This report expresses some of the recent research surrounding the Pythagorean Theorem and Pythagorean triples. Topics discussed include applications of the Pythagorean Theorem relating to recursion methods, acute and obtuse triangles, Pythagorean triangles in squares, as well as Pythagorean boxes. A short discussion on the depth of the Pythagorean Theorem taught in secondary schools is also included. / text

This report focuses on the discovery of Fermat numbers as well as the subsequent innovations in processes for finding factors of Fermat numbers. The property of the prime factors of Fermat numbers, as well as the connections between Fermat numbers and other areas of mathematics, is also discussed. / text

This thesis looks at the problem of reconstructing a single surface representation from multiple range images acquired from a terrestrial laser scanner. A solution to this problem is important to industries such as mining, where accurate spatial measurement is required for mapping and volumetric calculations. Laser scanners for 3D measurement are now commercially available and software for deriving useful information from the data these devices generate is essential. A reconstruction technique based on an implicit surface representation of the range images and a polygonisation algorithm called marching triangles has been implemented in software and its performance investigated. This work improves upon the existing techniques in that it takes into account the particular differences of terrestrial range data as compared with data from small scale laser scanners. The implementation is robust with respect to noisy data and environments and requires minimal user input. A new approach to 3D spatial indexing is also developed to allow rapid evaluation of the true closest point to a surface which is the basis of the signed distance function implicit surface representation. A new technique for locating step discontinuities in the range image is presented, which caters for the varying sampling densities of terrestrial range images. The algorithm is demonstrated using representative range images acquired for surface erosion monitoring and for underground mine surveying. The results indicate that this reconstruction technique represents an improvement over current techniques for this type of range data. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1169106 / Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 2005

This thesis looks at the problem of reconstructing a single surface representation from multiple range images acquired from a terrestrial laser scanner. A solution to this problem is important to industries such as mining, where accurate spatial measurement is required for mapping and volumetric calculations. Laser scanners for 3D measurement are now commercially available and software for deriving useful information from the data these devices generate is essential. A reconstruction technique based on an implicit surface representation of the range images and a polygonisation algorithm called marching triangles has been implemented in software and its performance investigated. This work improves upon the existing techniques in that it takes into account the particular differences of terrestrial range data as compared with data from small scale laser scanners. The implementation is robust with respect to noisy data and environments and requires minimal user input. A new approach to 3D spatial indexing is also developed to allow rapid evaluation of the true closest point to a surface which is the basis of the signed distance function implicit surface representation. A new technique for locating step discontinuities in the range image is presented, which caters for the varying sampling densities of terrestrial range images. The algorithm is demonstrated using representative range images acquired for surface erosion monitoring and for underground mine surveying. The results indicate that this reconstruction technique represents an improvement over current techniques for this type of range data. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1169106 / Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 2005

The topic of this research are the linkages between the descriptive and the substantive representation of women. The research questions seek to explore 1. in how far and on what basis women in parliament represent women and women's issues and how women in civil society perceive this 2. what obstacles to the work of women in congress are identified 3. to what extent women in Chilean congress work amongst each other and over organizational barriers with women's organizations and SERNAM and lastly 4. how the findings on the questions above relate and contribute to the broader debate on mediating factors between DRW and SRW and what conclusions on the potential impact of a quota on these factors they allow. For this purpose, this field study involved interviews conducted in April and May 2015 in Santiago de Chile and Valparaiso as well as one via Skype. The interviewees were seven current and recent female members of the Chilean congress as well as five representatives from reputable women's organizations. Furthermore, the extensive literature on the topic as well as reports on the issue of gender equality were reviewed. The method applied was qualitative and abductive. No theory-testing was involved, instead the approach was exploratory and theories and analytical frameworks were used as inspiration for interview questions in an abductive way. The results of this study shed light on six variables drawn from the research debate, the role of women's diversity, the impact of their attitudes towards the representation of women, tokenism, 'women's issues', feminist triangles and here also the relation of legislators to feminism, and finally the impact of numbers on all variables. Specifically feminist triangles reveal a wealth of interactions and potential for the promotion of enhancing SRW in ways contingent and non-contingent on DRW.

nÃo hà / Neste trabalho apresentamos algumas relaÃÃes trigonomÃtricas fundamentais e suas demonstraÃÃes. Tais relaÃÃes merecem destaque, pois sÃo essenciais na resoluÃÃo de problemas nas diversas Ãreas do conhecimento. Inicialmente fizemos um breve histÃrico da Trigonometria destacando a sua importÃncia no contexto da MatemÃtica. Em sequÃncia apresentamos as relaÃÃes fundamentais, dentre elas, enfatizamos as fÃrmulas da adiÃÃo de arcos, cujas demonstraÃÃes utilizamos Ãrea de figuras planas, a lei dos cossenos e a lei dos senos. AlÃm disso, mostramos como estas fÃrmulas se relacionam com o Teorema da Corda Quebrada e o Teorema de Ptolomeu da Geometria Plana. Explorando a relaÃÃes entre Trigonometria e Geometria Plana. Procuramos mostrar a importÃncia desta teoria no ensino mÃdio motivando alunos e professores a buscar um maior interesse pelo conhecimento em MatemÃtica. / The present research aims to present classic demonstrations of the fundamental relations of Trigonometry,with a simple approach, and exploring flat shapes. The intention is to make such demonstrations better known and provide a highlight for Trigonometry, since they are essential in solving problems of everyday life. For this purpose, we made a historical highlighting of the importance of trigonometry in the mathematical context. Since we know Trigonometry is loosing its status and not being considered essential in basic education anymore, such demonstrations, associated with the flat shapes, may be used as a model class. Therefore, we highlight the following fundamental relations: Basic Trigonometric relations, Derived Relations, Sine of the Sum and Difference of Two Arcs, Cosine of the Sum and Difference of Two Arcs, Double Arcs, Half Arc, Transformation in Product and Applications. For the demonstration ot these relations we used some area results, cosine law, Ptolemyâs theorem and the theorem of the broken chord Plane Geometry. We believe that Trigonometry is linked to the formation of these flat shapes. Thus, such demonstrationas associated to these flat shapes may serve to improve the Trigonometry teaching- learning and as motivator for students and teachers seeking to enhance their knowledge in mathematics.

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No description available.

Uma órbita bilhar em um triângulo é uma poligonal cujos segmentos começam e terminam nos lados do triângulo e que se refletem elasticamente nestes lados. É como o movimento de uma bola numa mesa de bilhar sem atrito (logo a bola tem velocidade constante e jamais para) cujas laterais formam um triângulo. Esta órbita é periódica se ela retorna infinitas vezes ao mesmo ponto com a mesma direção. A existência de órbitas bilhares periódicas em polígonos é uma questão aberta da matemática. Mesmo para um triângulo ainda não há resposta. Para triângulos agudos, a resposta é bem conhecida, pois o triângulo formato pelos pés das alturas do triângulo é uma órbita periódica. Para triângulos obtusos, em geral, pouco se sabe. O objetivo desta dissertação é coletar resultados e técnicas sobre órbitas bilhares periódicas em triângulos obtusos. Começamos introduzindo o trabalho de Vorobets, Galperin e Stepin, que no início dos anos 90 unificaram os casos conhecidos de triângulos que possuem órbita bilhar periódica, introduziram o conceito de estabilidade e mostraram novos resultados, como uma família infinita de órbitas estáveis. Temos também o teorema de 2000 de Halbeisen e Hungerbühler que estende as famílias de órbitas estáveis. Mencionamos em seguida os trabalhos de Schwartz de 2006 e 2009 que utilizam auxílio computacional para mostrar que todo triângulo com ângulos menores que $100\\degree$ possui órbita bilhar periódica. Depois temos os resultados de 2008 de Hooper e Schwartz sobre órbitas bilhares periódicas em triângulos quase isósceles e sobre estabilidade de órbitas em triângulos de Veech. Todos os casos abordados neste trabalho incluem uma vasta variedade de triângulos, mas a questão de existência de órbitas bilhares periódicas para todo triângulo está longe de ser totalmente contemplada. / A billiard orbit in a triangle is a polygonal with vertices at the boundary of the triangle such that its angles reflect elastically. It is similar to a moving ball on a billiard table without friction (so the ball has constant speed and never stops) whose sides form a triangle. This orbit is periodic if it returns infinitely to the same point with the same direction. The existence of periodic billiard orbits in polygons is an open problem in mathematics. Even for a triangle there is still no answer. For acute triangles the answer is well known since the triangle whose vertices are the base points of the three altitudes of the triangle is a periodic orbit. For obtuse triangles, in general, little is known. The aim of this thesis is to collect results and techniques on periodic billiard orbits in obtuse triangles. We start by introducing the work of Vorobets, Gal\'perin and Stepin, who unified in the early 1990s the known cases of triangles that have periodic billiard orbits, introduced the concept of stability and proved new results, such as an infinite family of stable orbits. We also have the theorem of Halbeisen and Hungerbühler of 2000 extending the families of stable orbits. Next, we mention the works of Schwartz of 2006 and 2009 that use computational assistance to prove that every triangle whose angles are at most $100\\degree$ have periodic billiard orbits. Then, we have the results of 2008 by Hooper and Schwartz on periodic billiard orbits in nearly isosceles triangles and on stability of billiard orbits in Veech triangles. All cases covered in this work include a wide variety of triangles, but the question of the existence of periodic billiard orbits for all triangles is far from being fully contemplated.

Uma órbita bilhar em um triângulo é uma poligonal cujos segmentos começam e terminam nos lados do triângulo e que se refletem elasticamente nestes lados. É como o movimento de uma bola numa mesa de bilhar sem atrito (logo a bola tem velocidade constante e jamais para) cujas laterais formam um triângulo. Esta órbita é periódica se ela retorna infinitas vezes ao mesmo ponto com a mesma direção. A existência de órbitas bilhares periódicas em polígonos é uma questão aberta da matemática. Mesmo para um triângulo ainda não há resposta. Para triângulos agudos, a resposta é bem conhecida, pois o triângulo formato pelos pés das alturas do triângulo é uma órbita periódica. Para triângulos obtusos, em geral, pouco se sabe. O objetivo desta dissertação é coletar resultados e técnicas sobre órbitas bilhares periódicas em triângulos obtusos. Começamos introduzindo o trabalho de Vorobets, Galperin e Stepin, que no início dos anos 90 unificaram os casos conhecidos de triângulos que possuem órbita bilhar periódica, introduziram o conceito de estabilidade e mostraram novos resultados, como uma família infinita de órbitas estáveis. Temos também o teorema de 2000 de Halbeisen e Hungerbühler que estende as famílias de órbitas estáveis. Mencionamos em seguida os trabalhos de Schwartz de 2006 e 2009 que utilizam auxílio computacional para mostrar que todo triângulo com ângulos menores que $100\\degree$ possui órbita bilhar periódica. Depois temos os resultados de 2008 de Hooper e Schwartz sobre órbitas bilhares periódicas em triângulos quase isósceles e sobre estabilidade de órbitas em triângulos de Veech. Todos os casos abordados neste trabalho incluem uma vasta variedade de triângulos, mas a questão de existência de órbitas bilhares periódicas para todo triângulo está longe de ser totalmente contemplada. / A billiard orbit in a triangle is a polygonal with vertices at the boundary of the triangle such that its angles reflect elastically. It is similar to a moving ball on a billiard table without friction (so the ball has constant speed and never stops) whose sides form a triangle. This orbit is periodic if it returns infinitely to the same point with the same direction. The existence of periodic billiard orbits in polygons is an open problem in mathematics. Even for a triangle there is still no answer. For acute triangles the answer is well known since the triangle whose vertices are the base points of the three altitudes of the triangle is a periodic orbit. For obtuse triangles, in general, little is known. The aim of this thesis is to collect results and techniques on periodic billiard orbits in obtuse triangles. We start by introducing the work of Vorobets, Gal\'perin and Stepin, who unified in the early 1990s the known cases of triangles that have periodic billiard orbits, introduced the concept of stability and proved new results, such as an infinite family of stable orbits. We also have the theorem of Halbeisen and Hungerbühler of 2000 extending the families of stable orbits. Next, we mention the works of Schwartz of 2006 and 2009 that use computational assistance to prove that every triangle whose angles are at most $100\\degree$ have periodic billiard orbits. Then, we have the results of 2008 by Hooper and Schwartz on periodic billiard orbits in nearly isosceles triangles and on stability of billiard orbits in Veech triangles. All cases covered in this work include a wide variety of triangles, but the question of the existence of periodic billiard orbits for all triangles is far from being fully contemplated.