Computer Algebra Systems (CAS) have been changing the mathematics instruction requirements for many years. Since the tendency of using CAS in mathematics instruction has been rising for decades and reports have often been positive, the implementation of notebook classes seems to be the consequent next step of mathematics instruction supported by computers. Experiences that have been made with the use of CAS in PC-rooms can be transformed directly into the classroom. Hence the use of CAS is no longer limited to certain rooms. The permanent availability of the notebook with installed CAS offers the chance to realize these concepts that have already been approved with the use of CAS so far. The following speech shall show what these concepts could look like and that the use of notebooks is not only the further development of teaching in PC-classes. Examples from personal experience in teaching will especially show meanders and thought-provoking impulses in order to support teachers finding their way into teaching mathematics instruction in notebook classes successfully. Please allow me to point out two things in the beginning: (1) Yes, I am a vehement supporter of the use of notebooks (and the use of CAS in particular) in mathematics instruction. (2) No, I do not believe that teachers who have chosen another path (or at least partly) are teaching badly.

Les décompositions en matrices triangulaires sont une brique de base fondamentale en calcul algébrique. Ils sont utilisés pour résoudre des systèmes linéaires et calculer le rang, le déterminant, l'espace nul ou les profiles de rang en ligne et en colonne d'une matrix. Le projet de cette thèse est de développer des implantations hautes performances parallèles de l'élimination de Gauss exact sur des machines à mémoire partagée.Dans le but d'abstraire le code de l'environnement de calcul parallèle utilisé, un langage dédié PALADIn (Parallel Algebraic Linear Algebra Dedicated Interface) a été implanté et est basé essentiellement sur des macros C/C++. Ce langage permet à l'utilisateur d'écrire un code C++ et tirer partie d’exécutions séquentielles et parallèles sur des architectures à mémoires partagées en utilisant le standard OpenMP et les environnements parallel KAAPI et TBB, ce qui lui permet de bénéficier d'un parallélisme de données et de taches.Plusieurs aspects de l'algèbre linéaire exacte parallèle ont été étudiés. Nous avons construit de façon incrémentale des noyaux parallèles efficaces pour les multiplication de matrice, la résolution de systèmes triangulaires au dessus duquel plusieurs variantes de l'algorithme de décomposition PLUQ sont construites. Nous étudions la parallélisation de ces noyaux en utilisant plusieurs variantes algorithmiques itératives ou récursives et en utilisant des stratégies de découpes variées.Nous proposons un nouvel algorithme récursive de l'élimination de Gauss qui peut calculer simultanément les profiles de rang en ligne et en colonne d'une matrice et de toutes ses sous-matrices principales, tout en étant un algorithme état de l'art de l'élimination de Gauss. Nous étudions aussi les conditions pour qu'un algorithme de l'élimination de Gauss révèle cette information en définissant un nouvel invariant matriciel, la matrice de profil de rang. / Triangular matrix decompositions are fundamental building blocks in computational linear algebra. They are used to solve linear systems, compute the rank, the determinant, the null-space or the row and column rank profiles of a matrix. The project of my PhD thesis is to develop high performance shared memory parallel implementations of exact Gaussian elimination.In order to abstract the computational code from the parallel programming environment, we developed a domain specific language, PALADIn: Parallel Algebraic Linear Algebra Dedicated Interface, that is based on C/C + + macros. This domain specific language allows the user to write C + + code and benefit from sequential and parallel executions on shared memory architectures using the standard OpenMP, TBB and Kaapi parallel runtime systems and thus providing data and task parallelism.Several aspects of parallel exact linear algebra were studied. We incrementally build efficient parallel kernels, for matrix multiplication, triangular system solving, on top of which several variants of PLUQ decomposition algorithm are built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies.We propose a recursive Gaussian elimination that can compute simultaneously therow and column rank profiles of a matrix as well as those of all of its leading submatrices, in the same time as state of the art Gaussian elimination algorithms. We also study the conditions making a Gaussian elimination algorithm reveal this information by defining a new matrix invariant, the rank profile matrix.

La combinatoire analytique étudie le comportement asymptotique des suites à travers les propriétés analytiques de leurs fonctions génératrices. Ce domaine a conduit au développement d’outils profonds et puissants avec de nombreuses applications. Au delà de la théorie univariée désormais classique, des travaux récents en combinatoire analytique en plusieurs variables (ACSV) ont montré comment calculer le comportement asymptotique d’une grande classe de fonctions différentiellement finies:les diagonales de fractions rationnelles. Cette thèse examine les méthodes de l’ACSV du point de vue du calcul formel, développe des algorithmes rigoureux et donne les premiers résultats de complexité dans ce domaine sous des hypothèses très faibles. En outre, cette thèse donne plusieurs nouvelles applications de l’ACSV à l’énumération des marches sur des réseaux restreintes à certaines régions: elle apporte la preuve de plusieurs conjectures ouvertes sur les comportements asymptotiques de telles marches,et une étude détaillée de modèles de marche sur des réseaux avec des étapes pondérées. / The field of analytic combinatorics, which studies the asymptotic behaviour ofsequences through analytic properties of their generating functions, has led to thedevelopment of deep and powerful tools with applications across mathematics and thenatural sciences. In addition to the now classical univariate theory, recent work in thestudy of analytic combinatorics in several variables (ACSV) has shown how to deriveasymptotics for the coefficients of certain D-finite functions represented by diagonals ofmultivariate rational functions. This thesis examines the methods of ACSV from acomputer algebra viewpoint, developing rigorous algorithms and giving the firstcomplexity results in this area under conditions which are broadly satisfied.Furthermore, this thesis gives several new applications of ACSV to the enumeration oflattice walks restricted to certain regions. In addition to proving several openconjectures on the asymptotics of such walks, a detailed study of lattice walk modelswith weighted steps is undertaken.

The two main parts of this doctoral thesis are a theorem that tight closure is contained in continuous closure via axes closure on the one hand and an algorithm to decide semistability of sheaves (or geometric vector bundles) via reduction to a linear algebra problem on the other hand. The sheaf stability algorithm was explicitly implemented by the author.

This diploma thesis deals with conict economic situations based on game theory. In the beginning, basic models of conict situations and current popular software tools are dened not only for the general support of student education or for science, but also for solving economic problems in game theory. Based on this analysis, the conicting situation of two competing rms is being solved. Gradually, work goes deeper into areas of delay dierential equations that better show the behavior of two players on the market. Subsequently, these delayed dierential equations are projected into the Cournot model, for which a critical value is identied that switches the stability of two rms on the market due to the delayed realization of their outputs.

For the problem of producing pedagogical step-by-step solutions to mathematical problems in education, standard methods and algorithms used in construction of computer algebra systems are often not suitable. A method of using rules to manipulate mathematical expressions in small steps is suggested and implemented. The problem of creating a step-by-step solution by choosing which rule to apply and when to do it is redefined as a graph search problem and variations of the A* algorithm are used to solve it. It is all put together into one prototype solver that was evaluated in a study. The study was a questionnaire distributed among high school students. The results showed that while the solutions were not as good as human-made ones, they were competent. Further improvements of the method are suggested that would probably lead to better solutions. / För problemet att producera pedagogiska steg-för-steg-lösningar till matematiska problem inom utbildning, är vanliga metoder och algoritmer som används i konstruktion av datoralgebrasystem ofta inte lämpliga. En metod som använder regler för att manipulera matematiska uttryck i små steg föreslås och implementeras. Problemet att välja vilka regler som ska appliceras och när de ska göra det för att skapa en steg-för-steg-lösning omdefineras som ett grafsökningsproblem och varianter av algoritmen A* används för att lösa det. Allt sätts ihop till en prototyp av en lösare vilken utvärderas i en studie. Studien var ett frågeformulär som delades ut till gymnasiestudenter. Resultaten visade att även fast lösningar skapade av programmet inte var lika bra som lösningar skapade av människor, så var de anständiga. Fortsatta föbättringar av metoden föreslås, vilka troligtvis skulle leda till bättre lösningar.

This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups. To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple product Property, and the construction of such groups via uniquely solvable puzzles. The higher order singular value decomposition is an important decomposition of tensors that retains some of the properties of the singular value decomposition of matrices. However, we have proven a novel negative result which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm that is no better than the standard algorithm. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta dissertação é apresentada uma modelagem analítica para o processo evolucionário formulado pela Teoria da Evolução por Endossimbiose representado através de uma sucessão de estágios envolvendo diferentes interações ecológicas e metábolicas entre populações de bactérias considerando tanto a dinâmica populacional como os processos produtivos dessas populações. Para tal abordagem é feito uso do sistema de equações diferenciais conhecido como sistema de Volterra-Hamilton bem como de determinados conceitos geométricos envolvendo a Teoria KCC e a Geometria Projetiva. Os principais cálculos foram realizados pelo pacote de programação algébrica FINSLER, aplicado sobre o MAPLE. / This work presents an analytical approach for modeling the evolutionary process formulated by the Serial Endosymbiosis Theory represented by a succession of stages involving different metabolic and ecological interactions among populations of bacteria considering both the population dynamics and production processes of these populations. In such approach we make use of systems of differential equations known as Volterra-Hamilton systems as well as some geometric concepts involving the KCC Theory and the Projective Geometry. The main calculations were performed by the computer algebra software FINSLER based on MAPLE.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.