O objetivo desta tese é analisar a criação doParque Nacional da Serra do Divisor (PNSD), no Acre, em 1989, como parte de um longo processo histórico de implantação de Unidades de Conservação no Brasil. Também procuramos avaliar como se deu a escolha da área onde se localiza o Parque e os procedimentos de elaboração e implantação do Plano de Manejo desta unidade. Antes de analisar o caso brasileiro, consideramos importante acompanhar como se deu a criação de áreas de conservação em outros países. Desde a criação do Parque Nacional de Yellowstone, em 1872, nos Estados Unidos, até a instalação da União Internacional pela Conservação da Natureza (UICN), em 1948, e sua atuação em defesa da criação de áreas protegidas nos vários continentes. A União também foi fundamental no estabelecimento de padrões internacionais para estas áreas. Pela pesquisa, pudemos observar que as primeiras Unidades de Conservação no Brasil, os Parques Nacionais dos anos 1930, seguiram o modelo norte-americano, que tinha como objetivo a preservação da natureza e a contemplação das belezas cênicas, não permitindo a presença de moradores. Já noano 2000, quando foi criado o Sistema Nacional de Unidades de Conservação SNUC, acompanhando os novos padrões internacionais, o país contava com diferentes categorias de Unidades de Conservação, agrupadas em dois grupos: as de uso integral, que não permitiam a presença de moradores, e as de uso sustentável, que tentavam conciliar a preservação da natureza com a presença de moradores. Para a realização do trabalho, além de amplo levantamento da bibliografia que trata das Unidades de Conservação no Brasile em outros países, fizemos levantamento exaustivo da legislação federal referente a questões ambientais e áreas de preservação. Para o estudo do PNSD-AC, também levantamos legislação estadual e documentação relativa a organismos civis e aos movimentos sociais de cunho ambiental que ocorreram no Acre a partir dos anos 1970. Também realizamos entrevistas com especialistas que atuaram na elaboração do Plano de Manejo do PNSD. Deacordo com os levantamentos realizados, uma das conclusões da pesquisa é que aescolha da região onde hoje está localizado o PNSD, não foi aleatória, tendo seguido a orientação de vários estudos que mostraram o potencial da região em termos de biodiversidade. / The objective of this thesis is to analyze the creation of the Serra do Divisor National Park (PNSD) in Acre in 1989 as part of a long historical process Conservation Units deployment in Brazil. We also seek to assess how was the choice of the area where is located the Park and procedures for the establishment and implementation of the Management Plan of the unit. Before analyzing the Brazilian case, we consider important to monitor how the creation of protected areas in other countries was. Since the creation of Yellowstone National Park in 1872, the United States, until the installation of the International Union for Conservation of Nature (IUCN) in 1948, and his defense in action the creation of protected areas in the various continents. The Union was also instrumental in establishing international standards for these areas. For the study, we observed that the first protected areas in Brazil, the National Parks of the 1930s, followed the US model, which aimed to the preservation of nature and the contemplation of scenic beauty, not allowing the presence of residents. Already in 2000, when it was created the National Protected Areas System -SNUC, following the new international standards, the country had different categories of protected areas, grouped into two groups: the full use, which did not allow the presence residents, and sustainable use, which attempted to reconcile the preservation of nature with the presence of residents. To carry out the work, plus extensive survey of the literature dealing with protected areas in Brazil and other countries, did exhaustive survey of federal legislation concerning environmental issues and conservation areas. To study the PNSD-AC, also raised state legislation and documentation of civil organizations and social movements of an environmental nature that occurred in Acre from the 1970s also conducted interviews with experts who worked in the preparation of PNSD Management Plan. According to surveys conducted one of the research findings is that the choice of the region where today is located the PNSD, was not random, having followed the guidance of several studies that have shown the potential of the region in terms of biodiversity.

Greatest common divisor domains, Bezout domains, valuation rings, and Prüfer domains are studied. Chapter One gives a brief introduction, statements of definitions, and statements of theorems without proof. In Chapter Two theorems about greatest common divisor domains and characterizations of Bezout domains, valuation rings, and Prüfer domains are proved. Also included are characterizations of a flat overring. Some of the results are that an integral domain is a Prüfer domain if and only if every overring is flat and that every overring of a Prüfer domain is a Prüfer domain.

Submitted by Cristiane Chim (cristiane.chim@ucpel.edu.br) on 2017-02-10T11:37:48Z No. of bitstreams: 1 pedro luis.pdf: 2493331 bytes, checksum: 38fdc4ec8b3fee0815ba222c508dc8d4 (MD5) / Made available in DSpace on 2017-02-10T11:37:48Z (GMT). No. of bitstreams: 1 pedro luis.pdf: 2493331 bytes, checksum: 38fdc4ec8b3fee0815ba222c508dc8d4 (MD5) Previous issue date: 2016-12-05 / The matrix inversion calculation is present in several applications in the area of Signal Processing. Among these applications, the adaptive filtering, based on the algorithm of Affine Projections, includes the calculation of matrix inversion, which adds a high computational complexity. There are several algorithms for calculating matrix inversion. The complexity of the algorithm is associated with the size of the matrix, which varies according to the target application. This dissertation proposes the implementation in dedicated hardware of the analytical algorithm of matrix inversion. This algorithm is most appropriate for the implementation of a 2x2 size matrix, which is the appropriate size for an implementation of the algorithm of Affine Projections for several practical applications. In the matrix inversion block, the divisor circuit is that adds the highest computational complexity. Among the division algorithms from the literature, algorithms based on functional iterations are considered the fastest, because they are able to take advantage of high speed multipliers to converge in a quadratic form to a result. Among the algorithms based on functional iterations, Newton-Raphson and Goldschmidt algorithms are the most used algorithms. However, the Goldschmidt algorithm has been more used in applications that demand high processing speed, because unlike the Newton-Raphson algorithm, where the multiplications are dependent on each other, in the Goldschmidt algorithm the multiplications are performed in parallel. In this work, it is proposed the hardware implementation of an efficient divisor circuit based on the Goldschmidt algorithm. The divider circuit uses a radix-4 multiplier from the literature, which is more efficient in terms of power dissipation, when compared to the divider circuit using the multiplier from the synthesis tool. The proposed divider circuit increases the range of operating values by using the Q7.8 standard, which allows values between -127.99609375 and +127.99609375, rather than the original Goldschmidt divider, which supports a narrow range of values between 1 and 2. The main results show that the use of the proposed efficient Goldschmidt divider circuit makes the matrix inverter circuit with a lower power dissipation, which becomes an attractive for a future implementation of the complete affine projections algorithm in dedicated hardware. / O cálculo de inversão de matrizes está presente em várias aplicações da área de Processamento de Sinais. Entre essas aplicações, a filtragem adaptativa, baseada no algoritmo de Projeções Afins, inclui o cálculo de inversão de matrizes, que agrega uma elevada complexidade computacional. Existem vários algoritmos para o cálculo de inversão de matrizes. A complexidade do algoritmo está associada ao tamanho da matriz, que varia de acordo com a aplicação alvo. Essa dissertação propõe a implementação em hardware dedicado do algoritmo analítico de inversão de matrizes. Esse algoritmo é o mais apropriado para a implementação de uma matriz de tamanho 2x2, que é o tamanho adequado para uma implementação do algoritmo de Projeções Afins para diversas aplicações práticas. No bloco de inversão de matriz, o circuito divisor é o que agrega a maior complexidade computacional. Dentre os algoritmos de divisão presentes na literatura, os algoritmos baseados em iterações funcionais são considerados os mais rápidos, pois são capazes de tirar proveito de multiplicadores de alta velocidade, para convergir de forma quadrática para um resultado. Dentre os algoritmos baseados em iterações funcionais, destacam-se os algoritmos de Newton-Raphson e de Goldschmidt. Entretanto, o algoritmo de Goldschmidt tem sido mais utilizado em aplicações que demandam alta velocidade de processamento, pois ao contrário do algoritmo Newton-Raphson, onde as multiplicações são dependentes umas das outras, no algoritmo Goldschmidt as multiplicações são realizadas em paralelo. Nesse trabalho, propõe-se a implementação em hardware de um circuito divisor eficiente baseado no algoritmo Goldschmidt. O circuito divisor usa um multiplicador na base 4 da literatura, que torna o divisor mais eficiente em termos de dissipação de potência, quando comparado ao circuito divisor usando o multiplicador da ferramenta de síntese. O circuito divisor proposto aumenta a faixa de valores de operação através do uso do padrão Q7.8, que permite valores entre -127.99609375 e +127.99609375, ao contrário do divisor Goldschmidt original, que admite uma estreita faixa de valores ente 1 e 2. Os principais resultados mostram que o uso do divisor Goldschmidt eficiente proposto torna o circuito inversor de matriz com uma menor dissipação de potência, o que se torna um atrativo para uma futura implementação da arquitetura completa do algoritmo de Projeções Afins.

This thesis discusses the Kaplansky zero divisor conjecture. The conjecture states that a group ring of a torsion-free group over a field has no nonzero zero divisors. There are situations for which this conjecture is known to hold, such as linearly orderable groups, unique product groups, solvable groups, and elementary amenable groups. This paper considers the possibility that the conjecture is false and there is some counterexample in existence. The approach to searching for such a counterexample discussed here is to first find a torsion-free group that has subsets A and B such that AB has no unique product. We do this by exhaustively searching for the subsets A and B with fixed small sizes. When |A| = 1 or 2 and |B| is arbitrary we know that AB contains a unique product, but when |A| is larger, not much was previously known. After an example is found we then verify that the sets are contained in a torsion-free group and further investigate whether the group ring yields a nonzero zero divisor. Together with Dr. Pace P. Nielsen, assistant math professor of Brigham Young University, we created code that was implemented in Magma, a computational algebra system, for the purpose of considering each size of A and B and running through each case. Along the way we check for the possibility of torsion elements and for other conditions that lead to contradictions, such as a decrease in the size of A or B. Our results are the following: If A and B are sets of the sizes below contained in a torsion-free group, then they must contain a unique product. |A| = 3 and |B| ≤ 16; |A| = 4 and |B| ≤ 12; |A| = 5 and |B| ≤ 9; |A| = 6 and |B| ≤ 7. We have continued to run cases of larger size and hope to increase the size of B for each size of A. Additionally, we found a torsion-free group containing sets A and B, both of size 8, where AB has no unique product. Though this group does not yield a counterexample for the Kaplansky zero divisor conjecture, it is the smallest explicit example of a non-uniqueproduct group in terms of the size of A and B.

The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of tau-factorization, studied extensively by A. Frazier and D.D. Anderson. Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements. In this thesis, we investigate several methods for extending the theory of tau-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. Agargun and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations. This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using tau_z-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using tau-U-factorization, we are able to answer many questions that arise when discussing direct products of rings. There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending tau-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.

The elasticity of a Krull domain R is equivalent to the elasticity of the block monoid B(G,S), where G is the divisor class group of R and S is the set of elements of G containing a height-one prime ideal of R. Therefore the elasticity of R can by studied using the divisor class group. In this dissertation, we will study infinite divisor class groups to determine the elasticity of the associated Krull domain. The results will focus on the divisor class groups Z, Z(p infinity), Q, and general infinite groups. For the groups Z and Z(p infinity), it has been determined which distributions of the height-one prime ideals will make R a half-factorial domain (HFD). For the group Q, certain distributions of height-one prime ideals are proven to make R an HFD. Finally, the last chapter studies general infinite groups and groups involving direct sums with Z. If certain conditions are met, then the elasticity of these divisor class groups is the same as the elasticity of simpler divisor class groups.

The interplay between spectrum and structure of graphs is the recurring theme of the three more or less independent chapters of this thesis. The first chapter provides a method to relate the eigensolutions of two matrices, one being the principal submatrix of the other, via an arbitrary annihilating polynomial. This is extended to lambda-matrices and to matrices the entries of which are rational functions in one variable. The extension may be interpreted as a possible generalization of other known techniques which aim at reducing the size of a matrix while preserving the spectral information. Several aspects of an application in order to reduce the computational costs of ordinary eigenvalue problems are discussed. The second chapter considers the straightforward extension of the well known concept of equitable partitions to weighted graphs, i.e. complex matrices. It provides a method to divide the eigenproblem into smaller parts corresponding to the front divisor and its complementary factor in an easy and stable way with complexity which is only quadratic in matrix size. The exploitation of several equitable partitions ordered by refinement is discussed and a suggestion is made that preserves hermiticity if present. Some generalizations of equitable partitions are considered and a basic procedure for finding an equitable partition of complex matrices is given. The third chapter deals with isospectral and unitary equivalent graphs. It introduces a construction for unitary equivalent graphs which contains the well known GM-switching as a special case. It also considers an algebra of graph matrices generated by the adjacency matrix that corresponds to the 1-dimensional Weisfeiler-Lehman stabilizer in a way that mimics the correspondence of the coherent closure and the 2-dimensional Weisfeiler-Lehman stabilizer. The algebra contains the degree matrix, the (combinatorial, signless and normalized) Laplacian and the Seidel matrix. An easy construction produces graph pairs that are simultaneously unitary equivalent w.r.t. that algebra.

在這篇論文裡，我們研究Baker-Norine的搬硬幣遊戲，並且把這個遊戲應用在離散型的熱帶因子上。特別地，我們去探討這個遊戲與等價熱帶因子之間的關係。最後我們證明了下面的定理:若$D, E$為熱帶曲線$\Gamma$上的離散型熱帶因子, 而$\overline{D}$, $\overline{E}$分別代表因子$D,E$在搬硬幣遊戲時的狀態，因子$D$與$E$等價,若且為若 $\overline{D}$可經搬硬幣遊戲變成$\overline{E}$。 / In this thesis, we study Baker-Norine's chip-firing game, and apply it to discrete tropical divisors. In particularly, we discuss the relationship between this game and the equivalence of divisors. Finally, we give a proof of the theorem: Let $D$ and $E$ be discrete tropical divisors of tropical curve $\Gamma$, and let $\overline{D}$ and $\overline{E}$ be corresponding configurations of the chip-firing game. The divisors $D$ and $E$ are equivalent if and only if $\overline{D}$ can be transformed into $\overline{E}$.

Let $a \in \mathbb{Z}$ be a non-zero integer. Let $p$ be a prime such that $p \nmid a$. Define the index of $a$ modulo $p$, denoted $i_{a}(p)$, to be the integer $i_{a}(p) := [(\mathbb{Z}/p\mathbb{Z})^{\ast}:\langle a \bmod{p} \rangle]$. Let $N_{a}(x) := \#\{p \le x:i_{a}(p)=1\}$. In 1927, Emil Artin conjectured that \begin{equation*} N_{a}(x) \sim A(a)\pi(x) \end{equation*} where $A(a)>0$ is a constant dependent only on $a$ and $\pi(x):=\{p \le x: p\text{ prime}\}$. Rewrite $N_{a}(x)$ as follows: \begin{equation*} N_{a}(x) = \sum_{p \le x} f(i_{a}(p)), \end{equation*} where $f:\mathbb{N} \to \mathbb{C}$ with $f(1)=1$ and $f(n)=0$ for all $n \ge 2$.\\ \indent We examine which other functions $f:\mathbb{N} \to \mathbb{C}$ will give us formul\ae \begin{equation*} \sum_{p \le x} f(i_{a}(p)) \sim c_{a}\pi(x), \end{equation*} where $c_{a}$ is a constant dependent only on $a$.\\ \indent Define $\omega(n) := \#\{p|n:p \text{ prime}\}$, $\Omega(n) := \#\{d|n:d \text{ is a prime power}\}$ and $d(n):=\{d|n:d \in \mathbb{N}\}$. We will prove \begin{align*} \sum_{p \le x} (\log(i_{a}(p)))^{\alpha} &= c_{a}\pi(x)+O\left(\frac{x}{(\log x)^{2-\alpha-\varepsilon}}\right) \\ \sum_{p \le x} \omega(i_{a}(p)) &= c_{a}^{\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \\ \sum_{p \le x} \Omega(i_{a}(p)) &= c_{a}^{\prime\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \end{align*} and \begin{equation*} \sum_{p \le x} d(i_{a}) = c_{a}^{\prime\prime\prime}\pi(x)+O\left(\frac{x}{(\log x)^{2-\varepsilon}}\right) \end{equation*} for all $\varepsilon > 0$.\\ \indent We also extend these results to finitely-generated subgroups of $\mathbb{Q}^{\ast}$ and $E(\mathbb{Q})$ where $E$ is an elliptic curve defined over $\mathbb{Q}$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-08-03 10:45:47.408

Nesta dissertação apresentamos os principais algoritmos para o cálculo do Máximo Divisor Comum de polinômios a uma variável: os Algoritmos Euclidianos e os Algoritmos Modulares. Obtemos uma nova cota superior para os coeficientes do M.D.C., bem como demonstramos os resultados necessários para a obtenção da cota atualmente utilizada pelos Algoritmos Modulares. Além disso, apresentamos uma classe de polinômios para os quais a nova cota é menor que a anterior. / In this thesis we present the main algorithms for computing the Greatest Common Divisor of two univariate polynomials: the Euclidean Algorithms and the Modular Algorithms. We obtain a new upper bound for the coefficients of the G.C.D., as well we prove the results that are necessary for obtaining the bound that has been used by the Modular Algorithms. Besides, we present a class of polynomials for which the new bound is smaller than the previos one.