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A new connection between combinatorics and noncommutative algebra is established by relating a certain class of directed graphs to noncommutative Koszul algebras. The directed graphs in this class are called full graphs and are defined by a set of criteria on the edges. The structural properties of full graphs are studied as they relate to the edge criteria. A method is introduced for generating a Koszul algebra Lambda from a full graph G. The properties of Lambda are examined as they relate to the structure of G, with special attention being given to the construction of a projective resolution of certain semisimple Lambda-modules based on the structural properties of G. The characteristics of the Koszul algebra Lambda that is derived from the product of two full graphs G' and G' are studied as they relate to the properties of the Koszul algebras Lambda' and Lambda' derived from G' and G'. / Ph. D.

Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C_b(E) → L^∞(μ) with Γ(f∘Π) = f for every f ∈ C_b(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L^∞(μ) with Γ(f∘Π) = f for every f in C_b(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. / Ph. D.

No / A methodology for shape morphing using partial differential equation (PDE) surfaces is presented in this work. The use of the PDE formulation shows how shape morphing can be based on a boundary-value approach by which intermediate shapes can be created. Furthermore, the mathematical properties of the method give rise to several alternatives in which morphing one shape into another can be achieved. Three of these alternatives are presented here. The first one is based on the gradual variation of the weighted sum of the boundary conditions for each surface, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. Examples showing the efficiency of these methodologies are presented. Thus, it is shown that the PDE based approach for morphing, when combined with a parametric variation of the boundary conditions, is capable of obtaining smooth intermediate surfaces automatically.

The purpose of this thesis is to present some theoretical elements used in a comparative research that studies two guerrilla groups. The contexts of study, Angola and Colombia, in long internal conflicts during the second half of the twentieth century, witnessed the apparition of two guerrilla groups: the ELN (Ejército de Liberación Nacional/National Army of Liberation, 1963) and the MPLA (Movimento Popular de Libertação de Angola/Popular Movement for the Liberation of Angola, 1956). The goal is to provide an interpretation of the rise, transformations and uses of specific forms of historical representations. In the form of discourses and practices, the ELN and the MPLA constituted historical representations with the purpose of building new political imaginaries, in whose analysis it is possible to explain features such as the structures of power, knowledge and language, how they are constantly changing, and how guerrillas gain legitimacy within a society by using ideological paradigms. For instance, the research suggests that internal crises in the MPLA and the ELN promoted the change from a national liberation discourse, towards a more explicit use of Marxist-Leninist ideological principles. Also, such transformations are associated with the persistence of social distinctions—ethnolinguistic in Angola; rural/urban in Colombia— and their reflection in embryonic institutions that by the middle of the 1970s where supposed to constitute a revolutionary New Society in both Angola and Colombia. In a paradox, the embryonic institutions created by revolutionary groups let infer that they assume the role of a New Establishment because the deployment of power implies mechanism of control, coercion and discipline, rituals, ceremonies, practices and discourses that create truth, law, and language. / Master of Arts

Every positive integer has a unique radix 2 representation which uses the digits {0,1}. However, if we allow digits other than 0 and 1, say {0,1,-1}, then a positive integer has many representations. Of these <i>redundant</i> representations, it is possible to choose one that has few nonzero digits. It is well known that using representations of integers with few nonzero digits allows certain algebraic operations to be done more quickly. This thesis is concerned with various representations of integers that are related to efficient implementations of algebraic operations in cryptographic algorithms. The topics covered here include: <ul> <li> <i>The width-w nonadjacent form (w-NAF)</i>. We prove that the <i>w</i>-NAF of an integer has a minimal number of nonzero digits; that is, no other representation of an integer, which uses the <i>w</i>-NAF digits, can have fewer nonzero digits than its <i>w</i>-NAF. </li> <li><i>A left-to-right analogue of the w-NAF</i>. We introduce a new family of radix 2 representations which use the same digits as the <i>w</i>-NAF, but have the property that they can be computed by sliding a window from left to right across the binary representation of an integer. We show these new representations have a minimal number of nonzero digits. </li> <li><i>Joint representations</i>. Solinas introduced a {0,1,-1}-radix 2 representation for pairs of integers called the joint sparse form. We consider generalizations of the joint sparse form which represent <i>r</i>&ge;2 integers and use digits other than {0,1,-1}. We show how to construct a {0,1,2,3}-joint representation that has a minimal number of nonzero columns. </li> <li><i>Nonadjacent digit sets</i>. It is well known that if <i>x</i> equals 3 or -1 then every nonnegative integer has a unique {0,1,<i>x</i>}-nonadjacent form; that is, a {0,1,<i>x</i>}-radix 2 representation with the property that, of any two consecutive digits, at most one is nonzero. We investigate what other values of <i>x</i> have this property. </li> </ul>

This thesis explores the relationship between development representations and diaspora audiences. It brings together literature on representations, with concepts of audience, diaspora and identity to provide an in-depth study of how and with what effects, visual representations of development in NGO fundraising campaigning that depict Africa, impact on Nigerian diaspora audiences. This study challenges the tendency in much of development literature in this field to homogenise British audiences of NGO communication. This has imagined audiences as some form of monocultural Western-situated community, coextensive with the 'general' British public. It further assumes audiences read, interpret and are impacted by NGO representations in very similar ways. This assumption precludes critical engagement with the complexities and particularities of audiences and is unable to reflect the multiple and differentiated ways in which audiences think, feel and behave in response to development representations. By using focus group discussions with UK Nigerian diaspora audiences, one-to-one interviews and online-ethnography as the methodological tool, and postcolonialism as an analytical framing, this thesis reveals the complex and contested ways that individual diaspora subjectivities, positionalities and life experiences are implicated in their construal of development representations and the perspicuity of their impact. One of the key findings of this study is that development representations impact African diaspora audiences in diverse and complicated ways, that both reproduce and contradict negative and, stereotypical 'ways of seeing' and knowing Africa. Furthermore, it highlights how diaspora ethno-racial/cultural identities affect, and are implicated in, the reading and interpretation of development representations of Africa. Indeed, diaspora audiences affirm and challenge their connections or, lack thereof, with their country of origin through these representations. Moreover, the study shows how NGO development representations provide symbolic spaces from which diaspora audiences can articulate their identities as well as, forge relationships among themselves and with their wider communities. This study builds on Stuart Hall's ([1973]1980) Encoding/Decoding theorisation on audiences, by demonstrating that Nigerian diaspora audiences of development representations are sophisticated, varied and paradoxical in how they interpret and decipher media representations. Indeed, their socio-cultural positioning, personal histories and lived-experiences inform and shape how they discursively construct perceptions and knowledge of their place of origin through representations. Furthermore, it contributes to postcolonial theorisations of hybridity in diaspora identities, by showing that Nigerians strategically adopt new and preferential ethnosymbolic identities, in response to representations. These re-configurations of the Diaspora 'Self' are neither stable or consistent but are nonetheless utilised by Nigerians to subvert development representations and harmful public perceptions and stereotypes about Africans that they shape.

O processo de abertura a outros universos culturais decorrente do estreitamento das relações entre as sociedades fez emergir mais fortemente a necessidade de reflexão sobre as noções de cultura e de representação. Tais noções tornam-se, no campo da Didática das Línguas Estrangeiras, de fundamental importância uma vez que o conhecimento das línguas representa o vínculo de aproximação entre as diferentes comunidades culturais, e favorece a compreensão das diversas formas de ver e de viver o mundo. O processo de ensino-aprendizagem de uma língua estrangeira deve, deste modo, buscar favorecer a reflexão sobre o diverso com vistas ao despertar do aprendiz a essa diversidade, evitando, sempre que possível, a folclorização das culturas. Através da análise do corpus extraído da imprensa francesa e cujo objeto do discurso é o Brasil, buscamos demonstrar objetivamente uma das vias de reflexão sobre o diverso, uma vez que nos vemos representados no e pelo discurso dos artigos selecionados. Isto nos parece enriquecedor, pois nos coloca, enquanto brasileiros, frente às representações do Outro sobre a nossa sociedade, e nos faz perceber e refletir sobre a existência de um processo inverso de construção de representações de uma outra cultura, onde nos encontramos como agentes. O objetivo é favorecer, através da objetivação dos fatos apresentados, a reflexão sobre a diversidade cultural, sobre a importância da relativização e sobre os perigos da generalização / The opening process to other cultural universes generated through more closeness between societies as created needs for a new thinking approach on culture and representation. These concepts are now put in the field of Foreign Language Didactic at a very important place because the understanding of languages is an essential element in the relationship between various cultural communities and therefore help the understanding of the diverse forms to see and living the world. The teaching-learning of a foreign language must also stimulate the reflection on the diversity of cultural facts therefore stimulating the awareness of the learning person that excessive folklorisation could damage badly the reality of theses cultures. Through the analysis of the French Press elements about Brazil, we have been looking for an objective approach on how to demonstrate possible diversities, after having acknowledged the manner we have been seen through the selected articles. That has been very interesting because, as Brazilian, we have been in front of how the Other sees our society and therefore how we should conduct our representation analysis of the Other. The final aim is to support, through a better objectivity of the given facts, a thinking process on cultural diversity, on the importance of relativism and on the danger of excessive generalisation

Approval voting is an election method in which voters may cast votes for as many candidates as they desire. This can be modeled mathematically by associating to each voter an approval region: a set of potential candidates they approve. In this thesis we add another level of approval somewhere in between complete approval and complete disapproval. More than one level of approval may be a better model for a real-life voter's complex decision making. We provide a new definition for intersection that supports multiple levels of approval. The case of pairwise intersection is studied, and the level of agreement among voters is studied under restrictions on the relative size of each voter's preferences. We derive upper and lower bounds for the percentage of agreement based on the percentage of intersection.

Every positive integer has a unique radix 2 representation which uses the digits {0,1}. However, if we allow digits other than 0 and 1, say {0,1,-1}, then a positive integer has many representations. Of these <i>redundant</i> representations, it is possible to choose one that has few nonzero digits. It is well known that using representations of integers with few nonzero digits allows certain algebraic operations to be done more quickly. This thesis is concerned with various representations of integers that are related to efficient implementations of algebraic operations in cryptographic algorithms. The topics covered here include: <ul> <li> <i>The width-w nonadjacent form (w-NAF)</i>. We prove that the <i>w</i>-NAF of an integer has a minimal number of nonzero digits; that is, no other representation of an integer, which uses the <i>w</i>-NAF digits, can have fewer nonzero digits than its <i>w</i>-NAF. </li> <li><i>A left-to-right analogue of the w-NAF</i>. We introduce a new family of radix 2 representations which use the same digits as the <i>w</i>-NAF, but have the property that they can be computed by sliding a window from left to right across the binary representation of an integer. We show these new representations have a minimal number of nonzero digits. </li> <li><i>Joint representations</i>. Solinas introduced a {0,1,-1}-radix 2 representation for pairs of integers called the joint sparse form. We consider generalizations of the joint sparse form which represent <i>r</i>&ge;2 integers and use digits other than {0,1,-1}. We show how to construct a {0,1,2,3}-joint representation that has a minimal number of nonzero columns. </li> <li><i>Nonadjacent digit sets</i>. It is well known that if <i>x</i> equals 3 or -1 then every nonnegative integer has a unique {0,1,<i>x</i>}-nonadjacent form; that is, a {0,1,<i>x</i>}-radix 2 representation with the property that, of any two consecutive digits, at most one is nonzero. We investigate what other values of <i>x</i> have this property. </li> </ul>