Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: (Math symbols have changed) Wavelet and subdivision techniques have developed, over the last two decades, into powerful mathematical tools, for example in signal analysis and geometric modelling. Both wavelet and subdivision analysis are based on the concept of a matrix–refinable function, i.e. a finitely supported matrix function which is self-replicating in the sense that it can be expressed as a linear combination of the integer shifts of its own dilation with factor 2: F = TAF = å k∈Z F(2 ・ −k)Ak. The coefficients Ak, k ∈ Z of d × d matrices, of this linear combination constitute the so-called matrix- mask sequence. Wavelets are in fact constructed as a specific linear combination of the integer shifts of the 2-dilation of a matrix- refinable function cf. [2; 9], whereas the convergence of the associated matrix- subdivision scheme c0 = c, cr+1 = SAcr, r ∈ Z+, SA : c = (ck : k ∈ Z) 7→ SAc = å ℓ∈Z Ak−2ℓ cℓ : k ∈ Z ! , subject to the necessary condition that rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å j∈Z Aǫ+2j, ǫ ∈ {0, 1}, ( cf. [26]) , implies the existence of a finitely supported matrix- function which is refinable with respect to the mask coefficients defining the refinement equation and the subdivision scheme. Throughout this thesis, we investigate in time–domain for a given matrix mask sequence, the related issues of the existence of a matrix–refinable function and the convergence of the corresponding matrix– cascade algorithm, and finally we apply some results to the particular research area of Hermite interpolatory subdivision schemes. The dissertation is organized as follows: In order to provide a certain flexibility or freedom over the project, we established in Chapter 1 the equivalence relation between the matrix cascade algorithm and the matrix subdivision scheme, subject to a well defined class of initial iterates. Despite the general noncommutativity of matrices, we make use in the full rank case Qǫ = I, ǫ ∈ {0, 1}, of a symbol factorization, to develop in Chapter 2 some useful tools, yielding a convergence result which comes as close to the scalar case as possible: we obtained a concrete sufficient condition on the mask sequence based on the matrix version of the generating function introduced in [3, page 22] for existence and convergence. Whilst the conjecture on nonnegative masks was confirmed in 2005 by Zhou [29], our result on scalar case provided a progress for general mask sequences. We then applied to obtain a new one-parameter family of refinable functions which includes the cardinal splines as a special case, as well as corresponding convergent subdivision schemes. With the view to broaden the class of convergent matrix-masks, we replaced in chapter 3 the full rank condition by the rank one condition Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, then improved the paper by Dubuc and Merrien [13] by using the theory of rank subdivision schemes by Micchelli and Sauer [25; 26], and end up this improvement with a generalization of [13, Theorem 13, p.8] in to the context of rank subdivision schemes. In Chapter 4, we translated the concrete convergence criteria of the general theory from Theorem 3.2, based on the r-norming factor introduced in [13, Definition 6, p.6], into the context of rank, factorization and spectral radius (cf. [26]), and presented a careful analysis of the relationship between the two concepts. We then proceed with generalizations and improvements: we classified the matrix cascade algorithms in term of rank = 1, 2, . . . , d, and provided a complete characterization of each class with the use of a more general r−norming factor namely τ(r)-norming factor. On the other hand, we presented numerical methods to determine, if possible, the convergence of each class of matrix cascade algorithms. In both the scalar and matrix cases above, we also obtained explicitly the geometric constant appearing in the estimate for the geometric convergence of thematrix-cascade algorithm iterates to the matrix- refinable function. This same geometric convergence rate therefore also holds true for the corresponding matrix–cascade algorithm. Finally, in Chapter 5, we apply the theory and algorithms developed in Chapter 4 to the particular research area of Hermite interpolatory subdivision schemes: we provided a new convergence criterium, and end up with new convergence ranges of the parameters’ values of the famous Hermite interpolatory subdivision scheme with two parameters, due to Merrien [23]. / AFRIKAANSE OPSOMMING :(Wiskundige simbole het verander) Golfie en subdivisietegnieke het oor die afgelope twee dekades ontwikkel in kragtige wiskundige gereedskap, byvoorbeeld in seinanalise en geometriesemodellering. Beide golfie en subdivisie analise is gebaseer op die konsep van ’n matriks-verfynbare funksie; oftewel ’n eindig-ondersteunde matriksfunksie F wat selfreproduserend is in die sin dat dit uitgedruk kan word as ’n lineêre kombinasie van die heelgetalskuiwe van F se eie dilasie met faktor 2: F = Σ F(2 · −α)A(α), met A(α), α ∈ Z, wat aandui die sogenaamde matriks-masker ry. Golfies kan dan gekonstrueer word as ’n spesifieke lineêre kombinasie van die funksie ry {F(2 · −α) : α ∈ Z} (sien [2; 9]), terwyl die konvergensie van die ooreenstemmende matriks-subdivisie skema cº = c, cr+1 =(Σ β∈Z A(α − 2β) cr(β) : α ∈ Z ! , r ∈ Z+, onderhewig aan die nodige voorwaarde dat rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å α∈Z A(ǫ + 2α), ǫ ∈ {0, 1}, (sien [27]) die bestaan impliseer van ’n eindig-ondersteunde matriksfunksie F wat verfynbaar ismet betrekking tot diemaskerko¨effisi¨entewat die subdivisieskema definieer, en in terme waarvan die limietfunksie F van die subdivisieskema uitgedruk kan word as F = å α∈Z F(· − α)c(α). Ons hoofdoel hier is om , in die tydgebied, en vir ’n gegewematriks-masker ry, die verwante kwessies van die bestaan van ’nmatriks-verfynbare funksie en die konvergensie van die ooreenstemmende matriks-kaskade algoritme, en matriks-subdivisieskema, te ondersoek, en om uiteindelik sommige van ons resultate toe te pas op die spesifieke kwessie van die konvergensie van Hermite interpolerende subdivisieskemas. Summary v Eerstens, in Hoofstuk 1, ondersoek ons die verwantskap tussen matriks-kaskade algoritmes en matriks-subdivisie skemas, met verwysing na ’n goedgedefinieerde klas van begin-iterate. Vervolgens beskou ons die volle rang geval Qǫ = I, ǫ ∈ {0, 1}, om, in Hoofstuk 2, nuttige gereedskap te ontwikkel, en wat daarby ’n konvergensie resultaat met ’n sterk konneksie ten opsigte van die skalaar-geval oplewer. Met die doelstelling om ons klas van konvergente matriks-maskers te verbreed, vervang ons, in Hoofstuk 3, die volle rang voorwaarde met die rang een voorwaarde Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, en verkry ons dan ’n verbetering op ’n konvergensieresultaat in die artikel [14] deur Dubuc en Merrien, deur gebruik te maak van die teorie van rang subdivisieskemas van Micchelli en Sauer [26; 27], waarna ons die resultaat [14, Stelling 13, page 8] na die konteks van rang subdivisieskemas veralgemeen. InHoofstuk 4 herlei ons die konkrete konvergensie kriteria van Stelling 3.2, soos gebaseer op die r-normerende faktor gedefinieer in [14, Definisie 6, page 6] , na die konteks van rang, faktorisering en spektraalradius (sien [27]), en gee ons ’n streng analise van die verwantskap tussen die twee konsepte. Verder stel ons dan bekend ’n nuwe klassifikasie van matriks-kaskade algoritmes ten opsigte van rang, en verskaf ons ’n volledige karakterisering van elke klasmet behulp van ’nmeer algemene r-normerende faktor, nl. die τ(r)-normerende faktor. Daarby gee ons doeltreffende numeriesemetodes vir die implementering van ons teoretiese resultate. Ons verkry ook eksplisiet die geometriese konstante wat voorkom in die afskatting van die geometriese konvergensie van die matriks-kaskade algoritme iterate na die matriks-verfynbare funksie. Ten slotte, in Hoofstuk 5, pas ons die teorie en algoritmes ontwikkel in Hoofstuk 4 toe om die konvergensie van Hermite-interpolerende subdivisieskemas te analiseer. Spesifiek lei ons ’n nuwe konvergensie kriterium af, wat ons dan toepas om nuwe konvergensie gebiede vir die parameter waardes te verkry vir die beroemde Hermite interpolerende subdivisieskema met twee parameters, soos toegeskryf aan Merrien [24].

L’objectif principal de ce travail de thèse a été de caractériser les effets de l’anti-VEGF Bevacizumab (Avastin) sur le réseau vasculaire tumoral in vivo, au cours du temps, à l’aide du modèle de la chambre dorsale chez la souris nude. Les images du réseau vasculaire tumoral acquises par microscopie intravitale ont été analysées par un algorithme de traitement d’images développé au sein de notre équipe, permettant de mettre en évidence les modifications morphologiques induites par le traitement et d’isoler des paramètres discriminants de la « normalisation » vasculaire, par comparaison à un réseau vasculaire sain. La période de « normalisation » vasculaire détectée par notre outil a été confortée par l’analyse de la fonctionnalité des vaisseaux sanguins au cours du temps, in vivo et par une analyse immunohistochimique des vaisseaux sanguins tumoraux et du tissu tumoral. A travers des essais préliminaires in vivo, en regard des résultats de ce travail concernant une fenêtre de "normalisation", nous avons cherché à vérifier l'hypothèse d'un bénéfice d'un traitement anti-VEGF préalablement à la thérapie photodynamique (PDT) sur des tumeurs de glioblastome xénogreffées en sous-cutané et en chambre dorsale. L'efficacité de la PDT est décrite comme étant dépendante d'une d'oxygénation tumorale suffisante et d'une distribution maximale de l'agent photosensibilisant au coeur des tumeurs. Parallèlement à ces travaux, nous avons cherché en équipe pluridiscilinaire à développer un modèle mathématique de la réponse au bevacizumab à partir de données biologiques réelles obtenues sur le même modèle in vivo et permettant pour l'avenir de simuler les réponses à différentes doses et différentes durées de traitement, toujours à des fins d'optimisation des protocoles thérapeutiques / The main aim of this work was to characterize the effects of the anti-VEGF Bevacizumab (Avastin) on the tumor vascular network, in vivo, over time, thanks to the skin fold chamber model on the nude mouse. Images of the vascular network obtained using intravital microscopy were analyzed par a dedicated image processing algorithm developed within our research team, allowing to highlight the morphological modifications induced by the treatment and to isolate discriminating parameters of the vascular "normalization", by comparison to healthy vascular networks. Le vascular "normalization" period detected with our tool was comforted by the analysis of the functionality of the blood vessels over time, in vivo and by an immunohistochemical analysis of the blood vessels and of the tumor tissue. In preliminary in vivo experiments, we tried to verify the hypothesis of the benefits of an anti-VEGF treatment prior to photodynamic therapy (PDT) on glioblastoma xenografts implanted subcutaneously or in the skin fold chamber. The efficacy of PDT is described as being dependent on tumor oxygenation and on the distribution of the photosensitizing agent within the tumor. In paralel to this work, we tried as a pluridisciplinary team to develop a mathematical model of the tumor response to bevacizumab using biological data obtained on the same in vivo model et that will allow in the future to simulate the response for different doses and different treatment durations, for the optimization of therapeutic protocols

Vi lever i ett samhälle där datorer och annan digitalteknik blir allt mer central i vår vardag. Sveriges regering har därför ålagt Skolverket att stärka elevernas digitala kompetens. Som en del av detta införs programmering som ett digitalt verktyg i matematikundervisningen både i grundskolan och på gymnasiet. Det krävs dock i nuläget inga kurser i programmering för att bli en legitimerad matematiklärare. Syftet med undersökningen som presenteras i denna rapport är att undersöka matematiklärares uppfattningar som uppkommit på grund av att Skolverkets revidering av läroplanerna i matematik. Denna revidering innebär att vissa matematikkurser på gymnasiet innefattar att programmering skall användas som problemlösningsverktyg. Underlaget till denna undersökning är en transkribering och tematisering av kvalitativa intervjuer med tio matematiklärare, samt tidigare forskning. Undersökningen fann en viss oro bland lärarna som till stor del handlade om bristande kunskap i programmering samt problematiken med att hinna med att få in ytterligare ett moment i undervisningen. Under intervjuerna framgick det att lärarna var allmänt fundersamma om vilka digitala verktyg de skulle använda för att lösa detta nya krav. En majoritet av lärarna förordade dock Excel och Geogebra. Det framkom ett visst missnöje med att detta nya krav infördes med mycket kort varsel. Många lärare förväntade sig och litade på att läroboksförfattarna skulle komma med en uppdatering av läroböckerna i matematik. En uppdatering som förväntades innefatta programmering och som därmed skulle lösa den nya pedagogiska utmaningen. / Computers and other digital technology are becoming increasingly important in our society. Due to that, the Swedish Government has instructed their National Agency for Education to strengthen the students' digital competence. One outcome of this was that programming become a part of teaching mathematics both in primary and upper secondary school. Programming is not a part of the mandatory studies needed to become a certified mathematics teacher. The purpose of this study is to investigate the ideas, attitudes and ideas of mathematics teachers that have arisen because of the National Agency for Education's revision of the curricula in mathematics. According to this revision of the curricula, students should use programming as a problem-solving tool. The basis for this study is a transcription of qualitative interviews with ten mathematics teachers and an examination of previous research. This study found that there was some concern among the teachers. Most of the concern was about lack of knowledge in programming. The majority of teachers preferred to use Excel and Geogebra as a digital tool to teach programming. Many teachers expressed spontaneously a general dissatisfaction with the impact that calculators already have in mathematics education. There was some dissatisfaction with the introduction of this new requirement at very short notice. Many teachers expected and trusted that the textbook authors would come up with an update of the textbooks in mathematics. An update that would thus solve their new educational challenge.