Submitted by Leonardo Cavalcante (leo.ocavalcante@gmail.com) on 2018-05-02T18:31:29Z No. of bitstreams: 1 Arquivototal.pdf: 790723 bytes, checksum: 7e439258b05733c014dcd3e7de230c1f (MD5) / Made available in DSpace on 2018-05-02T18:31:29Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 790723 bytes, checksum: 7e439258b05733c014dcd3e7de230c1f (MD5) Previous issue date: 2017-08-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this present work, principles and theorems associated to integers are returned, as well as eigenvalues and eigenvectors problems, highlighting a Hermitian matrix. Then it is emphasized to the Circulating Matrices, through which it is found the association to two well-defined polynomials: the representative and the characteristic. Later a brief account about the history of polynomial equations is made, drafting the Cardano-Tartaglia Formulas associated to them. Afterwards a unification is made in the resolution process of the polynomial equations of smaller degrees than the equal to 4, by means of the circulating matrices. The work is completed by proving a Fermat theorem for n = 3, using the Cardano-Tartaglia Formulas. / No presente trabalho, princípios e teoremas associados aos números inteiros são retomados, bem como problemas de autovalores e autovetores, sendo ressaltada a matriz Hermitiana. Em seguida é dado ênfase às Matrizes Circulantes, através das quais verifica-se a associação a dois polinômios bem definidos: o representante e o característico. Posteriormente realiza-se um breve relato acerca da história das equações polinomiais, destacandose as Fórmulas de Cardano-Tartaglia associadas às mesmas. Logo após é feita uma unificação no processo de resolução das equações polinomiais de graus menores do que o igual a 4, por meio das matrizes circulantes. O trabalho é finalizado, sendo provado o Teorema de Fermat para n = 3, recorrendo-se às Fórmulas de Cardano-Tartaglia.

As equações polinomiais são estudadas desde a antiguidade e atualmente são utilizadas, por exemplo, para modelar problemas do cotidiano nas mais variadas áreas do conhecimento. As técnicas de solução de equações polinomiais nem sempre são triviais, principalmente quando envolvem equações de alta ordem e raízes complexas. O ensino desse tema no Ensino Básico é limitado a equações de segundo ou terceiro grau e coeficientes inteiros, o que restringe a aplicação em problemas mais realistas. Assim, o objetivo deste trabalho é trazer uma contribuição aos estudantes, aos professores do Ensino Básico e aos demais interessados, apresentando um material que aborde técnicas de resolução para equação polinomial de diversas naturezas. Iniciamos por uma revisão dos números complexos e dos polinômios, suas operações e propriedades. Embasamos o trabalho com teoremas e permeamos de exemplos com um crescente grau de dificuldade. Dividimos as técnicas de resolução em analíticas e numéricas. Entre as primeiras, tratamos das relações de Girard, das fórmulas resolventes e de alguns casos particulares de equações. Entre as técnicas numéricas, estudamos o método de Newton, o método das secantes e o método de Newton-Bairstow, este último para encontrar raízes complexas. / Polynomial equations have been studied since antiquity and are currently used, for example, to model everyday problems in the most varied areas of knowledge. The solution techniques of polynomial equations are not always trivial, especially when they involve high order equations and complex roots. The teaching of this subject in Basic Education is limited to second or third degree equations and integer coefficients, which restricts the application to more realistic problems. Thus, the objective of this work is to bring a contribution to students, teachers of Basic Education and other interested parties, presenting a material that treats of resolution techniques for polynomial equation of different natures. We begin with a review of complex numbers and polynomials, their operations and properties. We support the work with theorems and permeate examples with an increasing degree of difficulty. We divide the techniques of resolution into analytical and numerical. Among the first, we deal with Girards relations, the resolvent formulas, and some particular cases of equations. Among numerical techniques, we studied the Newton method, the secant method, and the Newton-Bairstow method, the last one to find complex roots.

Made available in DSpace on 2014-12-17T15:04:59Z (GMT). No. of bitstreams: 1 LiceuLC_DISSERT.pdf: 582558 bytes, checksum: 77eef9f6ab00def7a0eb1adb59957b9f (MD5) Previous issue date: 2012-12-20 / The present study investigates how the inter-relationship of the content of polynomial equations works with structured activities and with the history of mathematics through a sequence of activities presented in an e-book, so that the result of this research will proceed will result in a didactic and pedagogic proposal for the teaching of polynomial equations in a historical approach via the reported e-book. Therefore, we have considered in theoretical and methodological assumptions of the History of Mathematics, in structured activities and new technologies with an emphasis on e-book tool. We used as a methodological approach the qualitative research, as our research object adjusts to the objectives of this research mode. As methodological instruments, we used the e-book as a synthesis tool of the sequence of activities to be evaluated, while the questionnaires, semi-structured interviews and participant observation were designed to register and analyze the evaluation made by the research, participants in the structured activities. The processing and analysis of data collected though the questionnaires were organized, classified and quantified in summary tables to facilitate visualization, interpretation, understanding, and analysis of these data. As for participant observation was used to contribute to the qualitative analysis of the quantified data. The interviews were synthetically transcribed and qualitatively analyzed. The analysis ratified our research objectives and contributed to improve, approve and indicate the use of e-book for the teaching of polynomial equations. Thus, we consider that this educational product will bring significant contributions to the teaching of mathematical content, in Basic Education / O presente estudo investiga como se d? a inter-rela??o do conte?do de Equa??es Polinomiais com Atividades Estruturadas e a Hist?ria da Matem?tica por meio de uma sequ?ncia de atividades apresentadas num e-book, de modo que o resultado dessa investiga??o resulte numa proposta did?tica e pedag?gica para o ensino de Equa??es Polinomiais numa abordagem hist?rica por meio do referido e-book. Para tanto, nos fundamentamos em pressupostos te?ricos e metodol?gicos da Hist?ria da Matem?tica, em Atividades Estruturadas e em novas tecnologias, com ?nfase na ferramenta e-book. Utilizamos como abordagem metodol?gica a pesquisa qualitativa, visto que o nosso objeto de pesquisa se ajusta aos objetivos dessa modalidade de pesquisa. Como instrumentos metodol?gicos, utilizamos o ebook como ferramenta s?ntese da sequ?ncia de atividades a ser avaliada, enquanto que os question?rios, entrevistas semiestruturadas e observa??o participante tiveram o intuito de registrar e analisar a avalia??o que os participantes da pesquisa, fizeram das atividades estruturadas. O tratamento e an?lise dos dados colhidos por meio dos question?rios foram organizados, classificados e quantificados em quadros sint?ticos para facilitar a visualiza??o, a interpreta??o, a compreens?o e a an?lise desses dados. Quanto ? observa??o participante, foi utilizada para contribuir com a an?lise qualitativa dos dados quantificados. As entrevistas foram transcritas sinteticamente e analisadas qualitativamente. A an?lise ratificou os nossos objetivos da pesquisa e contribuiu para aperfei?oar, aprovar e indicar o uso do e-book para o ensino de Equa??es Polinomiais. Assim, consideramos que esse produto educacional trar? importantes contribui??es para o ensino desse conte?do matem?tico, na Educa??o B?sica

The theory of Grobner bases has garnered the interests of a large number of researchers in computational algebra due to its applications not only in mathematics but also in areas like control systems, robotics, cryptography to name a few. It is well known that the computation of Grobner bases takes time doubly exponential in the number of indeterminates rendering it impractical in all but a few places.The current known algorithms for Grobner bases depend on the term order over which Grobner bases is computed. In this thesis, we study computational complexity of some problems in computational ideal theory. We also study the algebraic formulation of combinatorial optimization problems. Gritzmann and Sturmfels (1993) posed the following question: Given a set of generators, decide whether it is a Gr¨obner bases with respect to some term order. This problem, termed as the Grobner Basis Detection(GBD)problem, was introduced as an application of Minkowski addition of polytopes. It was shown by Sturmfels and Wiegelmann (1997) that GBD is NP-hard. We study the problem for the case of zero-dimensional ideals and show that the problem is hard even in this special case. We study the detection problem in the case of border bases which are an alternative to Grobner bases in the case of zero dimensional ideals. We propose the Border Basis Detection(BBD) problem which is defined as follows: Given a set of generators of an ideal, decide whether that set of generators is a border basis of the ideal with respect to some order ideal. It is shown that BBD is NP-complete. We also formulate the rainbow connectivity problem as a system of polynomial equations such that solving the polynomial system yields a solution to it. We give an alternate formulation of the rainbow connectivity problem as a membership problem in polynomial ideals.

The problem of polynomial representability of functions is central to many branches of mathematics. If the underlying set is a finite field, every function can be represented as a polynomial. In this thesis we consider polynomial representability over a special class of finite rings, namely, Zpn, where p is a prime and n is a positive integer. This problem has been studied in literature and the two notable results were given by Carlitz(1965) and Kempner(1921).While the Kempner’s method enumerates the set of distinct polynomial functions, Carlitz provides a necessary and sufficient condition for a function to be polynomial using Taylor series. Further, these results are existential in nature. The aim of this thesis is to provide an algorithmic characterization, given a prime p and a positive integer n, to determine whether a given function over Zpn is polynomially representable or not. Note that one can give an exhaustive search algorithm using the previous results. Our characterization involves describing the set of polynomial functions over Zpn with a ‘suitable’ generating set. We make use of this result to give an non-exhaustive algorithm to determine whether a given function over Zpn is polynomial representable.nβ

Η παρούσα διδακτορική διατριβή πραγματεύεται το θέμα της αποδοτικής και με βεβαιότητα εύρεσης όλων των ριζών της παραμετρικής εξίσωσης f(x;[p]) = 0, μιας συνεχώς διαφορίσιμης συνάρτησης f με [p] ένα διάνυσμα που περιγράφει όλες τις παραμέτρους της παραμετρικής εξίσωσης και τυποποιούνται με τη μορφή διαστημάτων. Για την επίλυση αυτού του προβλήματος χρησιμοποιήθηκαν εργαλεία της Διαστηματικής Ανάλυσης. Το κίνητρο για την ερευνητική ενασχόληση με το παραπάνω πρόβλημα προέκυψε μέσα από ένα κλασικό πρόβλημα αριθμητικής ανάλυσης: την αριθμητική επίλυση συστημάτων πολυωνυμικών εξισώσεων μέσω διαστηματικής ανάλυσης. Πιο συγκεκριμένα, προτάθηκε μια ευρετική τεχνική αναδιάταξης του αρχικού πολυωνυμικού συστήματος που φαίνεται να βελτιώνει σημαντικά, κάθε φορά, τον χρησιμοποιούμενο επιλυτή. Η ανάπτυξη, καθώς και τα αποτελέσματα αυτής της εργασίας αποτυπώνονται στο Κεφάλαιο 2 της παρούσας διατριβής. Στο επόμενο Κεφάλαιο 3, προτείνεται μια μεθοδολογία για την αποδοτική και αξιόπιστη επίλυση μη-γραμμικών εξισώσεων με διαστηματικές παραμέτρους, δηλαδή την αποδοτική και αξιόπιστη επίλυση διαστηματικών εξισώσεων. Πρώτα, δίνεται μια νέα διατύπωση της Διαστηματικής Αριθμητικής και αποδεικνύεται η ισοδυναμία της με τον κλασσικό ορισμό. Στη συνέχεια, χρησιμοποιείται η νέα διατύπωση της Διαστηματικής Αριθμητικής ως θεωρητικό εργαλείο για την ανάπτυξη μιας επέκτασης της διαστηματικής μεθόδου Newton που δύναται να επιλύσει όχι μόνο κλασικές μη-παραμετρικές μη-γραμμικές εξισώσεις, αλλά και παραμετρικές (διαστηματικές) μη-γραμμικές εξισώσεις. Στο Κεφάλαιο 4 προτείνεται μια νέα προσέγγιση για την αριθμητική επίλυση του προβλήματος της Ολικής Βελτιστοποίησης με περιορισμούς διαστήματα, χρησιμοποιώντας τα αποτελέσματα του Κεφαλαίου 3. Το πρόβλημα της ολικής βελτιστοποίησης, ανάγεται σε πρόβλημα επίλυσης διαστηματικών εξισώσεων, και γίνεται εφικτή η επίλυσή του με τη βοήθεια των θεωρητικών αποτελεσμάτων και της αντίστοιχης μεθοδολογίας του Κεφαλαίου 3. Στο τελευταίο Κεφάλαιο δίνεται μια νέα αλγοριθμική προσέγγιση για το πρόβλημα της επίλυσης διαστηματικών πολυωνυμικών εξισώσεων. Η νέα αυτή προσέγγιση, βασίζεται και γενικεύει την εργασία των Hansen και Walster, οι οποίοι πρότειναν μια μέθοδο για την επίλυση διαστηματικών πολυωνυμικών εξισώσεων 2ου βαθμού. / In this dissertation the problem of finding reliably and with certainty all the zeros a pa-rameterized equation f(x;[p]) = 0, of a continuously differentiable function f is considered, where [p] is an interval vector describing all the parameters of the Equation, which are formed with interval numbers. For this kind of problem, methods of Interval Analysis are used. The incentive to this scientific research was emerged from a classic numerical analysis problem: the numerical solution of polynomial systems of equations using interval analysis. In particular, a heuristic reordering technique of the initial polynomial systems of equations is proposed. This approach seems to improve significantly the used solver. The proposed technique, as well as the results of this publication are presented in Chapter 2 of this dissertation. In the next Chapter 3, a methodology is proposed for solving reliably and efficiently parameterized (interval) equations. Firstly, a new formulation of interval arithmetic is given and the equivalence with the classic one is proved. Then, an extension of interval Newton method is proposed and developed, based on the new formulation of interval arithmetic. The new method is able to solve not only classic non-linear equations but, non-linear parameterized (interval) equation too. In Chapter 4 a new approach on solving the Box-Constrained Global Optimization problem is proposed, based on the results of Chapter 3. In details, the Box-Constrained Global Optimization problem is reduced to a problem of solving interval equations. The solution of this reduction is attainable through the methodology developed in Chapter 3. In the last Chapter of this dissertation a new algorithmic approach is given for the problem of solving reliably and with certainty an interval polynomial equation of degree $n$. This approach consists in a generalization of the work of Hansen and Walster. Hansen and Walster proposed a method for solving only quadratic interval polynomial equations

This thesis covers detailed analysis of bond pricing function. It focuses on connections between mathematical definitions and financial practice and it points out advantages and drawbacks of currently used function. Well known properties of this function are extended to negative internal rate of return values. This topic is further discussed with internal rate of return polynomial equations solving. Taylor series approximation is also shown regarding duration and convexity of bonds.

This study investigates the possibilities and implications for the teaching of the solving of polynomial equations. It is historically directed and also focusses on the working procedures in algebra which target the cognitive and affective domains. The teaching implications of the development of representational styles of equations and their solving procedures are noted. Since concepts in algebra can be conceived as processes or objects this leads to cognitive obstacles, for example: a limited view of the equal sign, which result in learning and reasoning problems. The roles of sense-making, visual imagery, mental schemata and networks in promoting meaningful understanding are scrutinised. Questions and problems to solve are formulated to promote the processes associated with the solving of polynomial equations, and the solving procedures used by a group of college students are analysed. A teaching model/method, which targets the cognitive and affective domains, is presented. / Mathematics Education / M.A. (Mathematics Education)

This study investigates the possibilities and implications for the teaching of the solving of polynomial equations. It is historically directed and also focusses on the working procedures in algebra which target the cognitive and affective domains. The teaching implications of the development of representational styles of equations and their solving procedures are noted. Since concepts in algebra can be conceived as processes or objects this leads to cognitive obstacles, for example: a limited view of the equal sign, which result in learning and reasoning problems. The roles of sense-making, visual imagery, mental schemata and networks in promoting meaningful understanding are scrutinised. Questions and problems to solve are formulated to promote the processes associated with the solving of polynomial equations, and the solving procedures used by a group of college students are analysed. A teaching model/method, which targets the cognitive and affective domains, is presented. / Mathematics Education / M.A. (Mathematics Education)