Inaug.-diss.--Universität Zürich. / Lebenslauf. Bibliographical foot-notes.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-16T13:43:54Z No. of bitstreams: 1 arquivototal.pdf: 497004 bytes, checksum: f533fe667e534433904bf0bb58473fac (MD5) / Made available in DSpace on 2017-08-16T13:43:54Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 497004 bytes, checksum: f533fe667e534433904bf0bb58473fac (MD5) Previous issue date: 2015-08-28 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Let k be an algebraically closed eld of characteristic zero and consider the polynomial ring S = k[x1, . . . , xn] endowed with the standard grading. The Waring's problem for a form F ∈ S of degree d asks about the least integer s ≥ 1 for which there exist linear forms L1, . . . , Ls ∈ S satisfying F = Σs i=1 Ldi. Such integer is called Waring rank of F. In this dissertation, we present a solution to this problem { due to Carlini-Catalisano-Geramita { in the case of monomials, as an interesting application of a theorem (due to the same authors) that establishes a lower bound for the multiplicity of (standard) graded, nitely generated, reduced, 1-dimensional k-algebras. / Seja k um corpo algebricamente fechado de caracter stica zero e considere o anel de polin^omios S = k[x1, . . . , xn] munido da gradua c~ao padr~ao. O Problema de Waring para uma forma F ∈ S de grau d questiona a respeito do menor inteiro s ≥ 1 para o qual existem formas lineares L1, . . . , Ls ∈ S satisfazendo F = Σs i=1 Ldi. Tal inteiro e denominado posto de Waring de F. Nesta disserta c~ao, apresentamos uma solu c~ao deste problema { devida a Carlini-Catalisano-Geramita { no caso de mon^omios, como uma interessante aplica c~ao de um teorema (dos mesmos autores) que estabelece uma cota inferior para a multiplicidade de k- algebras graduadas (padr~ao) nitamente geradas, reduzidas e 1-dimensionais.

This thesis is divided into two chapters. First, we want to study particularclasses of power ideals, with particular attention to their relation with the Fröberg conjecture on the Hilbert series of generic ideals. In the second part,we study a generalization (introduced by Fröberg, Ottaviani, and Shapiro in 2012)of the classical Waring problem for polynomials about writing homogeneouspolynomials as sums of powers. We see also how the theories of fat points andsecant varieties of Veronese varieties play a crucial role in the relation betweenthose chapters and in providing tools to nd an answer to our questions. The main results are the computation of the Hilbert series of particularclasses of power ideals, which in particular give us a proof of the Fröberg conjecturefor generic ideals generated by eight homogeneous polynomials of thesame degree in four variables, and the solution of the generalized Waring problemin the case of sums of squares in three and four variables. We also beginthe study of the generalized Waring problem for monomials.

As one of the major data analysis methods, statistical models play an important role in traffic safety analysis. A common situation associated with crash data is the phenomenon known as overdispersion which has been discussed and investigated frequently in recent years. As such, researchers have proposed several models, such as the Poisson Gamma (PG) or Negative Binomial (NB), the Poisson-lognormal, or the Poisson-Weibull, to handle the overdispersion. Unfortunately, very few models have been proposed for specifically analyzing the sources of dispersions in the data. Better understanding of sources of variation and overdispersion could help in managing safety, such as establishing relationships and applying appropriate treatments or countermeasures, more efficiently. Given the limitations of existing models for exploring the source of overdispersion of crash data, this research examined a new model function that could be applied to explore sources of extra variability through the use of the Generalized Waring (GW) models. This model, which was recently introduced by statisticians, divides the observed variability into three components: randomness, internal differences between road segments or intersections, and the variances caused by other external factors that have not been included as covariates in the model. To evaluate these models, GW models were examined using both simulated and empirical crash datasets, and the results were compared to the most commonly used NB model and the recently developed NB-Lindley models. For model parameter estimation, both the maximum likelihood method and a Bayesian approach were adopted for better comparison. A simulation study was used to show the better performance of this model compared to NB model for overdispersed data, and then an application in the empirical crash data illustrates its capability of modeling data sets with great accuracy and exploring the source of overdispersion. The performances of hotspot identification for these two kinds of models (i.e., GW models and NB models) were also examined and compared based on the estimated models from the empirical dataset. Finally, bias properties related to the choice of prior distributions for parameters in GW model were examined by using a simulation study. In addition, the suggestions on the choice of minimum sample size and priors were presented for different kinds of datasets.

No description available.

Le cadre général de cette thèse est l'étude des polynômes comme objets de modèles de calcul. Cette approche permet de définir de manière précise la complexité d'évaluation d'un polynôme, puis de classifier des familles de polynômes en fonction de leur difficulté dans ce modèle. Dans cette thèse, nous nous intéressons en particulier au modèle AffPow des sommes de puissance de forme linéaire, i.e. les polynômes qui s'écrivent $f = \sum_{i = 1}^s \alpha_i \ell_i^{e_i}$, avec $\deg \ell_i = 1$. Ce modèle semble assez naturel car il étend à la fois le modèle de Waring $f = \sum \alpha_i \ell_i^d$ et le modèle du décalage creux $f = \sum \alpha_i \ell^{e_i}$, mais peu de résultats sont connus pour cette généralisation.Nous avons pu prouver des résultats structurels pour la version univarié de ce modèle, qui nous ont ensuite permis d'obtenir des bornes inférieures et des algorithmes de reconstruction, qui répondent au problème suivant : étant donné $f = \sum \alpha_i (x-a_i)^{e_i}$ par la liste de ses coefficients, retrouver les $\alpha_i, a_i, e_i$ qui apparaissent dans la décomposition optimale de $f$.Nous avons aussi étudié plus en détails la version multivarié du modèle, qui avait été laissé ouverte par nos précédents algorithmes de reconstruction, et avons obtenu plusieurs résultats lorsque le nombre de termes dans une expression optimale est relativement petit devant le nombre de variables ou devant le degré du polynôme. / The general framework of this thesis is the study of polynomials as objects of models of computation. This approach allows to define precisely the evaluation complexity of a polynomial, and then to classify families of polynomials depending on their complexity. In this thesis, we focus on the study of the model of sums of affine powers, that is polynomials that can be written as $f = \sum_{i = 1}^s \alpha_i \ell_i^{e_i}$, with $\deg \ell_i = 1$.This model is quite natural, as it extends both the Waring model $f = \sum \alpha_i \ell_i^d$ , and the sparsest shift model $f = \sum \alpha_i \ell^{e_i}$, but it is still not well known.In this work, we obtained structural results for the univariate variant of this model, which allow us to obtain lower bounds and reconstruction algorithms, that solve the following problem : given $f = \sum \alpha_i (x-a_i)^{e_i}$ as a list of its coefficient, find the values of the $\alpha_i$’s, $e_i$’s and $a_i$’s in the optimal decomposition of $f$.We also studied the multivariate case and obtained several reconstruction algorithms that work whenever the number of terms in the optimal expression is small in terms of the number of variable or the degree of the polynomial.

In the present thesis we analyze different types of additive decompositions of homogeneous polynomials. These problems are usually called Waring-type problems and their story go back to the mid-19th century and, recently, they received the attention of a large community of mathematicians and engineers due to several applications. At the same time, they are related to branches of Commutative Algebra and Algebraic Geometry. The classical Waring problem investigates decompositions of homogeneous polynomials as sums of powers of linear forms. Via Apolarity Theory, the study of these decompositions for a given polynomial F is related to the study of configuration of points apolar to F, namely, configurations of points whose defining ideal is contained in the ``perp'' ideal associated to F. In particular, we analyze which kind of minimal set of points can be apolar to some given polynomial in cases with small degrees and small number of variables. This let us introduce the concept of Waring loci of homogeneous polynomials. From a geometric point of view, questions about additive decompositions of polynomials can be described in terms of secant varieties of projective varieties. In particular, we are interested in the dimensions of such varieties. By using an old result due to Terracini, we can compute these dimensions by looking at the Hilbert series of homogeneous ideal. Hilbert series are very important algebraic invariants associated to homogeneous ideals. In the case of classical Waring problem, we have to look at power ideals, i.e., ideals generated by powers of linear forms. Via Apolarity Theory, their Hilbert series are related to Hilbert series of ideals of fat points, i.e., ideals of configurations of points with some multiplicity. In this thesis, we consider some special configuration of fat points. In general, Hilbert series of ideals of fat points is a very active field of research. We explain how it is related to the famous Fröberg's conjecture about Hilbert series of generic ideals. Moreover, we use Fröberg's conjecture to deduce the dimensions of several secant varieties of particular projective varieties and, then, to deduce results regarding some particular Waring-type problems for polynomials. In this thesis, we mostly work over the complex numbers. However, we also analyze the case of classical Waring decompositions for monomials over the real numbers. In particular, we classify for which monomials the minimal length of a decomposition in sum of powers of linear forms is independent from choosing the ground field as the field of complex or real numbers.

<p>A lane departure warning system relying exclusively on a camera has several shortcomings and tends to be sensitive to, e.g., bad weather and abrupt manoeuvres. To handle these situations, the system proposed in this thesis uses a dynamic model of the vehicle and integration of relative motion sensors to estimate the vehicle’s position on the road. The relative motion is measured using vision, inertial, and vehicle sensors. All these sensors types are affected by errors such as offset, drift and quantization. However the different sensors are sensitive to different types of errors, e.g., the camera system is rather poor at detecting rapid lateral movements, a type of situation which an inertial sensor practically never fails to detect. These kinds of complementary properties make sensor fusion interesting. The approach of this Master’s thesis is to use an already existing lane departure warning system as vision sensor in combination with an inertial measurement unit to produce a system that is robust and can achieve good warnings if an unintentional lane departure is about to occur. For the combination of sensor data, different sensor fusion models have been proposed and evaluated on experimental data. The models are based on a nonlinear model that is linearized so that a Kalman filter can be applied. Experiments show that the proposed solutions succeed at handling situations where a system relying solely on a camera would have problems. The results from the testing show that the original lane departure warning system, which is a single camera system, is outperformed by the suggested system.</p>

A lane departure warning system relying exclusively on a camera has several shortcomings and tends to be sensitive to, e.g., bad weather and abrupt manoeuvres. To handle these situations, the system proposed in this thesis uses a dynamic model of the vehicle and integration of relative motion sensors to estimate the vehicle’s position on the road. The relative motion is measured using vision, inertial, and vehicle sensors. All these sensors types are affected by errors such as offset, drift and quantization. However the different sensors are sensitive to different types of errors, e.g., the camera system is rather poor at detecting rapid lateral movements, a type of situation which an inertial sensor practically never fails to detect. These kinds of complementary properties make sensor fusion interesting. The approach of this Master’s thesis is to use an already existing lane departure warning system as vision sensor in combination with an inertial measurement unit to produce a system that is robust and can achieve good warnings if an unintentional lane departure is about to occur. For the combination of sensor data, different sensor fusion models have been proposed and evaluated on experimental data. The models are based on a nonlinear model that is linearized so that a Kalman filter can be applied. Experiments show that the proposed solutions succeed at handling situations where a system relying solely on a camera would have problems. The results from the testing show that the original lane departure warning system, which is a single camera system, is outperformed by the suggested system.

The Erdös-Turán Conjecture, posed in 1941 in, states that if a subset B of natural numbers is such that every positive integer n can be written as the sum of a bounded number of terms from B, then the number of such representations must be unbounded as n tends to infinity. The case for h = 2 was given a positive answer by Erdös in 1956. The case for arbitrary h was given by Erdös and Tetali in 1990. Both of these proofs use the probabilistic method, and so the result only shows the existence of such bases but such bases are not given explicitly. Kolountzakis gave an effective algorithm that is polynomial with respect to the digits of n to compute such bases. Borwein, Choi, and Chu showed that the number of representations cannot be bounded by 7. Van Vu showed that the Waring bases contain thin sub-bases. We will discuss these results in the following work.