Examensarbetet ger en bild av både lärares och forskares syn på matematiksvårigheter samt deras uppfattning om orsakerna bakom problemen. Vi har undersökt vilka områden i matematiken som elever med matematiksvårigheter har mest problem med samt hur läraren förklarar och underlättar matematiken för dessa elever. Både forskningen och de lärare vi intervjuat är överens om att begreppet matematiksvårigheter är väldigt komplext. Orsakerna kan vara av medicinsk/neurologisk, psykologisk, sociologisk och didaktisk karaktär. I våra intervjuer framkommer att positionssystemet, bråk, procent, enheter, multiplikation och division är de områden som kan ställa till mest problem för elever med matematiksvårigheter. Dessa områden nämns även inom forskningen som möjliga problemområden. För att underlätta matematiken för elever med matematiksvårigheter anser forskarna att det är viktigt att undervisningen utgår från elevernas erfarenheter och förkunskaper. De påpekar också vikten av att undervisningen varieras och bör innefatta såväl laborativa som teoretiska arbetssätt där även diskussioner och gruppuppgifter ska förekomma. För att se på vilken nivå lärarna börjar förklara för elever med matematiksvårigheter gav vi dem tre uppgifter som de fick förklara. Svaren placerades in i fyra kategorier: 1 Erfarenhet/vardag, 2. Konkret material, 3. Rita, 4. Räkna. Resultatet visar att lärarna oftast börjar sin förklaring i kategori fyra. Många hamnar i kategori tre och väldigt få hamnar i kategori ett och två.

Ill-posed mathematical problem occur in many interesting scientific and engineering applications. The solution of such a problem, if it exists, may not depend continuously on the observed data. For computing a stable approximate solution it is necessary to apply a regularization method. The purpose of this thesis is to investigate regularization approaches and develop numerical methods for solving certain ill-posed problems for parabolic partial differential equations. In thermal engineering applications one wants to determine the surface temperature of a body when the surface itself is inaccessible to measurements. This problem can be modelled by a sideways heat equation. The mathematical and numerical properties of the sideways heat equation with constant convection and diffusion coefficients is first studied. The problem is reformulated as a Volterra integral equation of the first kind with smooth kernel. The influence of the coefficients on the degree of ill-posedness are also studied. The rate of decay of the singular values of the Volterra integral operator determines the degree of ill-posedness. It is shown that the sign of the coefficient in the convection term influences the rate of decay of the singular values. Further a sideways heat equation in cylindrical geometry is studied. The equation is a mathematical model of the temperature changes inside a thermocouple, which is used to approximate the gas temperature in a combustion chamber. The heat transfer coefficient at the surface of thermocouple is also unknown. This coefficient is approximated via a calibration experiment. Then the gas temperature in the combustion chamber is computed using the convection boundary condition. In both steps the surface temperature and heat flux are approximated using Tikhonov regularization and the method of lines. Many existing methods for solving sideways parabolic equations are inadequate for solving multi-dimensional problems with variable coefficients. A new iterative regularization technique for solving a two-dimensional sideways parabolic equation with variable coefficients is proposed. A preconditioned Generalized Minimum Residuals Method (GMRS) is used to regularize the problem. The preconditioner is based on a semi-analytic solution formula for the corresponding problem with constant coefficients. Regularization is used in the preconditioner as well as truncating the GMRES algorithm. The computed examples indicate that the proposed PGMRES method is well suited for this problem. In this thesis also a numerical method is presented for the solution of a Cauchy problem for a parabolic equation in multi-dimensional space, where the domain is cylindrical in one spatial direction. The formal solution is written as a hyperbolic cosine function in terms of a parabolic unbounded operator. The ill-posedness is dealt with by truncating the large eigenvalues of the operator. The approximate solution is computed by projecting onto a smaller subspace generated by the Arnoldi algorithm applied on the inverse of the operator. A well-posed parabolic problem is solved in each iteration step. Further the hyperbolic cosine is evaluated explicitly only for a small triangular matrix. Numerical examples are given to illustrate the performance of the method.

Open-pit mining is an operation in which blocks from the ground are dug to extract the ore contained in them, and in this process a deeper and deeper pit is formed until the mining operation ends. Mining is often a highly complex industrial operation, with respect to both technological and planning aspects. The latter may involve decisions about which ore to mine and in which order. Furthermore, mining operations are typically capital intensive and long-term, and subject to uncertainties regarding ore grades, future mining costs, and the market prices of the precious metals contained in the ore. Today, most of the high-grade or low-cost ore deposits have already been depleted, and to obtain sufficient profitability in mining operations it is therefore today often a necessity to achieve operational efficiency with respect to both technological and planning issues. In this thesis, we study the open-pit design problem, the open-pit mining scheduling problem, and the open-pit design problem with geological and price uncertainty. These problems give rise to (mixed) discrete optimization models that in real-life settings are large scale and computationally challenging. The open-pit design problem is to find an optimal ultimate contour of the pit, given estimates of ore grades, that are typically obtained from samples in drill holes, estimates of costs for mining and processing ore, and physical constraints on mining precedence and maximal pit slope. As is well known, this problem can be solved as a maximum flow problem in a special network. In a first paper, we show that two well known parametric procedures for finding a sequence of intermediate contours leading to an ultimate one, can be interpreted as Lagrangian dual approaches to certain side-constrained design models. In a second paper, we give an alternative derivation of the maximum flow problem of the design problem. We also study the combined open-pit design and mining scheduling problem, which is the problem of simultaneously finding an ultimate pit contour and the sequence in which the parts of the orebody shall be removed, subject to mining capacity restrictions. The goal is to maximize the discounted net profit during the life-time of the mine. We show in a third paper that the combined problem can also be formulated as a maximum flow problem, if the mining capacity restrictions are relaxed; in this case the network however needs to be time-expanded. In a fourth paper, we provide some suggestions for Lagrangian dual heuristic and time aggregation approaches for the open-pit scheduling problem. Finally, we study the open-pit design problem under uncertainty, which is taken into account by using the concept of conditional value-atrisk. This concept enables us to incorporate a variety of possible uncertainties, especially regarding grades, costs and prices, in the planning process. In real-life situations, the resulting models would however become very computationally challenging.

Using fnite difference methods for partial differential equations, this thesis focuses on the problem of connecting overlapping solution domains in the context of a frst order hyperbolic problem. Especially the stability properties of such constructions is studied, and a stable general implementation of the the test problem is proposed. However, no energy estimate could be found, and indeed proven not to exist in the natural norm. Finally, an example is also put forward where the interface conditions derived are, for stability considerations, incompatible with the boundary conditions in a coupled system of hyperbolic equations.

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In this paper the reflection-extension of one-dimensional quasiminimizers is studied.A brief introduction to quasiminimizers, focused on the one-dimensional ones, is given.The main result of the study concerns the size of the quasiminimizing constant of theextended function relative to the unextended one. Previous work by O. Martio gives anupper bound for this relation. This bound is lowered, and the new bound is proven to besharp.Sharp quasiminimizer constants are calculated for a few simple functions and theirreflection-extensions. / I det här arbetet studeras reflektionsutvidgningen av endimensionella kvasiminimerare.En kortfattad introduktion till kvasiminimerare, fokuserad på de endimensionella, ges.Huvudresultatet av arbetet rör storleken av kvasiminimerarkonstanten för den utvidgade funktionen i förhållande till den outvidgade. Tidigare arbete av O. Martio ger en övre gräns för detta förhållande. Den  gränsen sänks, och den nya gränsen visas vara skarp.Skarpa kvasiminimerarkonstanter ges för ett par enkla funktioner och för deras reflektionsutvidgningar.

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