In this paper two-dimensional systems of dierential equations are consideredtogether with their stabilization by a hybrid feedback control. A stabilizinghybrid control for an arbitrary controlled system that belongs to acertain category within two-dimensional systems is constructed as a result ofthis study and some stabilization proprieties of the system with the obtainedhybrid control are presented.

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This paper documents that a portfolio consisting of buy recommended stocks from SEB, Nordea and Carnegie Investment Bank, with daily rebalancing and no longer than one month time horizon, yields an annual abnormal return significantly different from zero. Less frequent rebalancing causes the abnormal return to disappear in the shorter term, but to become significant and reduced on a longer time horizon. These results provide evidence of the presence of abnormal returns on the Stockholm Stock Exchange due to buy recommendations. The intention of the study is to provide investors with a foundation upon which they can build profitable investment strategies. However, the adversities for such purpose are the higher transaction costs that follow the high rebalancing that was needed to obtain significant abnormal returns. The outcome of this study suggests that the market is efficient when transaction costs are considered.

In this paper, we investigate dynamical systems which are given by f : x 7→ x2 + c modulo a prime and for which value of the constant c can we get the largest possible cycle. We get the main ideas for finding cycles in iteration functions by introducing Floyd’s algorithm. Next, we implement the algorithms and ideas for finding cycles in Mathematica and visualize the results. In addition, we study the theoretical bounds for the length of the largest cycle and the case of c = 0 in detail. At last, we have some short discussion to this effect.

För att elever med svenska som andraspråk ska få möjligheten att utveckla sina matematiska förmågor i allmänhet och förmågan att kommunicera matematik i synnerhet, behöver lärare vara medvetna och utforma undervisningen på ett sätt som gör att eleverna ges dessa möjligheter. Syftet med studien är att undersöka hur lärare arbetar med dessa frågor i matematikklassrummet och ta reda på hur elever med svenska som andraspråk själva upplever att de får delta i kommunikationen. Studien har genomförts genom klassrumsobservationer under matematiklektioner där elever med svenska som andraspråk befunnit sig. En lärare och en elev med svenska som andraspråk har intervjuats efteråt för att få syn på deras tankar och upplevelser. Resultatet visar att eleverna anser att de får viss möjlighet att utveckla de kommunikativa förmågorna i matematik, men skulle kunna utvecklas mer genom att få använda sitt modersmål, få svara på mer öppna frågor och att samarbeta mer. / To create opportunities for second-language pupils to develop their mathematical abilities in general, and the ability to communicate mathematics in particular, teachers need to be aware and design education in such a way that students are given these opportunities. The purpose of the study is to examine how teachers actually work with these issues in the mathematic classroom, and to find out how second-language pupils feel regarding if they are able to participate in the communication. The study is conducted through classroom observations during math lessons where second-language pupils are present. One teacher and one student have been interviewed afterwards to find out their thoughts and experiences. The result shows that the students are given some opportunity to develop their communication skills in mathematics, but could be developed further by letting them use their mother tongue, let them answer more open questions and to cooperate more with each other.

Alla elever ska få det stöd de behöver i skolan för att kunna utvecklas och nå sin fulla potential. Studien genomfördes med utgångspunkt att granska hur ett sådant stöd kan se ut och hur väl stödet fungerar. Genom observationer och intervjuer har en fallstudie med en särbegåvad elev i matematik genomförts i syfte att granska stödet, hur väl stödet utvecklar elevens matematiska förmågor samt hur eleven själv uppfattar stödet. Eleven undervisas av en mentor i en liten grupp tillsammans med en annan elev. Observationerna visar att stödet i stort uppfyller de utvecklingsområden som krävs för att utveckla de matematiska förmågorna hos eleven, trots bristande arbete med problemlösning och matematiska diskussioner. Eleven själv anser att mentorstödet fungerar och har bidragit till att matematiken återigen är utmanande och rolig. Med utgångspunkt från studien kan man ställa hypotesen att lärarens kompetens och anpassningar efter individens behov och önskningar har en speciellt viktig roll att spela för att stödet till särbegåvade elever ska fungera. / All students shall receive the support they need in school to be able to develop and reach their full potential. This study was conducted on the basis of examining what that kind of support can look like and how well it’s working. Through observations and interviews, a case study involving a gifted student in mathematics has been carried out in order to examine the support, how well the support is developing the student’s mathematical abilities as well as how the student herself perceive the support. The student is educated by a tutor in a small group along with another student. The observations shows that the support for most part meets the development areas needed to develop the student’s mathematical abilities, despite the lack of work on problem solving and mathematical discussions. The student considers that the support of the tutor has contributed to making mathematics challenging and fun again. The study shows that teachers' expertise and adaptations to the individual's needs and desires are the most important to get support for gifted students to work.

I detta projekt har jag kollat på ett spel som kallas för Frysstopp. Syftet var att hitta den bästa strategin för att vinna spelet med så stor sannolikhet som möjligt. Projektet är uppdelat i två större delar. Den första delen består av matematiska definitioner och resonemang för att kunna begränsa problemet. Den andra delen består utav simuleringar som använder sig av resonemanget från den första delen, för att kunna visa andra hypoteser.

Denna uppsats handlar om representationer och moduler över ändliga supergrupper. Klassisk representationsteori lägger grunden för detta arbete. Vi ser att representationer över grupper och CG-moduler över gruppalgebror är ekvivalenta, på grund av detta studerar vi moduler över supergrupper (SG_-moduler) genom CG-moduler. Uppsatsen landar i en klassi_cering av moduler till SG_ under begränsningen att en viss G-representation (V; _) är den endimensionella triviala representationen; vi bestämmer upp till isomor_ alla ouppdelbara ändligdimensionella moduler. Inte alla dessa moduler är enkla, vissa är reducibla men ouppdelbara. / This paper is about representations and modules over nite supergroups. Classic representation theory sets the foundation of this paper. We nd that representations over groups and CG-modules over groupalgebras are equivalent, because of this we will study modules over supergroups (SG-modules) by CG-modules. The paper ends in a classication of modules to SG under the constriction that a certain G-representation (V; ) is the one-dimensional trivial representation; we recognize, up to isomorphism, all the indecomposable nite dimensional modules. Not all these modules are simple, some are reducible but indecomposable.

This thesis deals with processes on integer partitions and their limit shapes, with focus on deterministic and stochastic variants on one such process called Bulgarian solitaire. The main scientific contributions are the following. Paper I: Bulgarian solitaire is a dynamical system on integer partitions of n which converges to a unique fixed point if n=1+2+...+k is a triangular number. There are few results about the structure of the game tree, but when k tends to infinity the game tree itself converges to a structure that we are able to analyze. Its level sizes turns out to be a bisection of the Fibonacci numbers. The leaves in this tree structure are enumerated using Fibonacci numbers as well. We also demonstrate to which extent these results apply to the case when k is finite. Paper II: Bulgarian solitaire is played on n cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, σ-Bulgarian solitaire, the number of cards you pick from a pile is some function σ of the pile size, such that you pick σ(h) < h cards from a pile of size h. Here we consider a special class of such functions. Let us call σ well-behaved if σ(1) = 1 and if both σ(h) and h − σ(h) are non-decreasing functions of h. Well-behaved σ-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of n cards exists it is unique. Moreover, if piles are sorted in order of decreasing size then a configuration is convex if and only if it is a stable configuration of some well-behaved σ-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions (σ1, σ2, ...) may tend to a limit shape Φ. We show that every convex Φ with certain properties can arise as the limit shape of some sequence of well-behaved σn. For the special case when σn(h) = ceil(qnh) for 0 < qn ≤ 1 (where ceil is the ceiling function rounding upward to the nearest integer), these limit shapes are triangular (in case qn2n → 0), or exponential (in case qn2n → ∞), or interpolating between these shapes (in case qn2n → C > 0). Paper III: We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn,qn ≤ 1), played on n cards distributed in piles. In each pile, a number of cards equal to the proportion qn of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability pn, independently of other candidate cards. This generalizes Popov's random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed p as n tends to infinity. Here we let both pn and qn vary with n. We show that under the conditions qn2pnn/log n → ∞ and pnqn → 0 as n → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limit shape. Paper IV: We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.

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