I denna uppsats lägger jag fokus på att undersöka de olika matematiska uttrycksformer someleverna tillämpar när de löser ett rikt problem. Svaret söks med hjälp av empirisk data. Syftetmed arbetet är att undersöka hur några elever som går första året på gymnasiet löser ett riktproblem. Två grupper elever som går i två olika program deltar i undersökningen. Analysengjordes med hjälp av ”KLAG-matrisen”, dvs. en matris som innehåller uttrycksformerna Konkret,Logisk/språklig, Algebraisk/aritmetisk samt Grafisk/geometrisk. Resultatet av litteratur- ochempiristudien visar att oavsett hur eleverna uttrycker sig i sina lösningsförslag innehåller det alltidnågon form av algebraisk/aritmetisk uttrycksform. Detta kan bero på att det för dessa elever ärlättare att kommunicera med algebraisk/aritmetisk uttrycksform än med någon annan. Resultatetvisar också vikten av att använda problemlösning som ett medel i en lärandeprocess även för attutveckla andra förmågor. Eleverna har olika uppfattningar och gör olika tolkningar av problemet.De har olika förutsättningar och använder varierande lösningsmetoder. Detta skulle kunna varaen förklaring till varför deras användning av uttrycksformer är olika.

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In the finance world, option pricing techniques have become an appealing topic among researchers, especially for pricing American options. Valuing this option involves more factors than pricing the European style one, which makes it more computationally challenging. This is mainly because the holder of American options has the right to exercise at any time up to maturity. There are several approaches that have been proved to be efficient and applicable for maximizing the price of this type of options. A common approach is the Least squares method proposed by Longstaff and Schwartz. The purpose of this thesis is to discuss and analyze the implementation of this approach under the Multiscale Stochastic Volatility model. Since most financial markets show randomly variety of volatility, pricing the option under this model is considered necessary. A numerical study is performed to present that the Least-squares approach is indeed effective and accurate for pricing American options.

This PhD thesis consists of an introduction and eight papers, which deal with questions of the validity of some new discrete Hardy type inequalities in weighted spaces of sequences and on the cone of non-negative monotone sequences, and their applications. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of the development of Hardy type inequalities is given. In Paper 1 we find necessary and sufficient conditions on weighted sequences $\omega_i$, $i=1, 2,...,n-1$, $u$ and $v$, for which the operator $$ (S_{n}f)_i=\sum\limits_{k_1=1}^i\omega_{1,k_1}\cdots\sum\limits_{k_{n-1}=1}^{k_{n-2}} \omega_{n-1,k_{n-1}}\sum\limits_{j=1}^{k_{n-1}}f_j,~~ i\geq 1,~~~~~(1) $$ is bounded from $l_{p,v}$ to $l_{q,u}$ for $1&lt;p\leq q&lt;\infty$. In Paper 2 we prove a new discrete Hardy-type inequality $$ \|Af\|_{q,u}\leq C\|f\|_{p,v},~~~~1&lt;p\leq q&lt;\infty,~~~~~~~~~~~(2) $$ where the matrix operator $A$ is defined by $\left(Af\right)_i:=\sum\limits_{j=1}^ia_{i,j}f_j,$ ~$a_{i, j}\geq 0$, where the entries $a_{i, j}$ satisfy less restrictive additional conditions than studied before. Moreover, we study the problem of compactness for the operator $A$, and also the dual result is stated, proved and discussed. In Paper 3 we derive the necessary and sufficient conditions for inequality (2) to hold for the case $1&lt;q&lt;p&lt;\infty$. In Papers 4 and 5 we obtain criteria for the validity of the inequality (2) for slightly more general classes of matrix operators $A$ defined by $\left(Af\right)_j:=\sum\limits_{i=j}^\infty a_{i,j}f_i,$ ~$a_{i, j}\geq 0$, when $1&lt;p, q&lt;\infty$. Moreover, we study the problem of compactness for the operator $A$ for the case $1&lt;p\leq q&lt;\infty$, also the dual result is established and here we also give some applications of the main results. In Paper 6 we state boundedness for the operator of multiple summation with weights (1) in weighted sequence spaces for the case $1&lt;q&lt;p&lt;\infty$. Paper 7 deals with new Hardy-type inequalities restricted to the cones of monotone sequences. The case $1&lt;q&lt;p&lt;\infty$ is considered. Also some applications related to H\"{o}lder's summation method are pointed out. In Paper 8 we obtain necessary and sufficient conditions for which three-weighted Hardy type inequalities hold. / <p>Godkänd; 2015; 20151021 (aintem); Nedanstående person kommer att disputera för avläggande av teknologie doktorsexamen. Namn: Ainur Temirkhanova Ämne: Matematik/Mathematics Avhandling: Estimates for Discrete Hardy-type Operators in Weighted Sequence Spaces Opponent: Professor Mikhail Goldman, Dept of Nonlinear Analysis and Optimization, Peoples’Friendship University of Russia, Moscow, Russia. Ordförande: Professor Peter Wall, Avd för matematiska vetenskaper, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet, Luleå. Tid: Tisdag 8 december kl 10.00 Plats: E246, Luleå tekniska universitet</p>

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In this work we take interest in frames and modulation spaces. On the basis of their properties, we show how frame expansions can be used to prove the boundedness of a particular class of Hilbert operators on modulation spaces taking advantage of the special category of piece-wise polynomial functions known as B-splines.

This thesis concerns the Wiener algebra of periodic functions with absolutely convergent Fourier series, and the "Wiener-Lévy functions", meaning functions that preserve absolute convergence of Fourier series under composition. Results regarding the properties of functions in the Wiener algebra are established, as well as properties of Wiener-Lévy functions. By the Wiener-Lévy Theorem, analyticity is a sufficient condition for a function to be a Wiener-Lévy function, and in this paper, we find the necessary condition of being real analytic in the real and imaginary variable, separately.