Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In this thesis we study a standard example of refinable functions, that is, functions which can be reproduced by the integer shifts of their own dilations. Using the cardinal B-spline as an introductory example, we prove some of its properties, thereby building a basis for a later extension to the vector setting. Defining a subdivision scheme associated to the B-spline refinement mask, we then present the proof of a well-known convergence result. Subdivision is a powerful tool used in computer-aided geometric design (CAGD) for the generation of curves and surfaces. The basic step of a subdivision algorithm consists of starting with a given set of points, called the initial control points, and creating new points as a linear combination of the previous ones, thereby generating new control points. Under certain conditions, repeated applications of this procedure yields a continuous limit curve. One important goal of this thesis is to study a particular extension of scalar subdivision to matrix subdivision ...

SANTOS, Tyssia Nogueira Maciel dos. Avaliação genética da curva de lactação de cabras saanen utilizando modelos de regressão aleatória. 2014. 51 f. : Dissertação (Mestrado) - Universidade Federal do Ceará, Centro de Ciências, Departamento de Zootecnia, Programa de Pós-Graduação em Zootecnia. Fortaleza-CE, 2014. / Submitted by Eric Santiago (erichhcl@gmail.com) on 2016-08-09T13:21:33Z No. of bitstreams: 1 2014_dis_tnmsantos.pdf: 879224 bytes, checksum: 96fe44313df6f0e17fee9e748d618407 (MD5) / Approved for entry into archive by Nádja Goes (nmoraissoares@gmail.com) on 2016-08-09T13:31:08Z (GMT) No. of bitstreams: 1 2014_dis_tnmsantos.pdf: 879224 bytes, checksum: 96fe44313df6f0e17fee9e748d618407 (MD5) / Made available in DSpace on 2016-08-09T13:31:08Z (GMT). No. of bitstreams: 1 2014_dis_tnmsantos.pdf: 879224 bytes, checksum: 96fe44313df6f0e17fee9e748d618407 (MD5) Previous issue date: 2014 / The aim of this study was to perform a genetic-quantitative evaluation of the lactation curve up to 305 days of Saanen goats, fitting different random regression models to the best fit on the fixed and random trajectories of this curve and thus estimate the (co) variances and genetic parameters associated to this period. 11,018test-day milk yieldof 950 goats belonging to herds supported by the Dairy Goats Breeding Program (Capragene®) were used. The fixed effects used in the analysis were contemporary group, sex of offspring, type of milking and management. 18 models with different orderswere evaluated, to verify the simultaneous best fit of the regressions for the fixed and random trajectories (direct additive genetic, permanent environmental and residual effects). Ordinary orthogonal polynomials and Legendre polynomials, of second order were used to fit the fixed regression. Ordinary polynomials, orthogonal Legendre polynomials and B-spline functions, ranging from second to fourth order, were used for fit the random regressions. The model with orthogonal Legendre polynomials of second order to the fixed part and the b-spline function of fourth order for the random part was the most suitable for the fitting of the data analyzed. In fixed path, this model estimated peak lactation between 52-61 days, with an average of 2.378 kg of milk / day. The curve began with an average yield of 2.298 kg / day, ending 305 days with an average yield of 0.758 kg / day. Heritability estimates ranged from 0.294 ± 0.135 to 0.794 ± 0.075. Greater genetic variability was estimated for the intermediate test-day yields in the trajectory of the lactation curve, between 79 and 180 days, with higher heritability estimates. Genetic correlations among the test-day yields ranged from 0.551 ± 0.125 and 0.998 ± 0.001, with higher values between milk yield in the next and subsequent days, and becoming smaller as the distanceamong the test-day yield increase. The estimated genetic variability indicates that it is possible to modify the lactation curve of Saanen goats of the study population. The selection should be performed between 79 and 180 days of lactation, after the peak of lactation, however where the greatest heritabilities were observed period. The selection on this point will change the curve as a whole on the basis of correlated genetic responses. / O objetivo deste estudo foi realizar uma avaliação genético-quantitativa da curva de lactação até 305 dias de cabras da raça Saanen, avaliando distintos modelos de regressão aleatória para o melhor ajuste das trajetórias fixa e aleatórias desta curva, e assim estimar as covariâncias e parâmetros genéticos associados a este período. Foram utilizados 11.018 controles leiteiros de 950 lactações de cabras pertencentes a rebanhos associados ao Programa de Melhoramento Genético de Caprinos Leiteiros (Capragene®). Os efeitos fixos utilizados nas análises foram de grupo de contemporâneos, sexo das crias, tipo de ordenha e manejo. Foram avaliados 18 modelos, com diferentes ordens, para verificar o melhor ajuste simultâneo das regressões para as trajetórias fixa e aleatórias (efeitos genéticos aditivos diretos, de ambiente permanente e residual). Para ajuste da regressão fixa, foram utilizados os polinômios ordinários e os polinômios ortogonais de Legendre, de segunda ordem. Para as regressões aleatórias, foram utilizados os polinômios ordinários, os polinômios ortogonais de Legendre e as funções b-spline, variando da segunda a quarta ordem. O modelo com os polinômios ortogonais de Legendre de segunda ordem para a parte fixa e as funções b-spline de quarta ordem para a parte aleatória foi o mais adequado para o ajuste dos dados analisados. Na trajetória fixa, este modelo estimou o pico de lactação entre 52-61 dias, com valor médio de 2,378 kg de leite/dia. A curva iniciou-se com uma produção média de 2,298 kg/dia, encerrando aos 305 dias com uma produção média de 0,758 kg/dia. As estimativas de herdabilidade variaram de 0,294 ± 0,135 a 0,794 ± 0,075. Maior variabilidade genética foi estimada para os controles leiteiros intermediários na trajetória da curva de lactação, entre 79 e 180 dias, com maiores estimativas de herdabilidade. As correlações genéticas entre os controles variaram entre 0,551 ± 0,125 e 0,998 ± 0,001, como maiores valores para produções de leite em dias próximos e subsequentes, e tornando-se menores à medida que os controles se distanciavam. A variabilidade genética estimada indica que é possível modificar por seleção a curva de lactação das cabras Saanen da população estudada. Esta seleção deve ser realizada entre 79 e 180 dias de lactação, período posterior ao pico de lactação, entretanto onde se observaram as maiores herdabilidades. A seleção neste ponto permitirá alterações na curva como um todo, em função das respostas genéticas correlacionadas.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / O objetivo deste estudo foi realizar uma avaliaÃÃo genÃtico-quantitativa da curva de lactaÃÃo atà 305 dias de cabras da raÃa Saanen, avaliando distintos modelos de regressÃo aleatÃria para o melhor ajuste das trajetÃrias fixa e aleatÃrias desta curva, e assim estimar as covariÃncias e parÃmetros genÃticos associados a este perÃodo. Foram utilizados 11.018 controles leiteiros de 950 lactaÃÃes de cabras pertencentes a rebanhos associados ao Programa de Melhoramento GenÃtico de Caprinos Leiteiros (CaprageneÂ). Os efeitos fixos utilizados nas anÃlises foram de grupo de contemporÃneos, sexo das crias, tipo de ordenha e manejo. Foram avaliados 18 modelos, com diferentes ordens, para verificar o melhor ajuste simultÃneo das regressÃes para as trajetÃrias fixa e aleatÃrias (efeitos genÃticos aditivos diretos, de ambiente permanente e residual). Para ajuste da regressÃo fixa, foram utilizados os polinÃmios ordinÃrios e os polinÃmios ortogonais de Legendre, de segunda ordem. Para as regressÃes aleatÃrias, foram utilizados os polinÃmios ordinÃrios, os polinÃmios ortogonais de Legendre e as funÃÃes b-spline, variando da segunda a quarta ordem. O modelo com os polinÃmios ortogonais de Legendre de segunda ordem para a parte fixa e as funÃÃes b-spline de quarta ordem para a parte aleatÃria foi o mais adequado para o ajuste dos dados analisados. Na trajetÃria fixa, este modelo estimou o pico de lactaÃÃo entre 52-61 dias, com valor mÃdio de 2,378 kg de leite/dia. A curva iniciou-se com uma produÃÃo mÃdia de 2,298 kg/dia, encerrando aos 305 dias com uma produÃÃo mÃdia de 0,758 kg/dia. As estimativas de herdabilidade variaram de 0,294  0,135 a 0,794  0,075. Maior variabilidade genÃtica foi estimada para os controles leiteiros intermediÃrios na trajetÃria da curva de lactaÃÃo, entre 79 e 180 dias, com maiores estimativas de herdabilidade. As correlaÃÃes genÃticas entre os controles variaram entre 0,551  0,125 e 0,998  0,001, como maiores valores para produÃÃes de leite em dias prÃximos e subsequentes, e tornando-se menores à medida que os controles se distanciavam. A variabilidade genÃtica estimada indica que à possÃvel modificar por seleÃÃo a curva de lactaÃÃo das cabras Saanen da populaÃÃo estudada. Esta seleÃÃo deve ser realizada entre 79 e 180 dias de lactaÃÃo, perÃodo posterior ao pico de lactaÃÃo, entretanto onde se observaram as maiores herdabilidades. A seleÃÃo neste ponto permitirà alteraÃÃes na curva como um todo, em funÃÃo das respostas genÃticas correlacionadas. / The aim of this study was to perform a genetic-quantitative evaluation of the lactation curve up to 305 days of Saanen goats, fitting different random regression models to the best fit on the fixed and random trajectories of this curve and thus estimate the (co) variances and genetic parameters associated to this period. 11,018test-day milk yieldof 950 goats belonging to herds supported by the Dairy Goats Breeding Program (CaprageneÂ) were used. The fixed effects used in the analysis were contemporary group, sex of offspring, type of milking and management. 18 models with different orderswere evaluated, to verify the simultaneous best fit of the regressions for the fixed and random trajectories (direct additive genetic, permanent environmental and residual effects). Ordinary orthogonal polynomials and Legendre polynomials, of second order were used to fit the fixed regression. Ordinary polynomials, orthogonal Legendre polynomials and B-spline functions, ranging from second to fourth order, were used for fit the random regressions. The model with orthogonal Legendre polynomials of second order to the fixed part and the b-spline function of fourth order for the random part was the most suitable for the fitting of the data analyzed. In fixed path, this model estimated peak lactation between 52-61 days, with an average of 2.378 kg of milk / day. The curve began with an average yield of 2.298 kg / day, ending 305 days with an average yield of 0.758 kg / day. Heritability estimates ranged from 0.294  0.135 to 0.794  0.075. Greater genetic variability was estimated for the intermediate test-day yields in the trajectory of the lactation curve, between 79 and 180 days, with higher heritability estimates. Genetic correlations among the test-day yields ranged from 0.551  0.125 and 0.998  0.001, with higher values between milk yield in the next and subsequent days, and becoming smaller as the distanceamong the test-day yield increase. The estimated genetic variability indicates that it is possible to modify the lactation curve of Saanen goats of the study population. The selection should be performed between 79 and 180 days of lactation, after the peak of lactation, however where the greatest heritabilities were observed period. The selection on this point will change the curve as a whole on the basis of correlated genetic responses.

L’Analyse isogéométrique (AIG) est une méthode innovante de résolution numérique des équations différentielles, proposée à l’origine par Thomas Hughes, Austin Cottrell et Yuri Bazilevs en 2005. Cette technique de discrétisation est une généralisation de l’analyse par éléments finis classiques (AEF), conçue pour intégrer la conception assistée par ordinateur (CAO), afin de combler l’écart entre la description géométrique et l’analyse des problèmes d’ingénierie. Ceci est réalisé en utilisant des B-splines ou des B-splines rationnelles non uniformes (NURBS), pour la description des géométries ainsi que pour la représentation de champs de solutions inconnus.L’objet de cette thèse est d’étudier la méthode isogéométrique dans le contexte des problèmes hyperboliques en utilisant les fonctions B-splines comme fonctions de base. Nous proposons également une méthode combinant l’AIG avec la méthode de Galerkin discontinue (GD) pour résoudre les problèmes hyperboliques. Plus précisément, la méthodologie de GD est adoptée à travers les interfaces de patches, tandis que l’AIG traditionnelle est utilisée dans chaque patch. Notre méthode tire parti de la méthode de l’AIG et la méthode de GD.Les résultats numériques sont présentés jusqu’à l’ordre polynomial p= 4 à la fois pour une méthode deGalerkin continue et discontinue. Ces résultats numériques sont comparés pour un ensemble de problèmes de complexité croissante en 1D et 2D. / Isogeometric Analysis (IGA) is a modern strategy for numerical solution of partial differential equations, originally proposed by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005. This discretization technique is a generalization of classical finite element analysis (FEA), designed to integrate Computer Aided Design (CAD) and FEA, to close the gap between the geometrical description and the analysis of engineering problems. This is achieved by using B-splines or non-uniform rational B-splines (NURBS), for the description of geometries as well as for the representation of unknown solution fields.The purpose of this thesis is to study isogeometric methods in the context of hyperbolic problems usingB-splines as basis functions. We also propose a method that combines IGA with the discontinuous Galerkin(DG)method for solving hyperbolic problems. More precisely, DG methodology is adopted across the patchinterfaces, while the traditional IGA is employed within each patch. The proposed method takes advantageof both IGA and the DG method.Numerical results are presented up to polynomial order p= 4 both for a continuous and discontinuousGalerkin method. These numerical results are compared for a range of problems of increasing complexity,in 1D and 2D.

In the first part I discuss ancestors of NURBS curves and surfaces, rather Ferguson, Beziere, Coons and B-spline curves and surfaces and furthermore B-spline functions. In the second part I devote to NURBS curves and surfaces, their description as a linear combination of B-spline functions in the projective space. I specify conical arcs more detailed, their submit in the projective space and NURBS surfasec given as tensor product of NURBS curves. Last part is devote to describtion programs for modeling conicals and NURBS surface.

Introduction : Le cancer de l’ovaire est le cancer gynécologique le plus fatal dans le monde et est associé à un fardeau économique considérable pour les systèmes de santé publique, les patients et leurs familles. Actuellement, la prévention de ce cancer passe par l’identification des facteurs de risque, dont l’inflammation. Le double rôle de l’inflammation dans la carcinogenèse (transformation néoplasique et stimulation de la croissance pour l’inflammation chronique, mais également l’inhibition de la croissance pour inflammation aiguë) a déjà été observé au 19ième siècle, par Rudolf Virchow et par l’allemand Bruns, respectivement. Plusieurs preuves suggèrent aussi que le cancer de l’ovaire pourrait être lié à l’inflammation chronique de l’épithélium ovarien d’où l’hypothèse selon laquelle les analgésiques ayant une action anti-inflammatoire comme les anti-inflammatoires non stéroïdiens (AINS) et l’acétaminophène pourraient prévenir le cancer de l’ovaire. Contrairement à l’inflammation chronique, un autre facteur intéressant qui pourrait jouer un rôle sur le cancer de l’ovaire par le biais d’une inflammation aiguë est la mastite puerpérale qui est la forme la plus courante de mastite. Cependant, la littérature existante, examinant l’usage des analgésiques (aspirine, AINS non aspirine et acétaminophène) et le risque de cancer ovarien, est incohérente avec des différences populationnelles (cohortes de naissance différentes) et méthodologiques : variations des définitions de l’utilisation régulière, des variables d’ajustement, mais aussi dans la prise en compte d’une possible causalité inverse. De plus, aucune étude n’a tenté d’évaluer l’association dépendante du temps entre l’utilisation régulière de ces médicaments et le risque de cancer ovarien. Pour la mastite puerpérale pendant l’allaitement, deux articles avaient évalué son association avec le risque de cancer épithélial de l’ovaire (CEO), mais avec des limites méthodologiques : violation de la positivité avec l’inclusion des femmes qui n’ont jamais eu de grossesse et sur-ajustement avec la durée d’allaitement qui est dans le chemin causal. Objectif : Cette thèse visait à atteindre deux objectifs généraux qui sont de fournir de nouvelles preuves concernant les associations entre : 1) l’utilisation régulière d’analgésiques et le risque de CEO ; 2) la mastite puerpérale et le risque de CEO. Méthode : Nous avons utilisé les données d’une étude cas-témoin populationnelle visant à documenter les facteurs pour la prévention du cancer de l’ovaire au Québec (Étude PROVAQ). Cette étude a été menée dans la grande région de Montréal, Canada, de mars 2011 à septembre 2016 avec 498 cas et 908 témoins. Notre approche méthodologique a été effectuée en trois étapes. Premièrement, nous avons utilisé l’ensemble des données de PROVAQ pour l’évaluation des associations entre l’utilisation régulière de types de médicaments analgésiques, et aussi selon l’indication et le risque de CEO. Deuxièmement, à partir des données de PROVAQ, nous avons évalué l’association dépendante du temps entre l’utilisation régulière d’un type de médicaments et le risque de CEO à l'aide d'un indice cumulatif pondéré flexible d'exposition dans des modèles de régression logistique conditionnelle. Enfin, nous avons évalué l’association entre la mastite puerpérale et le risque de CEO chez les femmes allaitantes (174 cas et 431 témoins). La régression logistique a été utilisée pour estimer ces associations. Résultats : Nos résultats suggèrent que l'utilisation régulière d’aspirine et d'AINS non aspirine était inversement associée au CEO avec des rapports de cotes (RC) ajustés de 0,81 (IC à 95 % : 0,57–1,12) et 0,74 (IC à 95 % : 0,54–1,00), respectivement. Pour l'utilisation régulière d'AINS non aspirine, les RCs ajustés des COX-2 non sélective et sélective étaient de 0,73 (IC à 95 % : 0,50–1,00) et de 0,83 (IC à 95 % : 0,48–1,40), respectivement. Des associations similaires ont été observées selon le niveau de durée cumulative à vie ou de quantité cumulative à vie de prises d’aspirine et d’AINS non aspirine. Cependant, les associations entre les types de médicaments analgésiques et le CEO peuvent différer selon leurs indications. Aucune association n’a été trouvée entre le moment de l'utilisation régulière d’un type de médicaments analgésiques au cours des 40 années précédant la date index et le CEO. Aucune association significative n’a été aussi trouvée entre la mastite puerpérale pendant l'allaitement et le CEO (RC = 1,15 ; IC à 95 % : 0,71–1,84). Conclusions : Cette thèse fournit des preuves qui appuient l'hypothèse selon laquelle l'utilisation régulière d'aspirine et d'AINS non aspirine sont inversement associées au CEO. Nos résultats suggèrent également l'importance de considérer les indications d'utilisation lors de l'examen des relations entre les types de médicaments analgésiques et le CEO. Elle n’a pas trouvé d'association entre le moment de l'utilisation régulière d’analgésiques et le CEO mais aussi entre la mastite puerpérale pendant l’allaitement et le CEO. Cependant, notre étude a manqué de puissance. / Introduction: Ovarian cancer is the most fatal gynecological cancer in the world and is associated with a considerable economic burden for public health systems, patients and their families. Currently, the prevention of this cancer requires the identification of risk factors including inflammation. The dual role of inflammation in carcinogenesis (neoplastic transformation and stimulation of cancer growth for chronic inflammation, but also inhibition of cancer growth for acute inflammation) has already been observed in the 19th century, by Rudolf Virchow and by the German Bruns, respectively. Several pieces of evidence also suggest that ovarian cancer could be linked to chronic inflammation of the ovarian epithelium, hence the hypothesis that analgesics with an anti-inflammatory action such as nonsteroidal anti-inflammatory drugs (NSAIDs) and acetaminophen could prevent ovarian cancer. Unlike chronic inflammation, another interesting factor that could play a role in ovarian cancer through acute inflammation is puerperal mastitis which is the most common form of mastitis. However, the existing literature examining the use of analgesics (aspirin, non-aspirin NSAIDs and acetaminophen) and the risk of ovarian cancer is inconsistent with population (different birth cohorts) and methodological differences: variations in definitions of regular use, adjustment variables but also in taking into account a possible reverse causality. In addition, no studies have attempted to assess the time-dependent association between regular use of these drugs and the risk of ovarian cancer. For puerperal mastitis during breastfeeding, two articles had assessed its association with the risk of epithelial ovarian cancer (EOC) but with methodological limitations: violation of positivity with the inclusion of women who never had of pregnancy and over-adjustment with the duration of breastfeeding which is in the causal path. Objective: This thesis aimed to achieve two general objectives which are to provide new evidence regarding the associations between: 1) the regular use of analgesics and the risk of EOC; 2) puerperal mastitis and the risk of EOC. Method: We used data from a population-based case-control study aimed at documenting factors for the prevention of ovarian cancer in Quebec (PROVAQ study). This study was conducted in the greater Montreal area, Canada, from March 2011 to September 2016 with 498 cases and 908 controls. Our methodological approach was carried out in three stages. First, we used the PROVAQ dataset to assess associations between regular use of analgesic drugs types, and also by indication and EOC risk. Second, from PROVAQ data, we evaluated the time-dependent association between regular use of a type of medication and the risk of EOC using a flexible weighted cumulative index of exposure in conditional logistic regression models. Finally, we evaluated the association between puerperal mastitis and the risk of EOC in lactating women (174 cases and 431 controls). Unconditional logistic regression was used to estimate associations between regular use of analgesic drugs types, puerperal mastitis during breastfeeding and EOC risk. Results: Our results suggest that regular use of aspirin and non-aspirin NSAIDs were inversely associated with EOC with adjusted ORs of 0.81 (95% CI: 0.57–1.12) and 0.74 (95% CI: 0.54–1.00), respectively. For regular non-aspirin NSAID use, the adjusted ORs for non-selective and selective COX-2 were 0.73 (95% CI: 0.50–1.00) and 0.83 (95% CI: 0.48–1.40), respectively. Similar associations were observed according to the level of lifetime cumulative duration or lifetime cumulative quantity of aspirin and non-aspirin NSAID. However, the associations between analgesic drug types and EOC may differ according to their indications. No association was found between the time of regular use of any type of analgesic medication in the 40 years prior to the index date and EOC. No significant association was also found between puerperal mastitis during breastfeeding and EOC (OR = 1.15; 95% CI: 0.71–1.84). Conclusions: This thesis provides evidence that supports the hypothesis that regular use of aspirin and non-aspirin NSAIDs are inversely associated with EOC. Our results also suggest the importance of considering indications for use when examining relationships between analgesic drug types and EOC. We found no association between the timing of regular analgesic use and EOC but also between puerperal mastitis during breastfeeding and EOC. However, our study was underpowered.