many results in real and complex analysis are the consequence of mean value properties and theorems. This is the case for harmonic and holomorphic functions as well. The mean value property builds the foundation for several properties of each set of functions. Using this property one can derive more properties like the maximum principle for harmonic functions and the maximum modulus principle for holomorphic functions. These results are then used to show other properties. The goal is to compare the theorems and proofs for harmonic and holomorphic functions and to understand why the results seem to be similar.

Mathematics could be described as a special kind of language, a discourse in which participants communicate abstract concepts. Communication and language are known to be important factors in teaching and learning but little is known about neither communication between students in relation to geometrical concepts, nor what traces of mathematical facts and understanding is incorporated into their communication. Such knowledge would contribute to teachers´ understanding of students’ knowledge of this aspect of mathematical thinking. The purpose of this study is to investigate young students’ communication in the Discourse of geometry. To accomplish this, Swedish students aged between ten and eleven were video-recorded while working with two assignments, one involving describing quadrilaterals to each other and the other sorting quadrilaterals. An analysis of the students’ communication has identified a number of issues that teachers could benefit from being aware of in their teaching of the concept of `geometrical figure´. Examples of issues identified being that conceptual characteristic remain too much in the background of what in the study has been called the visual shape. For example, a long and narrow shape of a geometrical figure seems to draw students’ attention. Students also seem to be having difficulty in understanding that naming a figure is a way of merging the figures´ definitions into one word, and there were problems with words that anchor in daily life. / Matematik kan beskrivas som en speciell typ av språk, en diskurs i vilken deltagarna producerar berättelser för att kommunicera abstrakta begrepp. Kommunikation och språk är viktiga faktorer i lärande och undervisning men lite är känt om hur berättelser skapade av elever i relation till geometriska begrepp bidrar till deras kunskap om denna aspekt av matematiskt tänkande. Syftet med denna studie är att undersöka berättelser producerade av unga elever i diskursen geometri. För att genomföra detta videofilmades svenska elever i åldern 10 till 11 år under arbetet med två uppgifter, en där uppgiften var att beskriva fyrhörningar för varandra och en där uppgiften var att sortera fyrhörningar. En analys av de berättelser eleverna producerade identifierade ett antal problem som lärare behöver vara medvetna om och hantera i undervisningen av begreppet `geometrisk figur´. Exempel på identifierade problem är att konceptuella kännetecken alltför mycket hamnar i bakgrunden av vad som i studien kallats för "Den visuella formen", till exempel så verkar en lång och smal form dra till sig elevernas uppmärksamhet. Eleverna verkar också ha problem med att förstå att namnet på en geometrisk figur innebär att man fogat samman figurens konceptuella kännetecken i ett enda ord. Studien visar också på problem med användningen av vardagsord. / <p>Godkänd; 2015; 20150915 (evju)</p>

Rogue waves are unexpectedly high waves of 2.5X the significant wave height and which occur in nearly all phases of nature, from  oceans, to fiber-optic cables and atmospheric air-masses. In the ocean, rogue waves pose a significant danger to shipping and fishing vessels and have been found to reach 27.8 meters in height, and attain velocities of up to 100 km/hr. Mechanisms on naval structures for the real-time prediction of rogue waves are currently non-existent, and their development requires a) a good equation for simulating rogue waves and b) a deep study of the wave-trains of rogue waves. In this work, we consider the time-series of four rogue wave trains collected from various sources, including the U.S. Coastal Data Information Program. The method of study encompasses the development piece-wise constant functions from the rogue wave readings by laser/buoys. We use these piece-wise constant functions to form regularized functions as Fourier series, which we consider as weak solutions to the stationary nonlinear Schrödinger equation. The resulting force functions are quantified and compared to physical data of the rogue wave trains. The results show that we obtain a good correlation between the norms of the obtained force functions and the rogue wave height $H_{max}$ and the wave-velocity. The methods developed in the study build a potentially useful foundation for the development of a prediction model in a future study.

This thesis deals with social and gender equity within mathematics education, two sub questions have been explored.Social equity is investigated in the paper Kapitaltillgångar enligt Bourdieu och synen på matematik (Capital assets according to Bourdieu and the perception of mathematics). The main research questionis the following: Are connections on the individual level between success in mathematics studies and the economic, cultural and social background mediated by the perceptions of mathematics and knowledge of mathematics?How are the various capital resources according to Bourdieu’s theories influencing the view of mathematics as a cultural, applied or theoretical subject? The study uses quantitative methods withdata from a survey among students at upper secondary level, all studying optional mathematics courses. In order to identify perceptions of mathematics three aspects were used: the history ofmathematics, mathematical modelling and the inner structure of mathematics. Capital resources wereestimated from the answers to certain key questions of the survey. A statistical analysis showed that the method could be used as a simplified way of evaluating the different dimensions of the capitalassets. The connection between capital assets and the perception of mathematics was analysed with statistical methods.The study indicates a positive relation between capital assets and success in mathematics studies, evaluated from insights into the three aspects.Gender equity is treated in the two conference papers Is mathematics still a male domain? and Mathematics – a male domain?. These build on data, analysis and results from the Gender and Mathematics project (GeMa). The main research issues in the project concern the perception ofmathematics as a male, female or gender neutral and gender differences in the perception. Aregendered views of mathematics as male or female common among students in lower and upper secondary level? Are there connections between such perceptions and the fact that girls to some extent avoid mathematics when choosing study program at upper secondary level and tertiary level? The study is based on data from two large-scale questionnaire studies among students in year nine (lowersecondary level) and year eleven (upper secondary level) with varying socioeconomic andgeographical and educational background in Sweden.The results show that a majority of students have a gender neutral view of mathematics. On the other hand, considerable minorities view mathematics as either male (most often) or female, e.g. that mathematics is perceived as more important for boys or that girls more often find mathematics difficult and boring. There is no connection between girls’ choice of study program and perceptions ofmathematics as male or female. Contrary to this, boys choosing mathematics intensive programs more often tend to perceive mathematics as a male domain than boys making other choices.The sub question of social equity is part of a research project called SOMA. In the introductory sectionof the thesis – the “kappa” – the SOMA-project dominates the text. As the sole author and project leader, I have chosen to write in more detail about SOMA. One additional reason is that the GeMa project is previously extensively reported in journal articles and reports. / Avhandlingen handlar om jämlikhet inom matematikutbildning oberoende av social bakgrund och kön. Inom dessa områden har två delfrågor studerats närmare. Jämlikhet oberoende av social bakgrund behandlas i artikeln Kapitaltillgångar enligt Bourdieu och synen på matematik, där den övergripande forskningsfrågan är följande. Förmedlas sambanden mellanmatematikframgång och den ekonomiska, kulturella och sociala bakgrunden genom synen på och relationen till matematikämnet och matematisk kunskap?Hur kommer de olika kapitaltillgångarna, i enlighet med Bourdieus teorier, till uttryck i synen på matematik som ett kulturellt, praktiskt eller teoretiskt ämne? I studien, som är kvantitativ, används data från en enkätundersökning bland elever i gymnasieskolan som alla valt att läsa extra matematikkurser.För att bedöma synen på matematik som kulturellt, praktiskt och/eller teoretiskt ämne, användes i undersökningen följande tre aspekter på ämnet: matematikens historia, matematisk modelleringsamt matematikens inre struktur. Kapitaltillgångarna uppskattades utifrån svar på vissa nyckelfrågor. Det visade sig vid den statistiska analysen att metoden kunde användas som en förenklad metod för att mäta de sökta dimensionerna i kapitaltillgångarna. Även sambandet mellan kapitaltillgångarna ochsynen på matematiken analyserades statistiskt. Studien visar på ett samband mellan positiva kapitaltillgångar och framgång, bedömd med insikter om de tre aspekterna. Jämlikhet oberoende av kön behandlas i de två konferensbidragen, Is mathematics still a male domain? och Mathematics – a male domain?. Dessa bygger på Gender and Mathematics (GeMaprojektets)undersökningar och resultat. GeMa-projektets övergripande forskningsfrågor handlar om flickors/kvinnors och pojkars/mäns syn på matematikämnet som manligt, kvinnligt eller könsneutralt.Finns det skillnader i synen på matematik som ett könsneutralt område? Går det att hitta samband mellan det faktum att flickor, när möjlighet ges att välja, studerar mindre matematik än pojkar – och en syn på matematikämnet som ett manligt område? Undersökningens fokus ligger på vilka attityder somkan urskiljas - inte varför olika grupper visar olika attityder. Studien omfattade två omfattande enkätundersökningar bland elever från olika delar av landet och från grundskolans högstadium och två teoretiska program vid gymnasieskolan.Resultaten visade att en majoritet av eleverna betraktade matematiken som könsneutral men att det fanns grupper av elever som könsmärkte delar av matematiken som manlig eller kvinnlig. Betydande minoriteter ansåg exempelvis att det är vanligare att pojkar tycker om utmanande matematikproblem, tycker att matematik är lätt och behöver matematik i det framtida yrkeslivet. Flickor/kvinnor som grupp associerades, främst av sig själva, med negativa påståenden som att tycka matematik är tråkigtoch svårt. Å andra sidan ansågs flickor/kvinnor arbeta hårt med matematiken. För flickor fanns inget samband mellan könsmärkning av ämnet och valet av gymnasieprogram. Däremot fanns det för pojkar ett samband mellan synen på matematiken som ett manligt område och valet av matematikintensivagymnasieprogram. Delfrågan om social bakgrund ingår i det så kallade SOMA-projektet, SOciala faktorer och MAtematik. I avhandlingens inledande kappa har SOMA-projektet fått större utrymme än GeMaprojektet.Detta motiveras dels av att studien är min egen dels av att GeMa-projektet utkommit med både rapporter och artiklar där tidigare forskning, teorier, resultat med mera finns utförligt beskrivna.

The aim of this study is to explore Swedish upper elementary school students’experiences of mathematics teachers’ assessment practices, with a focus on educationalgoals communicated between the teacher and the class. In this study, students’experiences in three cases are viewed from a holistic perspective, by adapting Visualmodel of the curriculum policy, design and enactment system by Remillards and Heck(2014) to a Swedish context. Student’s perceptions are analysed and compared with theteacher’s view on educational goals and the implemented teaching. I use a multiple-casestudy method with an ethnographic approach suitable in a qualitative explorative studythat investigates a phenomenon in depth within its real-life context, in this study bytriangulation the empirical data.Results of the study show the complexity of communication about educational goals andeach case’s unique context. There is considerable variation between the three casesaccording to communication about educational goals, in relation to the syllabus inmathematics. It appears that in cases in which teachers express lack of knowledge ofsyllabus content, students get less information about teaching goals based on the syllabuscontent. Instead students express that the teacher assesses such matters as how much theywork in class and if they have clear presentations and responses of tasks. This is alsocommunicated by the teachers during lessons. Students in all three cases express andshow a lack of knowledge of syllabus in mathematics, especially when it comes to themathematical abilities and knowledge requirements, even in the case in which studentsreceive a lot of information. It is obvious that the textbooks influence is larger than theexplicit impact from the syllabus. From a student perspective the textbook is consideredas a concretization and visualization of the syllabus content. Teachers’ expressed lack ofknowledge about the syllabus in mathematics is probably due to insufficientimplementation efforts of the curriculum, Lgr11. With this study I contribute with newknowledge of assessment practices in Swedish classrooms and thereby hopefully improvedecision makers’, teachers’ and students’ awareness and knowledge of teachers’assessment practice.

This Licentiate thesis deals with Hardy-type inequalities restricted to cones of monotone functions. The thesis consists of two papers (paper A and paper B) and an introduction which gives an overview to this specific field of functional analysis and also serves to put the papers into a more general frame.We deal with positive $\sigma $-finite Borel measures on ${\mathbbR}_{+}:=[0,\infty)$ and the class $\mathfrak{M}\downarrow $($\mathfrak{M}\uparrow $) consisting of all non-increasing(non-decreasing) Borel functions $f\colon{\mathbbR}_{+}\rightarrow[0,+\infty ]$.In paper A some two-sided inequalities for Hardy operators on thecones of monotone functions are proved. The idea to study suchequivalences follows from the Hardy inequality$$\left( \int_{[0,\infty)}f^pd\lambda\right)^{\frac{1}{p}}\le \left(\int_{[0,\infty)} \left( \frac{1}{\Lambda(x)} \int_{[0,x]}f(t)d\lambda(t)\right)^p d\lambda(x)\right)^{\frac{1}{p}}$$$$\leq \frac{p}{p-1}\left(\int_{[0,\infty)}f^pd\lambda\right)^{\frac{1}{p}},$$which holds for any $f\in \mathfrak{M}\downarrow$ and $1<p<\infty.$In the paper similar equivalences are found for some otherHardy-type operators for the full range of parameter $p,\, p\neq 0.$ As anapplication of one of the results, we also obtain a newcharacterization of the discrete inequality for one of the mostinteresting cases of parameters, namely when $0<q<p\leq 1.$The equivalences we have proved in paper A are used in paper B to obtainnecessary and sufficient conditions for some other Hardy-typeinequalities on cones of monotone functions. In particular, wegive a complete description for inequalities with Volterra integraloperators involving Oinarov's kernel for the parameters$0<p<\infty,\,\,1\leq q<\infty.$ We also study inequalities of the form$$\left(\int_{[0,\infty)}{\left(\int_{[x,\infty)}fd\lambda\right)}^q d\lambda(x)\right)^\frac{1}{q} \leq C\left(\int_{[0,\infty)}f^p d\mu\right)^\frac{1}{p},\,\,\,f \in \mathfrak{M}\downarrow,\, f\not\equiv 0$$and$$\left(\int_{[0,\infty]}\left(\int_{[0,x]}fd\lambda\right)^qd\lambda(x)\right)^\frac{1}{q}\leq C\left(\int_{[0,\infty)}f^pd\mu\right)^\frac{1}{p},\,\,\,f \in \mathfrak{M}\uparrow,\, f\not\equiv 0$$and find necessary and sufficient conditions not only for positive, but also for negativeparameters of summation.

The exploration of complex systems is a fundamental pursuit in various scientific disciplines, includingphysics, biology, finance and engineering. The inherent complexity and dynamics within these systemspose significant challenges for traditional analytical methods. In recent years, the emergence of Topological Data Analysis (TDA) has provided a promising framework for uncovering hidden structures andpatterns in dynamic data sets. This thesis investigates the application of Topological Data Analysis to analyze system dynamics,aiming to enhance our understanding of their behavior. Through a detailed review of existing literature,we examine the theoretical foundations of TDA and its relevance to discrete and continuous processes.We discuss conceptual underpinnings of persistent homology, a key technique in TDA, and its potentialfor capturing essential features of system dynamics. By applying TDA to two distinct models, thestochastic ODE and the discrete logistic equation, we demonstrate its effectiveness in revealing underlyingstructures that traditional methods might overlook, thereby offering new insights into the analysis ofstochastic and discrete dynamical systems.

Alla elever måste någon gång möta det matematiska språket i skrift och detta är något som på grund av språkförbristningar kan bli ett problem för andraspråksele-ver. Syftet med denna studie är att få kunskap om hur lärare i förberedelseklass resonerar om att göra matematiska textuppgifter tillgängliga för andraspråkselever. För att undersöka detta har metoderna läromedelsanalys samt fokusgruppintervju med lärare i förberedelseklass tillämpats. Det insamlade materialet har analyserats genom en innehållsanalys med koppling till Cummins (2000) teorier samt Noréns (2010) teorier om matematikundervisning för andraspråkselever. Resultatet visar att en väl genomtänkt planering av undervisningen är nödvändigt för att andra-språkselever ska få möjlighet att utveckla förståelsen för det textbaserade matema-tiska språket. Resultatet visar också att textuppgifternas utformning med text och bild är av betydelse för hur andraspråkselever uppfattar innehållet. Ett par slutsat-ser utifrån denna studies resultat är för det första att det textbaserade matematiska språket behöver lyftas muntligt för andraspråkselever. Den andra slutsatsen utifrån resultatet är vikten av lärares uppmärksamhet gällande de många hinder som kan finnas dolda i matematiska textuppgifter. / <p>matematik</p>