<p> We study two objects: an ideal of a Hecke algebra, and a pairing in Galois cohomology.</p> <p> Let h be the Hecke algebra of cusp forms of weight 2, level n, and a fixed Dirichlet character modulo n generated by all Hecke operators, where n is an odd prime p or a product of two distinct odd primes N and p. We study the Eisenstein I ideal of h. We wrote a computer program to test whether Up - 1 generates this ideal, where Up is the pth Hecke operator in h. We found many cases of n and the character so that Up - 1 alone generates I. On the other hand, we found one example with N = 3 and p = 331 where Up - 1 does not generate I.</p> <p> Let K = Q(μn) be the nth cyclotomic field. Let S be the set of primes above p in K, and let G_K,S be the Galois group of the maximal extension of K unramified outside S. We study a pairing on cyclotomic p-units that arises from the cup product on H1(G_K,S, μp). This pairing takes values in a Gal(K/Q)-eigenspace of the p-part of the class group of K. Sharifi has conjectured that this pairing is surjective. We studied this pairing in detail by imposing linear relations on the possible pairing values. We discovered many values of n and the character such that these relations single out a unique nontrivial possibility for the pairing, up to a possibly zero scalar.</p> <p> Sharifi showed in [S2] that, under an assumption on Bernoulli numbers, the element Up - 1 generates the Eisenstein ideal I if and only if pairing with the single element p is surjective. In particular, in the instances for which we found a unique nontrivial possibility for the pairing, then if Up - 1 generates I, we know that the scalar up to which it is determined cannot be zero.</p> / Thesis / Master of Science (MSc)

In the Neotropical realm, little is yet known about the distribution, ecology, and biodiversity of limno-terrestrial tardigrades. Tardigrades are understudied micrometazoans and, in Chile, their biogeography and the variables that are associated with their diversity have never been recorded. This study proposes to examine the assemblages (composition and abundance) of tardigrades in forests throughout southern Chile and relate the patterns found to latitude, altitude, temperature, precipitation, primary productivity, and land use cover. This novel study shows basic information on the biogeographical distribution and diversity of forest Chilean tardigrades and examines the potential influence of landscape variables on the composition and abundance of this little-known phylum. Using univariate and direct gradient analysis, it was found that tardigrade alpha diversity was mainly effected by precipitation, mean monthly minimum, and maximum temperature; also, the combined effects of precipitation and mean monthly maximum temperature, precipitation, and mean monthly minimum temperature, had an interactive effect on tardigrade alpha diversity. The environmental and geographic variables explained the variation in the community structure of tardigrades. Overall, this study has given first insight into Chilean tardigrade ecology.

The study investigated the effects of two generative learning activities on students’ academic achievement of the part-whole representation of rational numbers while using virtual manipulatives. Third-grade students were divided randomly in two groups to evaluate the effects of two generative learning activities: answering-questions and generating-examples while using two virtual manipulatives related to part-whole representation of rational numbers. The study employed an experimental design with pre- and post-tests. A 2x2 mixed analysis of variance (ANOVA) was used to determine any significant interaction between the two groups (answering questions and generating-examples) and between two tests (pre-test and immediate post-test). In addition, a 2x3 mixed analysis of variance (ANOVA) and a Bonferroni post-hoc analysis were used to determine the effects of the generative strategies on fostering comprehension, and to determine any significant differences between the two groups (answering-questions and generating-examples) and among the three tests (pre-test, immediate post-test, and delayed posttest). Results showed that an answering-questions strategy had a significantly greater effect than a generating-examples strategy on an immediate comprehension posttest. In addition, no significant interaction was found between the generative strategies on a delayed comprehension tests. However a difference score analysis between the immediate posttest scores and the delayed posttest scores revealed a significant difference between the answering-questions and the generating-examples groups suggesting that students who used generating-examples strategy tended to remember relatively more information than students who used the answering-questions strategy. The findings are discussed in the context of the related literature and directions for future research are suggested. / Ph. D.

This dissertation examines the complex story of Korah’s rebellion found in Numbers 16 and 17 utilizing narrative critical theory. This study is first grounded in the context of historical questions surrounding Israel’s emergence as a nation and the narrative’s potential for historical veracity. Many narrative critics do not feel the theoretical necessity to establish the connection between an autonomous text and a historical context. This study does seek to collaborate with historical research, but only as permitted by the data. Though only biblical and tangential evidence supports the historicity of the wilderness sojourn, the narrative accounts should not be repudiated because of philosophical bias or the lack of corroborative extra biblical evidence. Especially important to a literary interpretation of this narrative is the work of source critics who during their own enquiries have identified the fractures and transitions within the story. In considering the text of Numbers 16 and 17, the hermeneutical approach employed in this study carefully endorses a balanced incorporation of the theoretical constructs of the author, text, and reader in the interpretive enquiry. From this hermeneutical approach recent literary theory is applied to the texts of Numbers 16 and 17 focusing particular attention on three narrative themes. First, the narrator’s point of view is examined to determine the manner that information is relayed to the reader so as to demur the rebellion leaders. Though features of characterization are often meager in biblical narratives, there remains sufficient data in this rebellion story to support the aims of the Hebrew writers and does not undermine the reader’s engagement with the story’s participants. Finally, the three separate plotlines in this narrative sustain the dramatic effect upon the readership holding attention and judgment throughout and beyond the story. In sum, this dissertation highlights the powerful contours of this ancient narrative by appropriating the theoretical work of narrative critics. The strategies employed in the writing and editing of this story uniquely condemn the rebels and at the same time serve to elevate God’s chosen leader Moses. / Old Testament and Ancient Near Eastern Studies / D. Th. (Old Testament)

Esta dissertação apresenta uma experiência de aplicação de sequência de atividades cujo tema central é números relativos, envolvendo uso de jogos. Tal sequência foi aplicada em uma turma do sétimo ano de uma escola particular da cidade de Porto Alegre. A construção da sequência apoiou-se em literatura que afirma que jogos podem propiciar contextos que façam mais sentido aos alunos do que as situações ditas do cotidiano, comumente retratadas em livros didáticos. Buscou-se, por meio dos jogos, constituir contextos que provocassem a necessidade da representação de números de sentidos opostos e das operações com esses números, incluindo situações complexas como a do efeito resultante da retirada de um número negativo. Foram coletados registros da produção dos alunos e das discussões em sala de aula por meio de gravação em áudio, fotografias de cadernos e do quadro-negro, e questionários preenchidos pelos alunos após cada atividade. A análise desses materiais permitiu concluir que os jogos representaram uma sustentação para o pensamento lógico e operatório dos alunos, na construção dos esquemas mobilizados para somar e subtrair números positivos e negativos. / This work presents an experiment of an activity sequence centered on Relative Numbers and involving the use of games. Such sequence was applied in a seventh grade class of a Porto Alegre’s private school. The conception of the sequence was supported by literature that affirms that games can provide contexts that make more sense to the students than the so-called “daily situations” frequently presented in didactic books. It was sought, by the games, to constitute contexts that provoked the need for representation of numbers with opposite signs and the operations with those numbers, including more complex cases, like the result of subtracting a negative number. The data collected includes audio recordings of students’ production and discussions, notebooks and blackboard’s photographs, and questionnaires filled by students after each activity. The analysis of this data allowed me to conclude that games could support the logical and operational thinking of the students in the construction of mobilized schemes to add and subtract positive and negative numbers.

O presente trabalho tem como perspectiva a caracterização dos números Racionais e Irracionais, e a sua devida aplicabilidade e variações no que tange o aspecto algébrico e transcendental. Sabe-se que o Número e (de Euler), pode ser classificado como um número transcendental, isto é, aqueles que não são raízes de nenhum polinômio que possua coeficientes inteiros. Nesse pressuposto, o Número deve ser considerado existente e irracional. O objetivo desta pesquisa consiste em caracterizar os fatores que abrangem os Números Racionais e Irracionais, oferecendo a compreensão necessária referente ao Número e e a sua ação nos Números Algébricos e Transcendentes. Como recurso metodológico, utilizou-se uma revisão de literatura, com um crivo pautado nos fatores qualitativos e quantitativos, a fim de se refletir sobre a temática proposta. Assim, nesta presente pesquisa, buscouse apresentar informações dentro das melhores formas e possibilidades de favorecer a compreensão, considerando a dificuldade em torno deste respectivo tema, devido a sua característica abstrata, o que dificulta o entendimento por parte de muitos. Portanto, destacam-se as iniciativas e argumentos em torno deste princípio temático, como forma de, possivelmente, fomentar o interesse de muitos pelo mesmo, além de que, tal trabalho possa ser relevante às necessidades de investigação de outros desejosos por este universo de pesquisa. / The present work has as its perspective the characterization of Rational and Irrational numbers, and their due applicability and variations regarding the algebraic and transcendental aspects. It is known that the number e (of Euler) can be classified as a transcendental number, that is, those that are not roots of any polynomial that has integer coefficients. In this assumption, the Number should be considered existent and irrational. The objective of this research is to characterize the factors that comprise the Rational and Irrational Numbers, offering the necessary understanding regarding Number e and its action in Algebraic and Transcendent Numbers. As a methodological resource, a literature review was used, based on qualitative and quantitative factors, in order to reflect on the proposed theme. Thus, in this present research, we sought to present information within the best ways and possibilities to favor understanding, considering the difficulty around this respective theme, due to its abstract feature, which makes it difficult for many to understand. Therefore, we highlight the initiatives and arguments around this thematic principle as a way of possibly fostering the interest of many by the same, and that such work may be relevant to the research needs of others desirous by this universe of research.

We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

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Thesis (Ph. D.)--Washington State University. / Subtitle of caption title on p. iv.: A comprehensive trust model based on the TRA/TPB. Includes bibliographical references.

Esta dissertação apresenta uma experiência de aplicação de sequência de atividades cujo tema central é números relativos, envolvendo uso de jogos. Tal sequência foi aplicada em uma turma do sétimo ano de uma escola particular da cidade de Porto Alegre. A construção da sequência apoiou-se em literatura que afirma que jogos podem propiciar contextos que façam mais sentido aos alunos do que as situações ditas do cotidiano, comumente retratadas em livros didáticos. Buscou-se, por meio dos jogos, constituir contextos que provocassem a necessidade da representação de números de sentidos opostos e das operações com esses números, incluindo situações complexas como a do efeito resultante da retirada de um número negativo. Foram coletados registros da produção dos alunos e das discussões em sala de aula por meio de gravação em áudio, fotografias de cadernos e do quadro-negro, e questionários preenchidos pelos alunos após cada atividade. A análise desses materiais permitiu concluir que os jogos representaram uma sustentação para o pensamento lógico e operatório dos alunos, na construção dos esquemas mobilizados para somar e subtrair números positivos e negativos. / This work presents an experiment of an activity sequence centered on Relative Numbers and involving the use of games. Such sequence was applied in a seventh grade class of a Porto Alegre’s private school. The conception of the sequence was supported by literature that affirms that games can provide contexts that make more sense to the students than the so-called “daily situations” frequently presented in didactic books. It was sought, by the games, to constitute contexts that provoked the need for representation of numbers with opposite signs and the operations with those numbers, including more complex cases, like the result of subtracting a negative number. The data collected includes audio recordings of students’ production and discussions, notebooks and blackboard’s photographs, and questionnaires filled by students after each activity. The analysis of this data allowed me to conclude that games could support the logical and operational thinking of the students in the construction of mobilized schemes to add and subtract positive and negative numbers.