Currently, there exist several methods for finding roots of polynomial functions. From elementary processes such as the quadratic formula and the Rational Root Theorem to calculus-based ideas, choosing an appropriate means of solving often depends on the conditions of the given polynomial. This report will explore several solving methods and discuss their advantages as well as their limitations. / text

Generalized inverses of matrices are of great importance in the resolution of linear systems and have been extensively studied by many researchers. A collection of some results on generalized inverses of matrices over commutative rings has been provided by K. P. S. Bhaskara Rao (2002). In this thesis, we consider constructing algorithms for finding generalized inverses and generalizing the results collected in Rao's book to the non-commutative case. We first construct an algorithm by using the greatest common divisor to find a generalized inverse of a given matrix over a commutative Euclidean domain. We then build an algorithm for finding a generalized inverse of a matrix over a non-commutative Euclidean domain by using the one-sided greatest common divisor and the least common left multiple. Finally, we explore properties of various generalized inverses including the Moore-Penrose inverse, the group inverse and the Drazin inverse in the non-commutative case.

本文分為兩章. 第一章先介紹sieve-number(即s(H)) ,並將所有mixed hypergraph的最大著色數能用n-s(H)的圖形條件限制出來.再討論能用s(H)表示其最大著色數的圖形. 第二章主要是討論interval mixed hypergraph的著色方程式.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 110-116). Also available via the Internet.

In this thesis numerical integration in one and two dimensions is considered. In chapter two transformation methods are considered primarily for singular integrals and methods of computing the transformations themselves are derived. The well-known transformation based on the IMT rule and error function are extended to non-standard functions. The implementation of these rules and their performances are demonstrated. These transformations are then extended to two-dimensions and are used to develop accurate rules for integrating singular integrals. In addition to this, a polynomial transformation with the aim of the reduction in the number of function evaluations is also considered and the resultant product rule is applied to two-dimensional non-singular integrals. Finally, the use of monomials in the construction of integration rules for non-singular two-dimensional integrals is considered and some rules developed. In all these situations the rules developed are tested and compared with existing methods. The results show that the new rules compare favourably with existing ones.

Les matériaux composites à matrice organique sont de plus en plus utilisés dans divers domaines tels que l'aérospatiale ou les énergies marines renouvelables en raison de leurs excellentes propriétés spécifiques. Cependant, les procédés de fabrication des structures composites sont complexes et peuvent conduire à l'apparition de défauts, en particulier de plissement des plis et de porosité, qui affectent les propriétés mécaniques de la structure. Les pièces composites sont ainsi systématiquement soumises à des contrôles CND long et coûteux. En cas de résultats négatifs par rapport à des critères conservatifs, celles-ci peuvent être rejetées, avec des conséquences économiques non négligeables. L'objectif de cette étude est de quantifier l'impact des défauts observés et des incertitudes associées sur le comportement de pièce composite. Dans ce travail, nous adoptons une vision paramétrique des incertitudes consistant à représenter le contenu probabiliste à travers d’un ensemble fini de variables aléatoires. Nous nous concentrons sur la propagation des incertitudes basée sur des méthodes stochastiques spectrales. L'étude portant sur le défaut de porosités se fait à l’échelle microscopique puis macroscopique. Les paramètres aléatoires d'entrée sont liés à la géométrie des porosités et à leur taux. L'étude du défaut plissements à l'échelle mésoscopique est basée sur une représentation paramétrique de la géométrie du plissement. Les paramètres aléatoires d'entrée représentent alors la forme et la taille de ces défauts. Il est donc possible d'analyser l'impact de ces défauts à l'échelle structurelle par des grandeurs mécaniques classiques et des critères de rupture. / Composite materials are increasingly used in various fields such as aerospace or renewable marine energies due to their excellent specific properties. However, the manufacturing processes of the composite structures are complex, which can lead to the appearance of defects, particularly wrinkles and porosities, which affect the mechanical properties of the structure. Based on conservative criteria, a system of non-destructive testing of composite parts thus makes it possible to judge their conformity. In case of non-conformity, those components are rejected, with non-negligible economic consequences. The objective of this study is to quantify the impact of the defects and associated uncertainties on the behavior of composite parts. In this work, we adopt a parametric vision of the uncertainties consisting in representing the probabilistic content through a finite set of random variables. We focus on the propagation of uncertainties based on spectral stochastic methods. The study involving porosity is done at the micro-scale and then at the macro-scale. The random input parameters are related to the geometry of the porosities and their rates. The study of the wrinkle defect, done at the mesoscopic scale, is based on a parametric representation of the geometry of the wrinkle. The random input parameters then represent the shape and size of this defect. It is therefore possible to analyze the impact of these two manufacturing defects at a structural scale through classical mechanical quantities and check the failure of the structure with failure criteria.

Current methodologies for built-in test pattern generation usually employ a predetermined linear feedback shift register (LFSR) in order to generate or decompress deterministic test patterns. As a direct consequence, the test pattern computation and the fault coverage are constrained to the preselected architecture. Work has been done to determine desirable characteristics in the LFSR to be used. Also, work has been done in the use of these predefined architectures, in order to compact the test data. In general, these methodologies take advantage of the large amount of don't care bits present in the test patterns, to accommodate the few specified bits to the output generated by the predefined LFSR. This dissertation explores the design of the LFSR as a built-in mechanism for test pattern generation in integrated circuits. The advantage of designing such devices is that the test set generation process is not constrained to a predefined LFSR mechanism, and the fault coverage is not affected. The methodologies presented in this work are based on cryptography concepts and heuristics to perform its computation. First, it is shown how these concepts can be adapted to test pattern generation. After this, methodologies are presented to generate one-dimensional and two-dimensional test sets. For the case of two-dimensional test set, the design of phase shifters is included.

During the last twenty years the theory of linear algebraic and high-order differential equation systems has been greatly researched. Two commonly used types of system description are the so-called matrix fraction description (MFD) and the Rosenbrock system matrix (RSM); these are defined by polynomial matrices in one indeterminate. Many of the system's physical properties are encoded as algebraic properties of these polynomial matrices. The theory is well developed and the structure of such systems is well understood. Analogues of these 1-D realisations can be set up for many dimensional systems resulting in polynomial matrices in many indeterminates. The scarcity of detailed algebraic results for such matrices has limited the understanding of such systems.

In [ASTW] it was given a Gröbner reduction based division formula for entire functions by polynomial ideals. Here we give degree bounds where the input function can be truncated in order to compute approximations of the coeffcients of the power series appearing in the division formula within a given precision. In addition, this method can be applied to the approximation of the value of the remainder function at some point.

In [ASTW] it was given a Gröbner reduction based division formula for entire functions by polynomial ideals. Here we give degree bounds where the input function can be truncated in order to compute approximations of the coeffcients of the power series appearing in the division formula within a given precision. In addition, this method can be applied to the approximation of the value of the remainder function at some point.