Studie är baserad på Siegler & Ramani (2008) “Playing linear numerical board games promotes low-income child­ren´s numerical development”. Syfte är att undersöka om det med intensivträning (i form av ett linjärt tärningsspel) går att stimulera sexåriga barns talförståelse inom talområdet 1-10. Metoderna som valts är fältexperiment och observation. Fält­experimentet påvisar att intensivträning med ett tärningsspel, utfört under 4 stycken 15 minuters lektioner under en tvåveckors period, tydligt kan förbättra barns talförståelse medan observationen visar att utfallet påverkas av pedagogens olika yrkesverktyg samt undervis­ningens organisation (en-till-en undervisning). Slutsatsen är att tidig pedagogisk matematik­stimulans kan förbättra elevers förutsättningar inför skolstart. / This study is based on Siegler & Ramani (2008) “Playing linear numerical board games promotes low-income children's numerical development.” The purpose is to examine to which extent intensive training of a linear boardgame may stimulate the number concept 1-10 among six-year-olds and the factors conducive to such results. The chosen methods are experiment and  observation. The experiment shows that intensive training with a linear boardgame, done during four 15-minute lessons over a two week period clearly can improve the childrens numerical understanding. The observation also indicate the importance for outcome of the  teacher, her professional tools and the teaching organization (one-to-one teaching). The conclusion is that early maths teaching can improve the conditions for pupils beginning school

In this thesis we focus mainly on special finite differences and finite volume methods and apply them to the pricing of barrier options.The structure of this work is the following: in Chapter 1 we introduce the definitions of options and illustrate some properties of vanilla European options and exotic options.Chapter 2 describes a classical model used in the financial world, the  Black-Scholes model. We derive theBlack-Scholes formula and show how stochastic differential equations model financial instruments prices.The aim of this chapter is also to present the initial boundary value problem and the maximum principle.We discuss boundary conditions such as: the first boundary value problem, also called  Dirichlet problem that occur in pricing ofbarrier options and European options. Some kinds of put options lead to the study of a second boundary value problem (Neumann, Robin problem),while the Cauchy problem is associated with one-factor European and American options.Chapter 3 is about finite differences methods such as theta, explicit, implicit and Crank-Nicolson method, which are used forsolving partial differential equations.The exponentially fitted scheme is presented in Chapter 4. It is one of the new classesof a robust difference scheme that is stable, has good convergence and does not produce spurious oscillations.The stability is also advantage of the box method that is presented in Chapter 5.In the beginning of the Chapter 6 we illustrate barrier options and then we consider a novel finite volume discretization for apricing the above options.Chapter 7 describes discretization of the Black-Scholes equation by the fitted finite volume scheme. In  Chapter 8 we present and describe numerical results obtained by using  the finite difference methods illustrated in the previous chapters.

Ruttningen av trafik i IP-nätverk sker ofta med hjälp av bågvikter som bestämmer vilken väg trafiken tar (kortastevägruttning). Problemet här är att avgöra ifall det existerar en uppsättning vikter givet ett önskat ruttningsschema. Den hör rapporten undersöker prestandan hos ett antal modeller och optimeringsprogram avsedda att lösa denna typ av problem som ofta kallas inversa kortastevägruttningsproblemet. Undersökningen visar att det existerar en stor skillnad mellan modellerna och optimeringsprogrammen och att modellen baserad på cykelbaser löst med CPLEXdualopt lösaren är snabbast. / The routing of traffic in IP networks is often done with a set of weights that determinewhich way the traffic will go (shortest path routing). The problem here is todetermine if there exists a set of weights for a desired routing pattern. This thesis willinvestigate the performance of a number of different models and solvers for solvingthis type of problem which is usually called the inverse shortest path routing (ISPR)problem. The models tested are the same as described in [1]. The different solversused are mainly the linear CPLEX solvers but also a few multi commodity networksolvers. The tests showed that there is a big performance difference between the models andsolvers and that the cycle bases model solved with the CPLEX dualopt solver wasthe fastest overall.

Since the liberalisation of the energy market in Europe in the early 1990s, much opportunity to trade electricity as a commodity has arisen. One significant consequence of this movement is that market prices have become more volatile instead of its tradition constant rate of supply. Spot price markets have also been introduced, affecting the demand of electricity as companies now have the option to not only produce their own supply but also purchase this commodity from the market. Following the liberalisation of the energy market, hence creating a greater demand for trading of electricity and other types of energy, various types of options related to the sales, storage and transmission of electricity have consequently been introduced. Particularly, swing options are popular in the electricity market. As we know, swing-type derivatives are given in various forms and are mainly traded as over-the-counter (OTC) contracts at energy exchanges. These options offer flexibility with respect to timing and quantity. Traditionally, the Geometric Brownian Motion (GBM) model is a very popular and standard approach for modelling the risk neutral price dynamics of underlyings. However, a limitation of this model is that it has very few degrees of freedom, as it does not capture the complex behaviour of electricity prices. In short the GBM model is inefficient in the pricing of options involving electricity. Other models have subsequently been used to bridge this inadequacy, e.g. spot price models, futures price models, etc. To model risk-neutral commodity prices, there are basically two different methodologies, namely spot and futures or so-called term structure models. As swing options are usually written on spot prices, by which we mean the current price at which a particular commodity can be bought or sold at a specified time and place, it is important for us to examine these models in order to more accurately inculcate their effect on the pricing of swing options. Monte Carlo simulation is also a widely used approach for the pricing of swing options in the electricity market. Theoretically, Monte Carlo valuation relies on risk neutral valuation and the technique used is to simulate as many (random) price paths of the underlying(s) as possible, and then to average the calculated payoff for each path, discounted to today's prices, giving the value of the desired derivative. Monte Carlo methods are particularly useful in the valuation of derivatives with multiple sources of uncertainty or complicated features, like our electricity swing options in question. However, they are generally too slow to be considered a competitive form of valuation, if any analytical techniques of valuation exist. In other words, the Monte Carlo approach is, in a sense, a method of last resort. In this thesis, we aim to examine a numerical method involved in the pricing of swing options in the electricity market. We will consider an existing and widely accepted electricity price process model, use the finite volume method to formulate a numerical scheme in order to calibrate the prices of swing options and make a comparison with numerical solutions obtained using the theta-scheme. Further contributions of this thesis include a comparison of results and also a brief discussion of other possible methods.

In this thesis we investigate the case of monitoring of stocks havingjust been introduced for public trading on the nancial market. Theempirical distribution of the change-point for 20 assets for 60 days was calculated to check the support for the assumption that the priceinitially drop or rise to some steady level.The price process X = {Xt : t in Z} is assumed to be an AR(1) process with a shift in the mean value from a slope to a constant. The Shiryaev-Roberts, Shewhart, EWMA, Likelihood ratio and CUSUM proceduresfor detecting a change-point in such a process are derived. The expecteddelay of the motivated alarm according to these methods is achievedunder the assumptions of a Poisson, uniform, binomial and geometric distributed by means of simulations.

To understand and model the dynamics of the implied volatility smile is essential for trading, pricing and risk management portfolio. We suggest a  linear Kalman filter for updating of the Stochastic Volatility Inspired (SVI) model of the volatility. From a risk management perspective we generate the 1-day ahead forecast of profit and loss (P\&L) of option portfolios. We compare the estimation of the implied volatility using the SVI model with the cubic polynomial model. We find that the SVI Kalman filter has outperformed the  others.

In this paper the Heath, Jarrow and Morton (HJM) framework is applied in the programming language Java for the estimation of the future spot rate. The subcase of an exponential model for the diffusion coefficient (volatility) is used for the pricing of interest rate derivatives (caps).

This report consider a system describing three competing species with populations x, y and z. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.

Syftet med uppsatsen har varit att undersöka hur elever ser på ämnet matematik och matematikundervisning. Vår ambition har även varit att få syn på vad elever finner lustfyllt och mindre lustfyllt med matematiken och matematikundervisning. Utgångspunkten till det valda ämnet är att många elever som slutar grundskolan inte når upp till betyget Godkänd i matematik (SOU, 2004). Med hjälp av enkätfrågor ställda till 30 stycken elever i år 4, varav 7 elever som ingår i en mindre undervisningsgrupp under matematiklektioner, försökte vi finna svar på våra frågor. De 7 elever som ingår i den mindre undervisningsgruppen har även svarat på enkäten utifrån hur de upplevde matematiklektionerna när de var integrerade i den stora undervisningsgruppen. Vårt syfte med detta har varit att göra en jämförelse mellan de olika gruppernas upplevelse. Resultatet av enkätundersökningen visade att majoriteten av eleverna i den mindre undervisningsgruppen tycker att matematik är "så där", vilket handlar om att eleverna upplevde matematik både som roligt och tråkigt. Hälften av dessa elever tyckte inte om matematikundervisningen när de ingick i stor undervisningsgrupp, men som de däremot gör när de ingår i liten undervisningsgrupp. Drygt hälften av eleverna i stor undervisningsgrupp tycker om matematik och hälften av dessa elever gav svaret att de tycker om matematiklektionerna, andra hälften tyckte ”så där” om matematiklektionerna. I litteraturstudien kom vi bland annat i kontakt med Sjöbergs studie (2006) där 13 elever som efter avslutad grundskola inte nått upp till betyget godkänd i matematik, fick ge sin syn på varför de ansåg att de hamnat i matematiksvårigheter. En annan studie som också visade intresse var Engström (2006) som i sin avhandling har tittat på hur datorn kan användas som redskap i matematikundervisning. Det framkom även i litteraturstudien att orsakerna till elevers matematiksvårigheter kan orsakas av vilken förmåga som läraren har att förmedla matematikkunskaper till elever. Vi alla är olika och lär oss på olika sätt. Samt att miljön har en stor inverkan.

In this paper it is considered the Sircar-Papanicolaou model wich takesinto account a feedback effect of dynamic hedging strategies of pro-gramme traders. Using the Lie group analysis we describe the symmetrygroup of the main equation of the concerned model. We reduce this par-tial differential equation to the ordinary differential equations by usingcorresponding invariants of the subgroups of the main symmetry group.