Klockan har visat sig medföra svårigheter för eleverna att förstå eftersom den i grunden bygger på ett sexbas decimaltalsystem (talbas 60) och därmed frångår det 10-bassystem som är allmänt känt och förekommer i grundläggande skolmatematik. Detta ställer krav på dagens läromedel att tillhandahålla ett material som på olika sätt skapa förutsättningar för lärande inom området klockan. Syftet med studien är att belysa hur olika läromedel presentera klockan utifrån de fem representationsformerna konkret modell, bild, språk, symboler samt verklighet. Vidare syftar studien på att undersöka hur dessa i sin tur samverkar. Genom en innehållsanalys av läromedlet Favorit matematik tre böcker för årskurs ett till tre, har studien visat att läromedlet presenterar ämnesområdet klockan med varierande uppgifter där olika representationsformer används. Studien visar även att representationsformerna kombineras i större utsträckning ju längre i bokserien eleverna kommer. Med hjälp av detta kan läraren få kunskap om läromedlets upplägg och innehåll och där med har möjlighet att komplettera undervisning med klockan där representationsformerna är underrepresenterade.

This thesis is concerned with the problem of characterizing sums, differences, and products of two projections on a separable Hilbert space. Other objective is characterizing the Moore-Penrose and the Drazin inverse for pairs of operators. We use reasoning similar to one presented in the famous P. Halmos’ two projection theorem: using matrix representation of two orthogonal projection depending on the relations between their ranges and null-spaces gives us simpler form of their matrices and allows us to involve matrix theory in solving problems. We extend research to idempotents, generalized and hypergeneralized projections and their combinations.

Layer potential operators associated to elliptic partial differential equations have been an object of investigation for more than a century, due to their contribution in the solution of boundary value problems through integral equations. In this Licentiate thesis we prove the boundedness of the double layer potential operator on the Hilbert space of square integrable functions on the boundary, associated to second order uniformly elliptic equations in divergence form in the upper half-space, with real, possibly non-symmetric, bounded measurable coefficients, that do not depend on the variable transversal to the boundary. This uses functional calculus of bisectorial operators and is done through a series of four steps. The first step consists of reformulating the second order partial differential equation as an equivalent first order vector-valued ordinary differential equation in the upper halfspace. This ordinary differential equation has a particularly simple form and it is here that the bisectorial operator corresponding to the original divergence form equation appears as an infinitesimal generator. Solving this ordinary differential through functional calculus comprises the second step. This is done with the help of the holomorphic semigroup associated to the restriction of the bisectorial operator to an appropriate spectral subspace; the restriction of the operator is a sectorial operator and the holomorphic semigroup is well-defined on the spectral subspace. The third step is the construction of the fundamental solution to the original divergence form equation. The behaviour of this fundamental solution is analogous to the behaviour of the fundamental solution to the classical Laplace equation and its conormal gradient of the adjoint fundamental solution is used as the kernel of the double layer potential operator. This third step is of a different nature than the others, insofar as it does not involve tools from functional calculus. In the last step Green’s formula for solutions of the divergence form partial differential equation is used to give a concrete integral representation of the solutions to the divergence form equation. Identifying this Green’s formula with the abstract formula derived by functional calculus yields the sought-after boundedness of the double layer potential operator, for coefficients of the particular form mentioned above.

Let  be a bounded domain in Rn with a Lipschitz boundary Г divided into two parts Г0 and Г1 which do not intersect one another and have a common Lipschitz boundary. We consider the following Cauchy problem for the Helmholtz equation: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cbegin%7Bcases%7D%5CDelta%20u%20+%20k%5E2%20u%20=%200%20&amp;%20%5Cquad%20%5Cmbox%7Bin%7D%20%5Cquad%20%5COmega,%5C%5Cu%20=%20f%20&amp;%20%5Cquad%20%5Cmbox%7Bon%7D%20%5Cquad%20%5CGamma_0,%5C%5C%5Cpartial_%7B%5Cnu%7D%20u%20=%20g%20&amp;%20%5Cquad%20%5Cmbox%7Bon%7D%20%5Cquad%20%5CGamma_0,%5Cend%7Bcases%7D" /> where k, the wave number, is a positive real constant, аv denotes the outward normal derivative, and f and g are specified Cauchy data on Г0. This problem is ill–posed in the sense that small errors in the Cauchy data f and g may blow up and cause a large error in the solution. Alternating iterative algorithms for solving this problem are developed and studied. These algorithms are based on the alternating iterative schemes suggested by V.A. Kozlov and V. Maz’ya for solving ill–posed problems. Since these original alternating iterative algorithms diverge for large values of the constant k2 in the Helmholtz equation, we develop a modification of the alterating iterative algorithms that converges for all k2. We also perform numerical experiments that confirm that the proposed modification works.

This thesis focuses on the problem of estimating parameters in multivariate linear models where particularly the mean has a bilinear structure and the covariance matrix has a linear structure. Most of techniques in statistical modeling rely on the assumption that data were generated from the normal distribution. Whereas real data may not be exactly normal, the normal distributions serve as a useful approximation to the true distribution. The modeling of normally distributed data relies heavily on the estimation of the mean and the covariance matrix. The interest of considering various structures for the covariance matrices in different statistical models is partly driven by the idea that altering the covariance structure of a parametric model alters the variances of the model’s estimated mean parameters. The extended growth curve model with two terms and a linearly structured covariance matrix is considered. In general there is no problem to estimate the covariance matrix when it is completely unknown. However, problems arise when one has to take into account that there exists a structure generated by a few number of parameters. An estimation procedure that handles linear structured covariance matrices is proposed. The idea is first to estimate the covariance matrix when it should be used to define an inner product in a regression space and thereafter reestimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into three orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed and numerical examples are given. The models based on normally distributed random matrix are also studied in this thesis. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimation equations in a flip-flop relation are presented and numerical examples are given.

No description available.

The canonical stucture information, i.e. the elementary divisors and minimalindices of a matrix polynomial, is sensitive to perturbations of the matrixcoefficients of the polynomial, e.g., the eigenvalues may change or disappear.Passing to a strong linearization is a way to solve a number of problems formatrix polynomials, the linearization then has the same finite and infiniteelementary divisors and the change in minimal indices is known. However,when the linearization is perturbed by a full perturbation the correspondencebetween the linearization and matrix polynomial is lost, hence weseek a method to restore a matrix polynomial that corresponds to perturbedlinearization. Therefore we present a numerical method for computing theperturbation of a matrix polynomial from a full perturbation of its linearization.Our method is iterative and requires of solving a system of coupledSylvester equations. We limit the method to symmetric matrix polynomials.

Denna studie är ett examensarbete inom speciallärarprogrammet riktat mot matematikutveckling. Utifrån författarnas egna erfarenheter av matematikundervisning, identifierades ett problem i att elever ofta har svårt med matematisk problemlösning i textuppgifter. Detta gäller i synnerhet elever som befinner sig i matematiksvårigheter. En metod som i tidigare forskning har visat ge positiva resultat på matematisk problemlösning i textuppgifter är arbete med Blockmodellen. Studien har en kvalitativ ansats med syfte att skapa förståelse för hur blockmodellen kan användas för att främja elevernas problemlösningsförmåga i textuppgifter, samt genom denna förståelse utveckla undervisningen för elever i matematiksvårigheter. Deltagarna i studien var sex elever i årskurs fem, som samtliga befann sig i matematiksvårigheter. För att uppnå resultat som är direkt användbara i undervisningen valdes metoden designresearch där tre lektionstillfällen med Blockmodellen dokumenterades genom videoobservation, ljudupptagning och deltagande observation. Till studien valdes ett konceptuellt ramverk som bygger på flera erkända forskares teorier, bl a Vygotsky, Bruner, Skemp och Pòlya. Dessa teorier användes i analysen inför varje lektionstillfälle för att utveckla de designprinciper som är studiens resultat.

Stochastic volatility models have become essential for financial modelling and forecasting.The present thesis works with a two-factor stochastic volatility model that is reduced to four parameters. We start by making the case for the model that best fits data, use that modelto produce said parameters and then analyse the time series of these parameters. Suitable ARIMA models were then used to forecast the parameters and in turn, the implied volatilities.It was established that fitting the model for different groups of maturities produced better results. Moreover, we managed to reduce the forecasting errors by forecasting according to the different maturity groups.

Background: The last two decades have seen several new methods for analysing gene expression data. One such method is pathway analysis in which it’s possible to decipher response patterns between pathways (groups of interacting genes) and external stress factors. A critical issue in pathway analysis is often limited sample sizes resulting in undesirable batch affects. A method for reducing such effects, called Centralization Within Sub-Experiments (CSE), has recently been developed. This method makes it possible to predict pathways using data from experiments with a diversity of external conditions. Aims: We propose a method for identifying stress-responsive pathways predicted using CSE pre-processed expression data. Method: 27 CSE predicted pathways were analysed and tested for stress-responsiveness associated with specific external stress factors. Firstly, we screened the complete gene list for DE genes. Secondly, we analysed the occurrences of DE genes within each pathway and finally, an over-representation analysis was performed with the aim of identifying pathways with significantly larger portions of differently expressed genes categorised as stress-responsive. Results: The analysis resulted in a list of pathways with significantly larger proportions of DE genes. This was only when screened for salt-responsive pathways. We also found that these pathways contained several mitochondrial genes confirmed to be associated with salt stress. Conclusion: The results show great promise in using the method for extracting information regarding the stress responsiveness amongst pathways predicted using CSE-pre-processing expression data.