This dissertation uses methods from homological algebra and computational commutative algebra to study four problems. We use Hilbert function computations and classical homology theory and combinatorics to answer questions with a more applied mathematics content: splines approximation, hyperplane arrangements, configuration spaces and coding theory. In Chapter II we study a problem in approximation theory. Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree d and smoothness r. Schenck and Stiller conjectured that this formula holds for all d 2r + 1. In this chapter we show that there exists a simplicial complex such that for any r, the dimension of the spline space in degree d = 2r is not given by this formula. Chapter III is dedicated to formal hyperplane arrangements. This notion was introduced by Falk and Randell and generalized to formality by Brandt and Terao. In this chapter we prove a criteria for formal arrangements, using a complex constructed from vector spaces introduced by Brandt and Terao. As an application, we give a simple description of formality of graphic arrangements in terms of the homology of the flag complex of the graph. Chapter IV approaches the problem of studying configuration of smooth rational curves in P2. Since an irreducible conic in P2 is a P1 (so a line) it is natural to ask if classical results about line arrangements in P2, such as addition-deletion type theorem, Yoshinaga criterion or Terao's conjecture verify for such configurations. In this chapter we answer these questions. The addition-deletion theorem that we find takes in consideration the fine local geometry of singularities. The results of this chapter are joint work with H. Schenck. In Chapter V we study a problem in algebraic coding theory. Gold, Little and Schenck find a lower bound for the minimal distance of a complete intersection evaluation codes. Since complete intersections are Gorenstein, we show a similar bound for the minimal distance depending on the socle degree of the reduced zero-dimensional Gorenstein scheme. The results of this chapter are a work in progress.

Thesis (M.S.)--Brigham Young University. Dept. of Mechanical Engineering, 2008. / Includes bibliographical references (p. 85-86).

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Proefschrift Universiteit van Amsterdam. / Met lit. opg. - Met samenvatting in het Nederlands.

A method for compressing surfaces associated with C1 cubic splines defined on triangulated quadrangulations is described. The method makes use of hierarchical bases, and does not require the construction of wavelets.

[pt] Neste trabalho, a técnica de interpolação de superfícies definidas numericamente por pseudo-splines é aplicada a antenas refletoras offset e com alto grau de modelagem. É feito um estudo comparativo com técnica de interpolação alternativa levando-se em conta os diversos métodos de análise de antenas refletoras. / [en] In this work, the pseudo-spline technique for numerically defined surfaces is applied to the approximation of single and dual-shaped offset reflector antennas. A comparative study with an usual alternative technique is undertaken, emphasizing the implications observed when different methods are employed for the evaluation of the antenna radiation field.

Depuis quelques années un engouement pour les méthodes de modélisation bayésienne a été observé dans divers domaines comme l'environnement, la médecine. Les études en recherche clinique s'appuient sur la modélisation mathématique et l'inférence statistique. Le but de cette thèse est d'étudier les applications possibles d'une telle modélisation dans le cadre de la recherche clinique. En effet, les jugements, les connaissances des experts (médecins, biologistes..) sont nombreux et importants. Il semble donc naturel de vouloir prendre en compte toutes ces connaissances a priori dans le modèle statistique. Après un rappel sur les fondamentaux des statistiques bayésiennes, des travaux préliminaires dans le cadre de la théorie de la décision sont présentés ainsi qu'un état de l'art des méthodes d'approximation. Une méthode de Monte Carlo par chaînes de Markov à sauts réversibles a été mise en place dans le contexte de modèles bien connus en recherche clinique : le modèle de Cox et le modèle logistique. Une approche de sélection de modèle est proposée comme une alternative aux critères classiques dans le cadre de la régression spline. Enfin différentes applications des méthodes bayésiennes non paramétriques sont développées. Des algorithmes sont adaptés et implémentés afin de pouvoir appliquer de telles méthodes. Cette thèse permet de mettre en avant les méthodes bayésiennes de différentes façons dans le cadre de la recherche clinique au travers de plusieurs jeux de données. / For some years a craze for the methods of bayesian modelling was observed in diverse domains as the environment, the medicine. The studies in clinical research lies on the modelling mathematical and the statistical inference. The purpose of this thesis is to study the possible applications of such a modelling within the framework of the clinical research. Indeed, judgments, knowledge of the experts (doctors, biologists) are many and important. It thus seems natural to want to take into account all these knowledge a priori in the statistical model. After a background on the fundamental of the bayesian statistics, preliminary works within the framework of the theory of the decision are presented as well as a state of the art of the methods of approximation. A MCMC method with reversible jumps was organized in the context of models known well in clinical research : the model of Cox and the logistic model. An approach of selection of model is proposed as an alternative in the classic criteria within the framework of the regression spline. Finally various applications of the nonparametric bayesian methods are developed. Algorithms are adapted and implemented to be able to apply such methods. This thesis allows to advance the bayesian methods in various ways within the framework of the clinical research through several sets of data.

This thesis provides a survey study on applications of spline functions to statistics. We start with a brief history of splines. Then, we discuss the application of splines to statistics as they are applied today. Several topics included in the discussion are splines, spline regression, spline smoothing, and estimating the smoothing parameter for spline regression. Also, we give a very brief discussion of multivariate splines in statistics and wavelets in statistics. Both of these topics are currently subjects for continuing research by many mathematicians.