In this thesis we explore the topic of resultants, focusing on the resultants of binomial sys-tems. The study aims to provide an introduction to resultants and explore some of the morerecent research on the relationship between cycles in graphs and the resultant of binomialsystems. This will be done by going over some general theory from algebraic geometry suchas polynomials and ideals, in addition to studying the graphs of the binomial systems and thesurrounding theory. The research includes a comprehensive analysis of these graphs and theircycles, leading to findings that the cycles that appear are not restricted in length. This workprovides exploration of a potential extension of the recent research and encourages furtherresearch on the topic.

This work explores the unsolvability of the general quintic equation through the lens of Galois theory. We begin by providing a historical perspective on the problem. This starts with the solution of the general cubic equation derived by Italian mathematicians. We then move on to Lagrange's insights on the importance of studying the permutations of roots. Finally, we discuss the critical contributions of Évariste Galois, who connected the solvability of polynomials to the properties of permutation groups. Central to our thesis is the introduction and motivation of key concepts such as fields, solvable groups, Galois groups, Galois extensions, and radical extensions. We rigorously develop the theory that connects the solvability of a polynomial to the solvability of its Galois group. After developing this theoretical framework, we go on to show that there exist quintic polynomials with Galois groups that are isomorphic to the symmetric group S5. Given that S5 is not a solvable group, we establish that the general quintic polynomial is not solvable by radicals. Our work aims to provide a comprehensive and intuitive understanding of the deep connections between polynomial equations and abstract algebra.

Thesis (PhD)–Stellenbosch University, 2014. / ENGLISH ABSTRACT: See pdf for abstract / AFRIKAANSE OPSOMMING: Sien PDF vir die opsomming

Η τόσο συχνή εμφάνιση της ακολουθίας Fibonacci στη φύση καθώς και ο συσχετισμός της με πλείστους τομείς της μαθηματικής επιστήμης έδωσε αφορμή να ενταθεί η έρευνα στην περιοχή αυτή. Και τούτο ιδιαίτερα τις τελευταίες δύο δεκαετίες. Τα πολυώνυμα Fibonacci k-τάξης αποτελούν μία από τις ευρύτερες γενικεύσεις της ακολουθίας Fibonacci. Η μελέτη των πολυωνύμων αυτών και η σύνδεσή τους με την πιθανότητα είναι το κύριο αντικείμενο της διατριβής αυτής. Η κατανομή πιθανότητας της τ. μ. Χk, όπου Xk το πλήθος των επαναλήψεων σε ένα πείραμα δοκιμών Bernoulli ώσπου να προκύψουν k διαδοχικές επιτυχίες, έχει ονομασθεί "κατανομή πιθανότητας Fibonacci". Η σχέση της κατανομής Fibonacci με τα πολυώνυμα Fibonacci οδήγησε στις γενικευμένες κατανομές πιθανότητας που αποτέλεσε το δεύτερο άξονα της μελέτης αυτής. / The fact that Fibonacci sequences appear so frequently in nature together with their interrelationship with almost any branch of mathematics, has resulted in an intesive research in this area particularly during the last two decades. One of the most wide extensions of the Fibonacci sequence is provided by the Fibonacci polynomials of order k. The study of these polynomials and thier relation with probability is the main part of this dissertation. The probability distribution of the r.v. Xk, where Xk denotes the number of trials until the occurrence of the kth consecutive success in indipendent trials, thas been called "Fibonacci Probability Distribution". The relation between the Fibonacci Distribution and the Fibonacci polynomials led to generalized probability distributions (Geometric, Negative binomial, Poisson and Compound poisson) which consists the second major part of this study.

The special functions and, particularly, the classical orthogonal polynomials encountered in many branches of applied mathematics and mathematical physics satisfy a second order differential equation, which is known as the equation of the hypergeometric type. The variable coefficients in this equation of the hypergeometric type are of special structures. Depending on the coefficients the classical orthogonal polynomials associated with the names Jacobi, Laguerre and Hermite can be derived as solutions of this equation. In this thesis, these well known classical polynomials as well as another class of polynomials, which receive less attention in the literature called Bessel polynomials have been studied.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-07-21Bitstream added on 2014-06-13T19:55:24Z : No. of bitstreams: 1 milani_ff_me_sjrp.pdf: 539043 bytes, checksum: b4613024414cd9fa758d64376a046176 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / O objetivo deste trabalho é estudar os polinômios de Szegõ, que são ortogonais no círculo unitário, e suas relações com certas frações contínuas de Perron-Carathéodory e quadratura no círculo unitário, afim de resolver o problema de momento trigonométrico. Além disso, estudar a utilização dos polinômios de Szegõ na determinação das freqüências de um sinal trigonométrico em tempo discreto xN(m). Para isso, investigamos os polinômios de Szegõ gerados por uma medida N definida através do sinal trigonométrico xN(m), para m = 0, 1, 2,...N -1, e o comportamento dos zeros desses polinômios quando N_8. / The purpose here is to study the orthogonal polynomials on the unit circle, known as Szegõ polynomials, and the relations to Perron- Carathéodory continued fractions, and quadratures on the unit circle in order to solve the trigonometric moment problem. Another purpose is to study how the Szegõ polynomials can be used to determine the frequencies from a discrete time trigonometric signal xN(m). We investigate the Szegõ polynomials associated with a measure N defined by the trigonometric sinal xN(m), m = 0, 1, 2, ...N -1. We study the behaviour of zeros of these polynomials when N 8.

<div> <br></div><div> We consider the behavior of zeros of random polynomials of the from</div><div> \begin{equation*}</div><div> P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z)</div><div> \end{equation*}</div><div> as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.</div>

No description available.

Indiana University-Purdue University Indianapolis (IUPUI) / In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\ep}}))$ for any $\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\th_4(n\om)F^{n^2}(1+O(n^{-1}))$, where $\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials. / Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer. / Mestre em Matemática