Este trabalho define um ambiente que auxilia o usuário a determinar as raízes complexas de um polinômio, dados apenas seus coeficientes complexos. Para tanto é definida uma interface que habilita a comunicação e a apresentação de resultados de forma mais expressiva do que as habitualmente usadas, permitindo que o usuário perceba mais rapidamente as informações que a ferramenta lhe apresenta. Para capacitar o usuário a determinar as soluções o sistema faz uso de uma abordagem baseada em imagens, obtidas a partir da interpretação do polinômio sendo estudado como se fosse uma superfície originada por uma função de dual variáveis. Através do uso de imagens o usuário pode orientar o sistema para determinar as soluções de seu interesse particular, e pode adicionalmente perceber de forma muito clara detalhes do polinômio que não são apresentados por outras ferramentas. São definidas operações que facilitam a manipulação das imagens obtidas, podendo-se efetuar uma serie de computações que eventualmente dispensam os métodos de determinação numérica de raízes, obtendo-se de uma forma alternativa possíveis aproximações para as soluções. Esta abordagem apoiada em imagens é muito útil na determinação de raízes em regiões onde a avaliação do polinômio se mostra instável ou sujeita a erros grosseiros, como por exemplo raízes múltiplas. Quando são geradas imagens pode ser percebida claramente a presença de erros de avaliação, permitindo que o usuário tome a atitude que julgar adequada. Outros sistemas não permitem que tais instabilidades sejam notadas, fornecendo respostas que podem lido ter qualquer nexo com a realidade. Para uso na ferramenta que implementa a solução proposta, existe uma serie de comandos definidos que capacitam o usuário a fazer quaisquer operações de seu interesse. Estes comandos foram já testados no período da implementação do sistema e permitem que sejam feitas operações sobre o polinômio, as imagens geradas e as raízes determinadas. Em todo o projeto do sistema existe a preocupação de tornar claro o que está acontecendo a cada momento com o polinômio ou com os dados dele obtidos, para que o usuário não se canse em demasia, seja interpretando dados do sistema ou procurando decidir o que fazer em seguida. Assim, o use da ferramenta d bastante natural, e tenta atingir a intuição daqueles que a irão usar. Exemplos são apresentados, bem como descrições dos comandos utilizados na interface. De especial interesse são os algoritmos usados na construção das imagens, pois estes permitem uma grande economia nas avaliações do polinômio quando da geração das cenas, reduzindo sobremaneira o tempo de espera do usuário. Quanto as imagens, são fornecidos três tipos de representações: imagens coloridas em duas dimensões, curvas de nível e curvas de sinal. O uso de três dimensões nas imagens foi descartado, e as razões para tanto estão descritas no texto. Todos os tipos de imagens permitem que se determine e identifique com facilidade as soluções desejadas, e são construídas a partir de uma transformação sobre o polinômio original. Esta transformação é explicada em detalhe no transcorrer do texto. / This work describes an environment that helps the user to find the complex solutions of a polynomial, given its coefficients. To do so, a quite expressive user interface is defined providing easy communication between man and machine and also allowing users to perceive much more rapidly the informations given by the tool. The system uses an image-based interface, obtained from a two-variable surface defined by the polynomial. By using these pictures, it is possible to direct the tool, finding any solutions of special meaning to the user. In addition, some details and features of the polynomial can be very easily seen. To manipulate images a set of operations was defined, and this set has so many features that the use of some operations can even avoid the effort of numerically computing the zeros of the polynomial, also reducing the numerical error embedded in these computations. This image-based approach is very useful when finding domains where the evaluation of the polynomial is unstable or prone to errors, e. g., near multiple roots. In pictures, any evaluation problems are easily detected, and the user can decide how to take the answers given by the system. Other tools usually do not provide any means to detect such behavioral oscillations, and can give nonsense answers. There is a set of commands embedded on the tool that implements the proposed solution, and they allow the user to perform any operations of his (her) interest. Such commands were tested at implementation time, and all were useful at some moment. They perform operations on the polynomial, on the pictures and on the detected zeros. All over the design phase there was the care of making clear what is happening to the polynomial or any data, so that the user can always work without being stressed by trying to find out what is happening. So, we try to build a natural approach to this interfacing problem, trying to reach user's intuition. Examples and descriptions of the commands are given, and also there is a description of a quite important algorithm, the one that builds the pictures themselves. This algorithm needs quite few evaluations of the function to generate pictures, thus reducing the waiting times. For the images, there are three presentation types: color images in two dimensions, height curves or signal curves. Three-dimensional pictures were discarded, and the reasons to such decision are described on the text. All kinds of images provide easy and simple identification of solutions. The images are produced from a simple transformation of the starting polynomial. This transformation is also explained in detail.

Este trabalho define um ambiente que auxilia o usuário a determinar as raízes complexas de um polinômio, dados apenas seus coeficientes complexos. Para tanto é definida uma interface que habilita a comunicação e a apresentação de resultados de forma mais expressiva do que as habitualmente usadas, permitindo que o usuário perceba mais rapidamente as informações que a ferramenta lhe apresenta. Para capacitar o usuário a determinar as soluções o sistema faz uso de uma abordagem baseada em imagens, obtidas a partir da interpretação do polinômio sendo estudado como se fosse uma superfície originada por uma função de dual variáveis. Através do uso de imagens o usuário pode orientar o sistema para determinar as soluções de seu interesse particular, e pode adicionalmente perceber de forma muito clara detalhes do polinômio que não são apresentados por outras ferramentas. São definidas operações que facilitam a manipulação das imagens obtidas, podendo-se efetuar uma serie de computações que eventualmente dispensam os métodos de determinação numérica de raízes, obtendo-se de uma forma alternativa possíveis aproximações para as soluções. Esta abordagem apoiada em imagens é muito útil na determinação de raízes em regiões onde a avaliação do polinômio se mostra instável ou sujeita a erros grosseiros, como por exemplo raízes múltiplas. Quando são geradas imagens pode ser percebida claramente a presença de erros de avaliação, permitindo que o usuário tome a atitude que julgar adequada. Outros sistemas não permitem que tais instabilidades sejam notadas, fornecendo respostas que podem lido ter qualquer nexo com a realidade. Para uso na ferramenta que implementa a solução proposta, existe uma serie de comandos definidos que capacitam o usuário a fazer quaisquer operações de seu interesse. Estes comandos foram já testados no período da implementação do sistema e permitem que sejam feitas operações sobre o polinômio, as imagens geradas e as raízes determinadas. Em todo o projeto do sistema existe a preocupação de tornar claro o que está acontecendo a cada momento com o polinômio ou com os dados dele obtidos, para que o usuário não se canse em demasia, seja interpretando dados do sistema ou procurando decidir o que fazer em seguida. Assim, o use da ferramenta d bastante natural, e tenta atingir a intuição daqueles que a irão usar. Exemplos são apresentados, bem como descrições dos comandos utilizados na interface. De especial interesse são os algoritmos usados na construção das imagens, pois estes permitem uma grande economia nas avaliações do polinômio quando da geração das cenas, reduzindo sobremaneira o tempo de espera do usuário. Quanto as imagens, são fornecidos três tipos de representações: imagens coloridas em duas dimensões, curvas de nível e curvas de sinal. O uso de três dimensões nas imagens foi descartado, e as razões para tanto estão descritas no texto. Todos os tipos de imagens permitem que se determine e identifique com facilidade as soluções desejadas, e são construídas a partir de uma transformação sobre o polinômio original. Esta transformação é explicada em detalhe no transcorrer do texto. / This work describes an environment that helps the user to find the complex solutions of a polynomial, given its coefficients. To do so, a quite expressive user interface is defined providing easy communication between man and machine and also allowing users to perceive much more rapidly the informations given by the tool. The system uses an image-based interface, obtained from a two-variable surface defined by the polynomial. By using these pictures, it is possible to direct the tool, finding any solutions of special meaning to the user. In addition, some details and features of the polynomial can be very easily seen. To manipulate images a set of operations was defined, and this set has so many features that the use of some operations can even avoid the effort of numerically computing the zeros of the polynomial, also reducing the numerical error embedded in these computations. This image-based approach is very useful when finding domains where the evaluation of the polynomial is unstable or prone to errors, e. g., near multiple roots. In pictures, any evaluation problems are easily detected, and the user can decide how to take the answers given by the system. Other tools usually do not provide any means to detect such behavioral oscillations, and can give nonsense answers. There is a set of commands embedded on the tool that implements the proposed solution, and they allow the user to perform any operations of his (her) interest. Such commands were tested at implementation time, and all were useful at some moment. They perform operations on the polynomial, on the pictures and on the detected zeros. All over the design phase there was the care of making clear what is happening to the polynomial or any data, so that the user can always work without being stressed by trying to find out what is happening. So, we try to build a natural approach to this interfacing problem, trying to reach user's intuition. Examples and descriptions of the commands are given, and also there is a description of a quite important algorithm, the one that builds the pictures themselves. This algorithm needs quite few evaluations of the function to generate pictures, thus reducing the waiting times. For the images, there are three presentation types: color images in two dimensions, height curves or signal curves. Three-dimensional pictures were discarded, and the reasons to such decision are described on the text. All kinds of images provide easy and simple identification of solutions. The images are produced from a simple transformation of the starting polynomial. This transformation is also explained in detail.

In this thesis a high order finite difference scheme is derived and implemented solving the shallow water equations using the SBP-SAT method. This method was tested against various benchmark problems were convergence was verified. The shallow water equations were also solved on a multi-block setup representing a tsunami approaching a shoreline from the ocean. Experiments show that a bottom topography with many spikes provides a dispersing effect on the incoming tsunami wave. Higher order convergence is not guaranteed for the multi-block simulations and could be investigated further in a future study.

Este trabalho define um ambiente que auxilia o usuário a determinar as raízes complexas de um polinômio, dados apenas seus coeficientes complexos. Para tanto é definida uma interface que habilita a comunicação e a apresentação de resultados de forma mais expressiva do que as habitualmente usadas, permitindo que o usuário perceba mais rapidamente as informações que a ferramenta lhe apresenta. Para capacitar o usuário a determinar as soluções o sistema faz uso de uma abordagem baseada em imagens, obtidas a partir da interpretação do polinômio sendo estudado como se fosse uma superfície originada por uma função de dual variáveis. Através do uso de imagens o usuário pode orientar o sistema para determinar as soluções de seu interesse particular, e pode adicionalmente perceber de forma muito clara detalhes do polinômio que não são apresentados por outras ferramentas. São definidas operações que facilitam a manipulação das imagens obtidas, podendo-se efetuar uma serie de computações que eventualmente dispensam os métodos de determinação numérica de raízes, obtendo-se de uma forma alternativa possíveis aproximações para as soluções. Esta abordagem apoiada em imagens é muito útil na determinação de raízes em regiões onde a avaliação do polinômio se mostra instável ou sujeita a erros grosseiros, como por exemplo raízes múltiplas. Quando são geradas imagens pode ser percebida claramente a presença de erros de avaliação, permitindo que o usuário tome a atitude que julgar adequada. Outros sistemas não permitem que tais instabilidades sejam notadas, fornecendo respostas que podem lido ter qualquer nexo com a realidade. Para uso na ferramenta que implementa a solução proposta, existe uma serie de comandos definidos que capacitam o usuário a fazer quaisquer operações de seu interesse. Estes comandos foram já testados no período da implementação do sistema e permitem que sejam feitas operações sobre o polinômio, as imagens geradas e as raízes determinadas. Em todo o projeto do sistema existe a preocupação de tornar claro o que está acontecendo a cada momento com o polinômio ou com os dados dele obtidos, para que o usuário não se canse em demasia, seja interpretando dados do sistema ou procurando decidir o que fazer em seguida. Assim, o use da ferramenta d bastante natural, e tenta atingir a intuição daqueles que a irão usar. Exemplos são apresentados, bem como descrições dos comandos utilizados na interface. De especial interesse são os algoritmos usados na construção das imagens, pois estes permitem uma grande economia nas avaliações do polinômio quando da geração das cenas, reduzindo sobremaneira o tempo de espera do usuário. Quanto as imagens, são fornecidos três tipos de representações: imagens coloridas em duas dimensões, curvas de nível e curvas de sinal. O uso de três dimensões nas imagens foi descartado, e as razões para tanto estão descritas no texto. Todos os tipos de imagens permitem que se determine e identifique com facilidade as soluções desejadas, e são construídas a partir de uma transformação sobre o polinômio original. Esta transformação é explicada em detalhe no transcorrer do texto. / This work describes an environment that helps the user to find the complex solutions of a polynomial, given its coefficients. To do so, a quite expressive user interface is defined providing easy communication between man and machine and also allowing users to perceive much more rapidly the informations given by the tool. The system uses an image-based interface, obtained from a two-variable surface defined by the polynomial. By using these pictures, it is possible to direct the tool, finding any solutions of special meaning to the user. In addition, some details and features of the polynomial can be very easily seen. To manipulate images a set of operations was defined, and this set has so many features that the use of some operations can even avoid the effort of numerically computing the zeros of the polynomial, also reducing the numerical error embedded in these computations. This image-based approach is very useful when finding domains where the evaluation of the polynomial is unstable or prone to errors, e. g., near multiple roots. In pictures, any evaluation problems are easily detected, and the user can decide how to take the answers given by the system. Other tools usually do not provide any means to detect such behavioral oscillations, and can give nonsense answers. There is a set of commands embedded on the tool that implements the proposed solution, and they allow the user to perform any operations of his (her) interest. Such commands were tested at implementation time, and all were useful at some moment. They perform operations on the polynomial, on the pictures and on the detected zeros. All over the design phase there was the care of making clear what is happening to the polynomial or any data, so that the user can always work without being stressed by trying to find out what is happening. So, we try to build a natural approach to this interfacing problem, trying to reach user's intuition. Examples and descriptions of the commands are given, and also there is a description of a quite important algorithm, the one that builds the pictures themselves. This algorithm needs quite few evaluations of the function to generate pictures, thus reducing the waiting times. For the images, there are three presentation types: color images in two dimensions, height curves or signal curves. Three-dimensional pictures were discarded, and the reasons to such decision are described on the text. All kinds of images provide easy and simple identification of solutions. The images are produced from a simple transformation of the starting polynomial. This transformation is also explained in detail.

A proper understanding of biological processes requires an understanding of genetics and evolutionary mechanisms. The vast amounts of genetical information that can routinely be extracted with modern technology have so far not been accompanied by an equally extended understanding of the corresponding processes. The relationship between a single gene and the resulting properties, phenotype of an individual is rarely clear. This thesis addresses several computational challenges regarding identifying and assessing the effects of quantitative trait loci (QTL), genomic positions where variation is affecting a trait. The genetic information available for each individual is rarely complete, meaning that the unknown variable of the genotype in the loci modelled also needs to be addressed. This thesis contains the presentation of new tools for employing the information that is available in a way that maximizes the information used, by using hidden Markov models (HMMs), resulting in a change in algorithm runtime complexity from exponential to log-linear, in terms of the number of markers. It also proposes the introduction of inferred haplotypes to further increase the power to assess these unknown variables for pedigrees of related genetically diverse individuals. Modelling consequences of partial genetic information are also treated. Furthermore, genes are not directly affecting traits, but are rather expressed in the environment of and in concordance with other genes. Therefore, significant interactions can be expected within genes, where some combination of genetic variation gives a pronounced, or even opposite, effect, compared to when occurring separately. This thesis addresses how to perform efficient scans for multiple interacting loci, as well as how to derive highly accurate empirical significance tests in these settings. This is done by analyzing the mathematical properties of the objective function describing the quality of model fits, and reformulating it through a simple transformation. Combined with the presented prototype of a problem-solving environment, these developments can make multi-dimensional searches for QTL routine, allowing the pursuit of new biological insight. / eSSENCE

Numerical simulation of stochastic biochemical reaction networks has received much attention in the growing field of computational systems biology. Systems are frequently modeled as a continuous-time discrete space Markov chain, and the governing equation for the probability density of the system is the (chemical) master equation. The direct numerical solution of this equation suffers from an exponential growth in computational time and memory with the number of reacting species in the model. As a consequence, Monte Carlo simulation methods play an important role in the study of stochastic chemical networks. The stochastic simulation algorithm (SSA) due to Gillespie has been available for more than three decades, but due to the multi-scale property of the chemical systems and the slow convergence of Monte Carlo methods, much work is currently being done in order to devise more efficient approximate schemes. In this thesis we review recent work for the solution of the chemical master equation by direct methods, by exact Monte Carlo methods and by approximate and hybrid methods. We also describe two conceptually different numerical methods to reduce the computational time when studying models using the SSA. A hybrid method is proposed, which is based on the separation of species into two subsets based on the variance of the copy numbers. This method yields a significant speed-up when the system permits such a splitting of the state space. A different approach is taken in an algorithm that makes use of low-discrepancy sequences and the method of uniformization to reduce variance in the computed density function.

The chemical master equation (CME) describes the probability for the discrete molecular copy numbers that define the state of a chemical system. Each molecular species in the chemical model adds a dimension to the state space. The CME is a difference-differential equation which can be solved numerically if the state space is truncated at an upper limit of the copy number in each dimension. The size of the truncated CME suffers from an exponential growth for an increasing number of chemical species. In this thesis the chemical master equation is approximated by a continuous Fokker-Planck equation (FPE) which makes it possible to use sparser computational grids than for CME. FPE on conservative form is used to compute steady state solutions by computation of an extremal eigenvalue and the corresponding eigenvector as well as time-dependent solutions by an implicit time-stepping scheme. The performance of the numerical solution is compared to a standard Monte Carlo algorithm. The computational work for a solutions with the same estimated error is compared for the two methods. Depending on the problem, FPE or the Monte Carlo algorithm will be more efficient. FPE is well suited for problems in low dimensions, especially if high accuracy is desirable.

Numerical approximations using high order finite differences on summation-byparts (SBP) form are investigated for discontinuous and fully nonlinear systems of partial differential equations. Stability and conservation properties of the approximations are obtained through a weak imposition of interface and boundary conditions with the simultaneous-approximation-term (SAT) technique. The SBP-SAT approximations replicate the continuous integration by parts rule. From this property, well-posedness and integral properties of the continuous problem are mimicked, and energy estimates leading to stability are obtained. The first part of the thesis focuses on the simulations of discontinuous linear advection problems. An artificial interface is introduced, separating parts of the spatial domain characterized by different wave speeds. A set of flexible stability conditions at the interface are derived, which can be adapted to yield conservative or non-conservative approximations. This model can be interpreted as a simplified version of nonlinear problems involving jumps at shocks, or as a prototypical of wave propagation through different materials. In the second part of the thesis, the vorticity/stream function formulation of the nonlinear momentum equation for an incompressible inviscid fluid is considered. SBP operators are used to derive a new Arakawa-like Jacobian with mimetic properties by combining different consistent approximations of the convection terms. Energy and enstrophy conservation is obtained for periodic problems using schemes with arbitrarily high order of accuracy. These properties are crucial for long-term numerical calculations in climate and weather forecasts or ocean circulation predictions. The third and final contribution of the thesis is dedicated to the incompressible Navier-Stokes problem. First, different completely general formulations of energy bounding boundary conditions are derived for the nonlinear equations. The boundary conditions can be used at both far field and solid wall boundaries. The discretisation in time and space with weakly imposed initial and boundary conditions using the SBP-SAT framework is proved to be stable and the divergence free condition is approximated with the design order of the scheme. Next, the same formulations are considered in a linearised setting, whereupon the spectra associated with the initial boundary value problem and its SBP-SAT discretisation are derived using the Laplace-Fourier technique. The influence of different boundary conditions on the spectrum and in particular the convergence to steady state is studied. / Numeriska approximationer av ekvationer som styr fysikaliska lagar är avgörande i många tillämpningar. Förutom en matematisk modell som kan fånga huvuddragen i ett verkligt problem är det nödvändigt att kunna utföra tillförlitliga simuleringar. Denna avhandling behandlar numeriska approximationer som med hög noggrannhet bevarar både rent matematiska aspekter av ekvationerna så väl som viktiga egenskaper hos modellen. Dessutom ges särskild uppmärksamhet åt modeller med diskontinuiteter och icke-linjära beteenden. Den första delen av avhandlingen handlar om diskontinuerliga problem. Det fysiska rummet kan ha olika egenskaper i olika regioner, något som kan resultera i instabila lösningar. Tillvägagångssättet består av att införa artificiella gränssnitt som skiljer dessa regioneråt. På detta sätt kan varje region behandlas separat, men på liknande sätt. Exempel på naturliga tillämningsområden är vågutbredning genom olika material och jordbävningssimuleringar. I den andra delen av avhandlingen visar vi att om den numeriska approximationen imiterar partiell integration, då följer också de väsentliga egenskaperna hos modellen på ett naturligt sätt. Att fysikaliska egenskaper bevaras är nödvändigt för att bibehålla stabilitet under långa simuleringstider för bland annat geofysiska problem. Den sista delen av avhandlingen är ägnasåt en av de mest använda modellerna inom strömningsmekanik, nämligen Navier-Stokes ekvationer. Studien fokuserar på härledningen av randvillkor som garanterar att lösningen inte växer på ett oförutsett och okontrollerat vis. Slutligen visas att de härledda randvillkoren på ett korrekt och noggrant sätt återskapar den dissipativa mekanism som ger upphov till jämviktstillstånd.

Prioriteringsreglerna är en överenskommen konvention som beskriver räkneoperationers ordningsföljd. Reglerna har visat sig vara något som ställer till problem för elever. I många engelskspråkiga länder används så kallade minnesregler för att komma ihåg prioriteringsreglerna. Minnesregler används för att komma ihåg information med hjälp av akronymer. Trots att detta används frekvent i många läromedel har vetenskaplig forskning visat att det kan medföra missuppfattningar. Syftet med studien är att undersöka hur matematdidaktisk forskning beskriver elevers tillvägagångssätt vid beräkning av numeriska uttryck där prioriteringsregler behöver tillämpas. Denna litteraturstudie baseras på databassökning efter vetenskapliga artiklar. Forskningen som analyserats visar att elever upplever svårigheter vid beräkningar av numeriska uttryck där flera operationer förekommer. Forskning har även visat att det finns missuppfattningar bland elever vad gäller prioriteringsregler och struktur. Det kan beskrivas som att elever missbrukar, missuppfattar, ignorerar, glömmer bort eller inte har kunskap om de regler och konventioner som grundar strukturen för numeriska uttryck.

Cancer is a widespread type of diseases that each year affects millions of people. It is mainly treated by chemotherapy, surgery or radiation therapy, or a combination of them. One modality of radiation therapy is high dose-rate brachytherapy, used in treatment of for example prostate cancer and gynecologic cancer. Brachytherapy is an invasive treatment in which catheters (hollow needles) or applicators are used to place the highly active radiation source close to or within a tumour. The treatment planning problem, which can be modelled as a mathematical optimization problem, is the topic of this thesis. The treatment planning includes decisions on how many catheters to use and where to place them as well as the dwell times for the radiation source. There are multiple aims with the treatment and these are primarily to give the tumour a radiation dose that is sufficiently high and to give the surrounding healthy tissue and organs (organs at risk) a dose that is sufficiently low. Because these aims are in conflict, modelling the treatment planning gives optimization problems which essentially are multiobjective. To evaluate treatment plans, a concept called dosimetric indices is commonly used and they constitute an essential part of the clinical treatment guidelines. For the tumour, the portion of the volume that receives at least a specified dose is of interest while for an organ at risk it is rather the portion of the volume that receives at most a specified dose. The dosimetric indices are derived from the dose-volume histogram, which for each dose level shows the corresponding dosimetric index. Dose-volume histograms are commonly used to visualise the three-dimensional dose distribution. The research focus of this thesis is mathematical modelling of the treatment planning and properties of optimization models explicitly including dosimetric indices, which the clinical treatment guidelines are based on. Modelling dosimetric indices explicitly yields mixedinteger programs which are computationally demanding to solve. The computing time of the treatment planning is of clinical relevance as the planning is typically conducted while the patient is under anaesthesia. Research topics in this thesis include both studying properties of models, extending and improving models, and developing new optimization models to be able to take more aspects into account in the treatment planning. There are several advantages of using mathematical optimization for treatment planning in comparison to manual planning. First, the treatment planning phase can be shortened compared to the time consuming manual planning. Secondly, also the quality of treatment plans can be improved by using optimization models and algorithms, for example by considering more of the clinically relevant aspects. Finally, with the use of optimization algorithms the requirements of experience and skill level for the planners are lower. This thesis summary contains a literature review over optimization models for treatment planning, including the catheter placement problem. How optimization models consider the multiobjective nature of the treatment planning problem is also discussed.