Elements of Non-Euclidean geometry ...

Mode of access: Internet.

The elements of non-Euclidean geometry,

Mode of access: Internet.

A produção matemática em um ambiente virtual de aprendizagem: o caso da geometria euclidiana espacial

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Husserl on Geometry and Spatial Representation

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl's ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl.

Perspex machine VII: The universal perspex machine

The perspex machine arose from the unification of projective geometry with the Turing machine. It uses a total arithmetic, called transreal arithmetic, that contains real arithmetic and allows division by zero. Transreal arithmetic is redefined here. The new arithmetic has both a positive and a negative infinity which lie at the extremes of the number line, and a number nullity that lies off the number line. We prove that nullity, 0/0, is a number. Hence a number may have one of four signs: negative, zero, positive, or nullity. It is, therefore, impossible to encode the sign of a number in one bit, as floating-, point arithmetic attempts to do, resulting in the difficulty of having both positive and negative zeros and NaNs. Transrational arithmetic is consistent with Cantor arithmetic. In an extension to real arithmetic, the product of zero, an infinity, or nullity with its reciprocal is nullity, not unity. This avoids the usual contradictions that follow from allowing division by zero. Transreal arithmetic has a fixed algebraic structure and does not admit options as IEEE, floating-point arithmetic does. Most significantly, nullity has a simple semantics that is related to zero. Zero means "no value" and nullity means "no information." We argue that nullity is as useful to a manufactured computer as zero is to a human computer. The perspex machine is intended to offer one solution to the mind-body problem by showing how the computable aspects of mind and. perhaps, the whole of mind relates to the geometrical aspects of body and, perhaps, the whole of body. We review some of Turing's writings and show that he held the view that his machine has spatial properties. In particular, that it has the property of being a 7D lattice of compact spaces. Thus, we read Turing as believing that his machine relates computation to geometrical bodies. We simplify the perspex machine by substituting an augmented Euclidean geometry for projective geometry. This leads to a general-linear perspex-machine which is very much easier to pro-ram than the original perspex-machine. We then show how to map the whole of perspex space into a unit cube. This allows us to construct a fractal of perspex machines with the cardinality of a real-numbered line or space. This fractal is the universal perspex machine. It can solve, in unit time, the halting problem for itself and for all perspex machines instantiated in real-numbered space, including all Turing machines. We cite an experiment that has been proposed to test the physical reality of the perspex machine's model of time, but we make no claim that the physical universe works this way or that it has the cardinality of the perspex machine. We leave it that the perspex machine provides an upper bound on the computational properties of physical things, including manufactured computers and biological organisms, that have a cardinality no greater than the real-number line.

The Role of Geometry in Wordsworth's "Science of Feelings"

Thesis (Ph.D.)--University of Washington, 2016-06

Geometry and topography in James Joyce's Ulysses and Finnegans Wake

Following the development of non-Euclidean geometries from the mid-nineteenth century onwards, Euclid’s system had come to be re-conceived as a language for describing reality rather than a set of transcendental laws. As Henri Poincaré famously put it, ‘[i]f several geometries are possible, is it certain that our geometry [...] is true?’. By examining Joyce’s linguistic play and conceptual engagement with ground-breaking geometric constructs in Ulysses and Finnegans Wake, this thesis explores how his topographical writing of place encapsulates a common crisis between geometric and linguistic modes of representation within the context of modernity. More specifically, it investigates how Joyce presents Euclidean geometry and its topographical applications as languages, rather than ideally objective systems, for describing visual reality; and how, conversely, he employs language figuratively to emulate the systems by which the world is commonly visualised. With reference to his early readings of Giordano Bruno, Henri Poincaré and other critics of the Euclidean tradition, it investigates how Joyce’s obsession with measuring and mapping space throughout his works enters into his more developed reflections on the codification of visual signs in Finnegans Wake. In particular, this thesis sheds new light on Joyce’s developing fascination with the ‘geometry of language’ practised by Bruno, whose massive influence on Joyce is often assumed to exist in Joyce studies yet is rarely explored in any great detail.

Fractais - Aplicações em Engenharia

Num universo despovoado de formas geométricas perfeitas, onde proliferam superfícies irregulares, difíceis de representar e de medir, a geometria fractal revelou-se um instrumento poderoso no tratamento de fenómenos naturais, até agora considerados erráticos, imprevisíveis e aleatórios. Contudo, nem tudo na natureza é fractal, o que significa que a geometria euclidiana continua a ser útil e necessária, o que torna estas geometrias complementares. Este trabalho centra-se no estudo da geometria fractal e na sua aplicação a diversas áreas científicas, nomeadamente, à engenharia. São abordadas noções de auto-similaridade (exata, aproximada), formas, dimensão, área, perímetro, volume, números complexos, semelhança de figuras, sucessão e iterações relacionadas com as figuras fractais. Apresentam-se exemplos de aplicação da geometria fractal em diversas áreas do saber, tais como física, biologia, geologia, medicina, arquitetura, pintura, engenharia eletrotécnica, mercados financeiros, entre outras. Conclui-se que os fractais são uma ferramenta importante para a compreensão de fenómenos nas mais diversas áreas da ciência. A importância do estudo desta nova geometria, é avassaladora graças à sua profunda relação com a natureza e ao avançado desenvolvimento tecnológico dos computadores.

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data, or with compositional data, like percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data, or with compositional data, like percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

Fractal based techniques for classification of mammograms and identification of microcalcifications

After skin cancer, breast cancer accounts for the second greatest number of cancer diagnoses in women. Currently the etiologies of breast cancer are unknown, and there is no generally accepted therapy for preventing it. Therefore, the best way to improve the prognosis for breast cancer is early detection and treatment. Computer aided detection systems (CAD) for detecting masses or micro-calcifications in mammograms have already been used and proven to be a potentially powerful tool , so the radiologists are attracted by the effectiveness of clinical application of CAD systems. Fractal geometry is well suited for describing the complex physiological structures that defy the traditional Euclidean geometry, which is based on smooth shapes. The major contribution of this research include the development of • A new fractal feature to accurately classify mammograms into normal and normal (i)With masses (benign or malignant) (ii) with microcalcifications (benign or malignant) • A novel fast fractal modeling method to identify the presence of microcalcifications by fractal modeling of mammograms and then subtracting the modeled image from the original mammogram. The performances of these methods were evaluated using different standard statistical analysis methods. The results obtained indicate that the developed methods are highly beneficial for assisting radiologists in making diagnostic decisions. The mammograms for the study were obtained from the two online databases namely, MIAS (Mammographic Image Analysis Society) and DDSM (Digital Database for Screening Mammography.

Total absolute horospherical curvature of submanifolds in hyperbolic space

We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of M, which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern-Lashof inequality for surfaces in 3-space and the horospherical Fenchel and Fary-Milnor`s theorems.

Um Sistema para aprendizagem de demonstrações dedutivas em geometria euclidiana

O objetivo do presente trabalho é realizar a concepção de um sistema para a aprendizagem de demonstrações da Geometria Euclidiana Plana e a implementação de um protótipo deste sistema, denominado LEEG - Learning Environment on Euclidean Geometry, desenvolvido para validar as idéias utilizadas em sua especificação. Nos últimos anos, tem-se observado uma crescente evolução dos sistemas de ensino e aprendizagem informatizados. A preocupação com o desenvolvimento de ambientes cada vez mais eficientes, tanto do ponto de vista computacional quanto pedagógico, tem repercutido em um salto de qualidade dos software educacionais. Tais sistemas visam promover, auxiliar e motivar a aprendizagem das mais diversas áreas do conhecimento, utilizando técnicas de Inteligência Artificial para se aproximarem ao máximo do comportamento de um tutor humano que se adapte e atenda às necessidades de cada aluno. A Geometria pode ser vista sob dois aspectos principais: considerada como uma ciência que estuda as representações do plano e do espaço e considerada como uma estrutura lógica, onde a estrutura matemática é representada e tratada no mais alto nível de rigor e formalismo. Entretanto, o ensino da Geometria, nos últimos anos, abandonou quase que totalmente sua abordagem dedutiva. Demonstrações de teoremas geométricos não são mais trabalhadas na maioria das escolas brasileiras, o que repercute em um ensino falho da Matemática, que não valoriza o desenvolvimento de habilidades e competências relacionadas à experimentação, observação e percepção, realização de conjecturas, desenvolvimento de argumentações convincentes, entre outras. Levando-se em conta este cenário, desenvolveu-se o LEEG, um sistema para a aprendizagem de demonstrações geométricas que tem como objetivo auxiliar um aprendiz humano na construção de demonstrações da Geometria Euclidiana Plana. O sistema foi modelado sobre uma adaptação do protocolo de aprendizagem MOSCA, desenvolvido para suportar ambientes de ensino informatizados, cuja aprendizagem é baseada na utilização de exemplos e contra-exemplos. Este protocolo propõe um ambiente de aprendizagem composto por cinco agentes, dentre os quais um deles é o aprendiz e os demais assumem papéis distintos e específicos que completam um quadro de ensino-aprendizagem consistente. A base de conhecimento do sistema, que guarda a estrutura lógica-dedutiva de todas as demonstrações que podem ser submetidas ao Aprendiz, foi implementada através do modelo de autômatos finitos com saída. A utilização de autômatos com saída na aplicação de modelagem de demonstrações dedutivas foi extremamente útil por permitir estruturar os diferentes raciocínios que levam da hipótese à tese da proposição de forma lógica, organizada e direta. As demonstrações oferecidas pelo sistema são as mesmas desenvolvidas por Euclides e referem-se aos Fundamentos da Geometria Plana. São demonstrações que priorizam e valorizam a utilização de objetos geométricos no seu desenvolvimento, fugindo das demonstrações que apelam para a simples manipulação algébrica e que não oferecem uma construção significativa do ponto de vista da Geometria. Porém, mesmo sendo consideradas apenas as demonstrações contidas em Elements, todos os diferentes raciocínios para uma mesma demonstração são aceitos pelo sistema, dando liberdade ao aprendiz no processo de construção da demonstração.

Compositional analysis for an unbiased measure of soil aggregation

Soil aggregation is an index of soil structure measured by mean weight diameter (MWD) or scaling factors often interpreted as fragmentation fractal dimensions (D-f). However, the MWD provides a biased estimate of soil aggregation due to spurious correlations among aggregate-size fractions and scale-dependency. The scale-invariant D-f is based on weak assumptions to allow particle counts and sensitive to the selection of the fractal domain, and may frequently exceed a value of 3, implying that D-f is a biased estimate of aggregation. Aggregation indices based on mass may be computed without bias using compositional analysis techniques. Our objective was to elaborate compositional indices of soil aggregation and to compare them to MWD and D-f using a published dataset describing the effect of 7 cropping systems on aggregation. Six aggregate-size fractions were arranged into a sequence of D-1 balances of building blocks that portray the process of soil aggregation. Isometric log-ratios (ilrs) are scale-invariant and orthogonal log contrasts or balances that possess the Euclidean geometry necessary to compute a distance between any two aggregation states, known as the Aitchison distance (A(x,y)). Close correlations (r>0.98) were observed between MWD, D-f, and the ilr when contrasting large and small aggregate sizes. Several unbiased embedded ilrs can characterize the heterogeneous nature of soil aggregates and be related to soil properties or functions. Soil bulk density and penetrater resistance were closely related to A(x,y) with reference to bare fallow. The A(x,y) is easy to implement as unbiased index of soil aggregation using standard sieving methods and may allow comparisons between studies. (C) 2012 Elsevier B.V. All rights reserved.