A obra de triangulis omnimodis libri quinque de Johann Müller Regiomontanus (1436 1476): uma contribuição para o desenvolvimento da trigonometria

Trigonometry, branch of mathematics related to the study of triangles, developed from practical needs, especially relating to astronomy, Surveying and Navigation. Johann Müller, the Regiomontanus (1436-1476) mathematician and astronomer of the fifteenth century played an important role in the development of this science. His work titled De Triangulis Omnimodis Libri Quinque written around 1464, and published posthumously in 1533, presents the first systematic exposure of European plane and spherical trigonometry, a treatment independent of astronomy. In this study we present a description, translation and analysis of some aspects of this important work in the history of trigonometry. Therefore, the translation was performed using a version of the book Regiomontanus on Triangles of Barnabas Hughes, 1967. In it you will find the original work in Latin and an English translation. For this study, we use for most of our translation in Portuguese, the English version, but some doubt utterance, statement and figures were made by the original Latin. In this work, we can see that trigonometry is considered as a branch of mathematics which is subordinated to geometry, that is, toward the study of triangles. Regiomontanus provides a large number of theorems as the original trigonometric formula for the area of a triangle. Use algebra to solve geometric problems and mainly shows the first practical theorem for the law of cosines in spherical trigonometry. Thus, this study shows some of the development of the trigonometry in the fifteenth century, especially with regard to concepts such as sine and cosine (sine reverse), the work discussed above, is of paramount importance for the research in the history of mathematics more specifically in the area of historical analysis and critique of literary sources or studying the work of a particular mathematician

Análise da bacia hidrográfica do Rio Punaú-Rio Grande do Norte utilizando ferramentas de geoprocessamento

Provide data and information on watershed becomes important since the knowledge of their physical characteristics, land use, etcetera, allows for better planning and sustainable use of economically, socially and environmentally in this area. The investigation of the physical environment has been commonly given with the use of geoprocessing, which has proved a very efficient tool. Within this context, this research aims at analyzing the river basin Punaú (located in the cities of Touros, Rio do Fogo and Pureza, state of Rio Grande do Norte) in several aspects, using geoprocessing as a tool of work, to provide information about the entire watershed. Specifically, this study aimed to update pre-existing maps, such as geological, geomorphological and land use, generating map of environmental vulnerability, under the aspect of erosion susceptibility of the area, generating map of legal incompatibility, identifying areas that are already being employed in breach of environmental legislation; propose solutions to the occupation of the river basin Punaú, focused on environmental planning. The methodology was based on the use of geoprocessing tools for data analysis and to make maps of legal incompatibility and environmental vulnerability. For the first map was taken into account the environmental legislation regarding the protection of watersheds. For the vulnerability analysis, the generated map was the result of crossing the maps of geology, geomorphology, soils and land use, having been assigned weights to different attributes of thematic maps, generating a map of environmental vulnerability in relation to susceptibility to erosion. The analysis results indicate that agriculture is the most significant activity in the basin, in total occupied area, which confers a high degree of environmental vulnerability in most of the basin, and some agricultural areas eventually develop in a manner inconsistent with Brazilian environmental legislation. It is proposed to consider deploying a measure of revitalization of the watershed in more critical areas and conservation through mitigation measures on the causes of environmental degradation, such as protection of water sources, protection and restoration of riparian vegetation, protection of permanent preservation areas, containment of erosion processes in general, and others listed or not in specific laws, and even the establishment of a committee of basins in the area

Confinamento de fónons ópticos em estruturas piezoelétricas periódicas e quasiperiódicas

We study the optical-phonon spectra in periodic and quasiperiodic (Fibonacci type) superlattices made up from III-V nitride materials (GaN and AlN) intercalated by a dielectric material (silica - SiO2). Due to the misalignments between the silica and the GaN, AlN layers that can lead to threading dislocation of densities as high as 1010 cm−1, and a significant lattice mismatch (_ 14%), the phonon dynamics is described by a coupled elastic and electromagnetic equations beyond the continuum dielectric model, stressing the importance of the piezoelectric polarization field in a strained condition. We use a transfer-matrix treatment to simplify the algebra, which would be otherwise quite complicated, allowing a neat analytical expressions for the phonon dispersion relation. Furthermore, a quantitative analysis of the localization and magnitude of the allowed band widths in the optical phonon s spectra, as well as their scale law are presented and discussed

Espectro de polariton de plasmons e propriedades ópitcas de super-redes tipo cantor

Conselho Nacional de Desenvolvimento Científico e Tecnológico

Ondas de spin em super-redes ferromagnéticas com campo de anisotropia

Conselho Nacional de Desenvolvimento Científico e Tecnológico

Excitações em cristais fotônicos unidimensionais

In this work, we present a theoretical study of the propagation of electromagnetic waves in multilayer structures called Photonic Crystals. For this purpose, we investigate the phonon-polariton band gaps in periodic and quasi-periodic (Fibonacci-type) multilayers made up of both positive and negative refractive index materials in the terahertz (THz) region. The behavior of the polaritonic band gaps as a function of the multilayer period is investigated systematically. We use a theoretical model based on the formalism of transfer matrix in order to simplify the algebra involved in obtaining the dispersion relation of phonon-polaritons (bulk and surface modes). We also present a quantitative analysis of the results, pointing out the distribution of the allowed polaritonic bandwidths for high Fibonacci generations, which gives good insight about their localization and power laws. We calculate the emittance spectrum of the electromagnetic radiation, in THZ frequency, normally and obliquely incident (s and p polarized modes) on a one-dimensional multilayer structure composed of positive and negative refractive index materials organized periodically and quasi-periodically. We model the negative refractive index material by a effective medium whose electric permittivity is characterized by a phonon-polariton frequency dependent dielectric function, while for the magnetic permeability we have a Drude like frequency-dependent function. Similarity to the one-dimensional photonic crystal, this layered effective medium, called polaritonic Crystals, allow us the control of the electromagnetic propagation, generating regions named polaritonic bandgap. The emittance spectra are determined by means of a well known theoretical model based on Kirchoff s second law, together with a transfer matrix formalism. Our results shows that the omnidirectional band gaps will appear in the THz regime, in a well defined interval, that are independent of polarization in periodic case as well as in quasiperiodic case

Polaritons em materiais magnéticos nanoestruturados

In this work we present a theoretical study about the properties of magnetic polaritons in superlattices arranged in a periodic and quasiperiodic fashíons. In the periodic superlattice, in order to describe the behavior of the bulk and surface modes an effective medium approach, was used that simplify enormously the algebra involved. The quasi-periodic superlattice was described by a suitable theoretical model based on a transfer-matrix treatment, to derive the polariton's dispersion relation, using Maxwell's equations (including effect of retardation). Here, we find a fractal spectra characterized by a power law for the distribution of the energy bandwidths. The localization and scaling behavior of the quasiperiodic structure were studied for a geometry where the wave vector and the external applied magnetic field are in the same plane (Voigt geometry). Numerical results are presented for the ferromagnet Fe and for the metamagnets FeBr2 and FeCl2

Conjuntos numéricos

In this work, we present a text on the Sets Numerical using the human social needs as a tool for construction new numbers. This material is intended to present a text that reconciles the correct teaching of mathmatics and clarity needed for a good learning

Princípio da Indução Matemática no Ensino Médio

We developed this dissertation aiming its in the process of teaching and learning of the Principle of Mathematical Induction and we set our efforts so that the students of the first year of the high school can assimilate the content having the knowledge seen in the basic education as foreknowledge. With this, we seek to awake in the student the interest on proofs, showing how much it s needed in examples that involve contents that he is already seen

Em direção a uma representação para equações algébricas :uma lógica equacional local

The intervalar arithmetic well-known as arithmetic of Moore, doesn't possess the same properties of the real numbers, and for this reason, it is confronted with a problem of operative nature, when we want to solve intervalar equations as extension of real equations by the usual equality and of the intervalar arithmetic, for this not to possess the inverse addictive, as well as, the property of the distributivity of the multiplication for the sum doesn t be valid for any triplet of intervals. The lack of those properties disables the use of equacional logic, so much for the resolution of an intervalar equation using the same, as for a representation of a real equation, and still, for the algebraic verification of properties of a computational system, whose data are real numbers represented by intervals. However, with the notion of order of information and of approach on intervals, introduced by Acióly[6] in 1991, the idea of an intervalar equation appears to represent a real equation satisfactorily, since the terms of the intervalar equation carry the information about the solution of the real equation. In 1999, Santiago proposed the notion of simple equality and, later on, local equality for intervals [8] and [33]. Based on that idea, this dissertation extends Santiago's local groups for local algebras, following the idea of Σ-algebras according to (Hennessy[31], 1988) and (Santiago[7], 1995). One of the contributions of this dissertation, is the theorem 5.1.3.2 that it guarantees that, when deducing a local Σ-equation E t t in the proposed system SDedLoc(E), the interpretations of t and t' will be locally the same in any local Σ-algebra that satisfies the group of fixed equations local E, whenever t and t have meaning in A. This assures to a kind of safety between the local equacional logic and the local algebras

Função quadrática

In general, the study of quadratic functions is based on an excessive amount formulas, all content is approached without justification. Here is the quadratic function and its properties from problems involving quadratic equations and the technique of completing the square. Based on the definitions we will show that the graph of the quadratic function is the parabola and finished our studies finding that several properties of the function can be read from the simple observation of your chart. Thus, we built the whole matter justifying each step, abandoning the use of decorated formulas and valuing the reasoning

Usando a história da resolução de alguns problemas para introduzir conceitos: sistemas lineares, determinantes e matrizes.

In this work we studied the method to solving linear equations system, presented in the book titled "The nine chapters on the mathematical art", which was written in the first century of this era. This work has the intent of showing how the mathematics history can be used to motivate the introduction of some topics in high school. Through observations of patterns which repeats itself in the presented method, we were able to introduce, in a very natural way, the concept of linear equations, linear equations system, solution of linear equations, determinants and matrices, besides the Laplacian development for determinants calculations of square matrices of order bigger than 3, then considering some of their general applications

Uma ferramenta didática para ajudar na fixação dos conceitos introdutórios de análise combinatória

Humans, as well as some animals are born gifted with the ability to perceive quantities. The needs that came from the evolution of societies and technological resources make the the optimization of such counting methods necessary. Although necessary and useful, there are a lot of diculties in the teaching of such methods.In order to broaden the range of available tools to teach Combinatorial Analysis, a owchart is presented in this work with the goal of helping the students to x the initial concepts of such subject via pratical exercises

Calculando distância em geometria espacial usando material manipulável como recurso didático

This work presents a proposal for introducing the teaching of Geometry Space study attempts to demonstrate that the use of manipulatives as a teaching resource can be an alternative learning facilitator for fixing the primitive concepts of geometry, the postulates and theorems, position relationships between points, lines and planes and calculating distances. The development makes use of a sequence of activities aimed at ensuring that students can build a more systematic learning and these are divided into four steps

Uma proposta alternativa para o ensino progressões relacionadas a funções

Generally, arithmetic and geometric progressions are taught separately from ane and exponential functions, only by the use of memorized formulas and without any concern of showing students how these contents are related. This paper aims at presenting a way of teaching such contents in an integrated way, starting with the definition of ane and exponential functions relating them to situations from the daily life of the students. Then, characteristics and graphics of those functions are presented and, subsequently, arithmetic and geometric progression are shown as a restriction of the ane and exponential functions. Thus, the study of the progressions is introduced based on the functions mentioned above using situations from students daily lives as examples