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In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be C-0(infinity)-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not to vanish at the singularity and have an explicit decay at infinity.The paper uses only elementary number theory and is available to readers with no number-theoretic background.

[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given.

[Abstract]: We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid (small meaning that the two surfaces cannot be tangent at two points simultaneously when one is moved with respect to the other). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.

Objectives: Neandertal femora are particularly known for having a marked sagittal femoral curvature. This study examined femoral curvature in Neandertals in comparison to a modern human population from Belgium by the use of three-dimensional (3D) quadric surfaces modeled from the bone surface. 3D models provide detailed information and enabled femoral curvature to be analyzed in conjunction with other morphological parameters. Materials and Methods: 3D models were created from CT scans of 75 modern human femora and 7 Neandertal femora. Quadric surfaces (QS) were created from the triangulated surface vertices in all areas of interest (neck, head, diaphyseal shaft, condyles) extracted from previously placed anatomical landmarks. The diaphyseal shaft was divided into five QS shapes and curvature was measured by degrees of difference between QS shapes. Each bone was placed in a local coordinate system enabling each bone to be analyzed in the same way. Results: The use of 3D quadric surface fitting allowed the distribution of curvature with similarly curved femora to be analyzed and the different patterns of curvature between the two groups to be determined. The Neandertals were shown to have a higher degree of femoral curvature and a more distal point of femoral curvature than the modern human population from Belgium. Conclusions: Morphological aspects of the Neandertal femur are different from this modern human population although mainly seem unrelated to femoral curvature. The relative lack of correlations with other femoral bony morphological factors suggests femoral curvature variations may be related to other aspects., SCOPUS: ar.j, info:eu-repo/semantics/published

Early anthropological examination of Neandertal skeletal material by Marcellin Boule gave rise to popular images of Neandertals as brutish, ape-like creatures who walked hunched over with bent knees and a shuffling gait. Today, it is generally thought that Neandertals moved in a similar manner to humans with locomotive patterns within human ranges of variation and a bipedal gait likely to be indistinguishable from that of modern humans However, this hypothesis has not been tested by using the Neandertal skeleton as a whole. There is no complete Neandertal skeleton in the fossil record. The aim of this thesis was to reconstruct a complete virtual skeleton of a Neandertal based on the Spy II remains for educational museology purposes and for biomechanical analysis. Comparative analyses of Neandertal and modern human bones (pelvis, femur, rib) using different reference populations from Belgium were also undertaken. These studies were performed with a view to further understanding Neandertal bone morphology in comparison to modern humans, to assist in the reconstruction of the Neandertal Spy II skeleton and to aid in biomechanical analysis. A study on the sex determination of the pelves showed that there was no difference between physical linear measurements and virtual measurements, which was an important validation. The lhpFusionBox musculoskeletal software, which was developed at ULB, was used to scale available Neandertal and modern human bones to reconstruct the skeleton. Previous methods in the literature have only scaled bones of the same nature. A novel and validated scaling method was used to scale the Kebara 2 pelvis to the dimensions of the Spy II femur (as Spy II only has a small piece of sacrum) via the Neandertal 1 femur and pelvis. The reconstruction of a complete Neandertal skeleton based on the Spy II remains enabled the validation of individual reconstructions of bones, reconstructed long bones to be compared together with other limb proportions of Nea, Historiquement, le travaux de Marcellin Boule ont donné lieu à des représentations populaires de Néandertaliens vus comme des créatures simiesques bestiales qui se déplaçaient courbés avec des genoux pliés en traînant les pieds. Aujourd'hui, il est généralement admis que les Néandertaliens se déplaçaient d'une manière similaire à l'homme moderne avec une locomotion bipède. Toutefois, cette hypothèse n'a pas été encore testée sur un squelette néandertalien dans son ensemble. Comme il n’existe aucun squelette de Néandertalien entier, le but de cette thèse était de reconstituer un squelette virtuel complet sur la base des restes de Spy II à des fins éducatives de muséologie et pour l'analyse biomécanique.Des analyses comparatives d’os de Néandertalien et d’humains modernes (bassin, fémur, côtes) en utilisant diverses populations de référence de Belgique ont également été menées. Ces études ont été réalisées en vue de comprendre la morphologie néandertalienne, pour aider à la reconstruction du squelette Spy II et à l'analyse biomécanique. Une étude sur la détermination du sexe à partir des pelvis a montré qu'il n'y avait pas de différence entre les mesures linéaires physiques et les mesures virtuelles, ce qui était une validation importante.Le logiciel musculosquelettique lhpFusionBox, développé à l'ULB, a été utilisé pour reconstruire le squelette. Un nouveau procédé de mise à l'échelle validé a été utilisé pour mettre à l'échelle le bassin de Kebara 2 aux dimensions du fémur de Spy II (comme Spy II ne possède qu’un petit fragment de sacrum) via le fémur et le bassin de Neandertal 1. La reconstruction d'un squelette complet de Néandertalien a apporté de nouvelles connaissances sur cette espèce dans différents domaines. Elle a permis la validation des reconstructions individuelles des os, de comparer les os longs reconstruits avec d'autres proportions des membres de Néandertaliens, de faire une estimation de stature, et de reconsidérer les attributions antérieures d'os, Doctorat en Sciences biomédicales et pharmaceutiques (Médecine), info:eu-repo/semantics/nonPublished

[EN] In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing., [ES] En esta comunicación presentamos la modelización matemática como herramienta de diseño en arquitectura. Nuestra tesis es que las matemáticas y un software adecuado como es el programa Mathematica pueden aportar nuevas ideas para la creación de formas en arquitectura. Para ello hemos diseñado nuevas obras arquitectónicas concebidas íntegramente desde las matemáticas. Los resultados han apoyado nuestra tesis, ya que los diseños creados con Mathematica nada tienen que envidiar a los que podemos encontrar habitualmente en arquitectura.

[EN] In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing., [ES] En esta comunicación presentamos la modelización matemática como herramienta de diseño en arquitectura. Nuestra tesis es que las matemáticas y un software adecuado como es el programa Mathematica pueden aportar nuevas ideas para la creación de formas en arquitectura. Para ello hemos diseñado nuevas obras arquitectónicas concebidas íntegramente desde las matemáticas. Los resultados han apoyado nuestra tesis, ya que los diseños creados con Mathematica nada tienen que envidiar a los que podemos encontrar habitualmente en arquitectura.

[EN] In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing., [ES] En esta comunicación presentamos la modelización matemática como herramienta de diseño en arquitectura. Nuestra tesis es que las matemáticas y un software adecuado como es el programa Mathematica pueden aportar nuevas ideas para la creación de formas en arquitectura. Para ello hemos diseñado nuevas obras arquitectónicas concebidas íntegramente desde las matemáticas. Los resultados han apoyado nuestra tesis, ya que los diseños creados con Mathematica nada tienen que envidiar a los que podemos encontrar habitualmente en arquitectura.

[EN] In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing., [ES] En esta comunicación presentamos la modelización matemática como herramienta de diseño en arquitectura. Nuestra tesis es que las matemáticas y un software adecuado como es el programa Mathematica pueden aportar nuevas ideas para la creación de formas en arquitectura. Para ello hemos diseñado nuevas obras arquitectónicas concebidas íntegramente desde las matemáticas. Los resultados han apoyado nuestra tesis, ya que los diseños creados con Mathematica nada tienen que envidiar a los que podemos encontrar habitualmente en arquitectura.

[EN] In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing., [ES] En esta comunicación presentamos la modelización matemática como herramienta de diseño en arquitectura. Nuestra tesis es que las matemáticas y un software adecuado como es el programa Mathematica pueden aportar nuevas ideas para la creación de formas en arquitectura. Para ello hemos diseñado nuevas obras arquitectónicas concebidas íntegramente desde las matemáticas. Los resultados han apoyado nuestra tesis, ya que los diseños creados con Mathematica nada tienen que envidiar a los que podemos encontrar habitualmente en arquitectura.

[EN] In this work we present mathematical modelling as a design tool in architecture. Our thesis is that mathematics joint with a suitable software, such as Mathematica, are a source of ideas for creating forms in architecture. In this sense we have designed new architectural volumes using mathematics as a main tool. The results support our thesis, since the volumes created with Mathematica have nothing to envy to others created with specific software for designing., [ES] En esta comunicación presentamos la modelización matemática como herramienta de diseño en arquitectura. Nuestra tesis es que las matemáticas y un software adecuado como es el programa Mathematica pueden aportar nuevas ideas para la creación de formas en arquitectura. Para ello hemos diseñado nuevas obras arquitectónicas concebidas íntegramente desde las matemáticas. Los resultados han apoyado nuestra tesis, ya que los diseños creados con Mathematica nada tienen que envidiar a los que podemos encontrar habitualmente en arquitectura.

The aim of this paper is to study what happens when a slight perturbation affects the coefficients of a quadratic equation defining a variety (a quadric) in R^n. Structurally stable quadrics are those a small perturbation on the coefficients of the equation defining them does not give rise to a "different" (in some sense) set of points. In particular we characterize structurally stable quadrics and give the "bifurcation diagrams" of the non stable ones (showing which quadrics meet all of their neighbourhoods), when dealing with the "affine" and "metric" equivalence relations. This study can be applied to the case where a set of points which constitute the set of solutions of a problem is deffined by a quadratic equation whose coefficients are given with parameter uncertainty.

This report is a user manual for the program QUADRAW which draws visible-line-projection pictures of objects bounded by sections of quadric surfaces. The report describes three versions of the program, one each for the CDC 6600, Univac 1110 and IBM System/360-370 computers. It also provides the necessary documentation to permit one to modify the program for any other computer. (Author)