This thesis treats different aspects of time-frequency analysis and pseudodifferential operators, with particular emphasis on techniques involving vector-valued functions and operator-valued symbols. The vector (Banach) space is either motivated by an application as in Paper I, where it is a space of stochastic variables, or is part of a general problem as in Paper II, or arises naturally from problems for scalar-valued operators and function spaces, as in Paper V. Paper III and IV fall outside this framework and treats algebraic aspects of time-frequency analysis and pseudodifferential operators for scalar-valued symbols and functions that are members of modulation spaces. Paper IV builds upon Paper III and applies the results to a filtering problem for second-order stochastic processes. Paper I treats the Wigner distribution of a Gaussian weakly harmonizable stochastic process defned on Rd. Paper II extends recent continuity results for pseudodifferential and localization operators, with symbols in modulation spaces, to the vector/operator-framework, where the vector space is a Hilbert or a Banach space. In Paper III we give algebraic results for the Weyl product acting on modulation spaces. We give suffcient conditions for a weighted modulation space to be an algebra under theWeyl product, and we also give necessary conditions for unweighted modulation spaces. In Paper IV we discretize the results of Paper III by means of a Gabor frame delined by a Gaussian function. Finally, Paper V deals with pseudodifferential operators with symbols that are almost periodic in the first variable. We show that such operators may be transformed to Fourier multiplier operators with operator- valued symbols such that the transformation preserves positivity and operator composition.

The research interest in this thesis concerns novice university students of mathematics. The aim of my thesis is to understand the transition between mathematics studies at upper secondary school and university from a student perspective. It is a qualitative study of five teacher students during their initial university studies in mathematics. The students were interviewed before beginning their mathematics studies, during their courses and observed during lectures, during problem solving lessons and while working with mathematics out of class both individually and with fellow students. The transcriptions from the preliminary interviews have been analyzed using methods inspired by Grounded Theory. The main result consists of three categories that describe students’ learning of mathematics. Mathematical learning objects refer to students’ conceptions of the main goals and/or contents of studying mathematics, for example their views of the subject, the usefulness of knowing mathematics and what is needed to learn mathematics. Students’ mathematical learning objects should be considered as an individual and relational phenomenon between the student and his or hers mathematics studies that can change over time. Mathematical resources relate to phenomena and objects that students use to approach and possibly acquire various mathematical learning objects. The teacher, fellow students, the textbook and own preliminary knowledge are some examples of entities that students may use as mathematical resources. These entities become mathematical resources only when the students use them in relation to mathematical learning objects.  The third category, student acting as a learner, captures students’ actions, intentions and conceptions in relation to learning mathematics. Students use different mathematical resources to acquire different mathematical learning objects. This is expressed in students’ actions and reasoning, which serve as empirical data for understanding the student acting as a learner. The three categories are theoretical concepts that elucidate essential aspects of learning and teaching of mathematics as seen from a student perspective and serve as a foundation for a theoretical framework that may be used to describe the transition. A theoretical description of the transition has been obtained by investigating how the respective categories, and the relations between the categories, change or endure during the transition. Results from the analysis of subsequent interviews and observations indicate that the transition can be understood in terms of an increasing gap between mathematical learning objects and mathematical resources, a reorientation of mathematical learning objects and a requirement of supplementing mathematical resources.

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Syftet med mitt arbete var att ta reda på pedagogernas syn på laborativ matematik i undervisningen, om de såg en kunskapsutveckling hos eleverna när de arbetade laborativt samt om de såg några fördelar respektive nackdelar med att arbeta med en laborativ arbetsmetod. Resultaten i arbetet kommer ifrån tre intervjuade pedagoger. De tre pedagogerna är alla ämneslärare i matematik och undervisar i ämnet i varierande årskurser. Alla tre pedagogerna jobbar med laborativ matematik i någon utsträckning. Pedagogerna tyckte att begreppet laborativ matematik var svårt att definiera då arbetsmetoden kan innehålla många olika saker. Alla tre pedagogerna var överrens om att det finns nackdelar med arbetssättet,dels för att det tar upp mycket tid och dels för att utfallet kanske inte blir som man har tänkt sig vid undervisningstillfället. Intervjuerna visade också att pedagogerna ser laborativ matematik som ett hjälpmedel för konkretisering och att man inte enbart arbetar med det abstrakta i läroböckerna.

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The purpose of this thesis is to study tautological rings of moduli spaces of curves. The moduli spaces of curves play an important role in algebraic geometry. The study of algebraic cycles on these spaces was started by Mumford. He introduced the notion of tautological classes on moduli spaces of curves. Faber and Pandharipande have proposed several deep conjectures about the structure of the tautological algebras. According to the Gorenstein conjectures these algebras satisfy a form of Poincaré duality. The thesis contains three papers. In paper I we compute the tautological ring of the moduli space of stable n-pointed curves of genus one of compact type. We prove that it is a Gorenstein algebra. In paper II we consider the classical case of genus zero and its Chow ring. This ring was originally studied by Keel. He gives an inductive algorithm to compute the Chow ring of the space. Our new construction of the moduli space leads to a simpler presentation of the intersection ring and an explicit form of Keel’s inductive result. In paper III we study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The Gorenstein conjecture is proved in this case as well. / QC 20110608

This thesis consist of an introduction and four research papers concerning numerical analysis for a certain class of free boundary problems. Paper I is devoted to the numerical analysis of the so-called two-phase membrane problem. Projected Gauss-Seidel method is constructed. We prove general convergence of the algorithm as well as obtain the error estimate for the finite difference scheme. In Paper II we have improved known results on the error estimates for a Classical Obstacle (One-Phase) Problem with a finite difference scheme. Paper III deals with the parabolic version of the two-phase obstacle-like problem. We introduce a certain variational form, which allows us to definea notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. In the last paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support. The proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions. / QC 20110510

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