Elaboração de uma sequência didática sobre os conceitos geométricos preliminares ao estudo da trigonometria

This study is qualitative, including literature search and preparation of teaching materials. Your goal is to report the study of geometric problems of character presented in the application of trigonometry and work on the preparation of detailed activities that help in overcoming these difficulties. For this, we read some papers on teaching and learning of trigonometry in order to identify the difficulties encountered during their journey. Then separate the geometric difficulties of character and prepare a list of geometric content and procedures associated with them. Thus, we can organize a notebook with activities that would address most of these concepts. Finally we present the specification of activities called Activity on introductory concepts to the study of trigonometry

Um estudo de práticas matemáticas históricas e sugestões de uso na matemática escolar

This dissertation aims to contribute on teaching of mathematics for enabling learning connected to the relationship among science, society, culture and cognition. To this end, we propose the involvement of our students with social practices found in history, since. Our intention is to create opportunities for school practices that these mathematical arising from professional practice historical, provide strategies for mathematical thinking and reasoning in the search for solutions to problematizations found today. We believe that the propose of producing Basic Problematization Units, or simply UBPs, in math teacher formation, points to an alternative that allows better utilization of the teaching and learning process of mathematics. The proposal has the aim of primary education to be, really forming the citizen, making it critical and society transformative agent. In this sense, we present some recommendations for exploration and use of these units for teachers to use the material investigated by us, in order to complement their teaching work in mathematics lessons. Our teaching recommendations materialized as a product of exploration on the book, Instrumentos nuevos de geometria muy necessários para medir distancia y alturas sem que interuengan numeros como se demuestra em la practica , written by Andrés de Cespedes, published in Madrid, Spain, in 1606. From these problematizations and the mathematics involved in their solutions, some guidelines for didactic use of the book are presented, so that the teacher can rework such problematizations supported on current issues, and thus use them in the classroom

Fractais e Percolação na Recuperação de Petróleo

The complex behavior of a wide variety of phenomena that are of interest to physicists, chemists, and engineers has been quantitatively characterized by using the ideas of fractal and multifractal distributions, which correspond in a unique way to the geometrical shape and dynamical properties of the systems under study. In this thesis we present the Space of Fractals and the methods of Hausdorff-Besicovitch, box-counting and Scaling to calculate the fractal dimension of a set. In this Thesis we investigate also percolation phenomena in multifractal objects that are built in a simple way. The central object of our analysis is a multifractal object that we call Qmf . In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, c, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability p. Depending on a parameter, ρ, characterizing the multifractal and the lattice size, L, the histogram can have two peaks. We observe that the probability of occupation at the percolation threshold, pc, for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent β. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation. The area and the number of neighbors of the blocks of Qmf show a non-trivial behavior. A general view of the object Qmf shows an anisotropy. The value of pc is a function of ρ which is related to its anisotropy. We investigate the relation between pc and the average number of neighbors of the blocks as well as the anisotropy of Qmf. In this Thesis we study likewise the distribution of shortest paths in percolation systems at the percolation threshold in two dimensions (2D). We study paths from one given point to multiple other points

Fractais e percolação na recuperação de Petróleo

The complex behavior of a wide variety of phenomena that are of interest to physicists, chemists, and engineers has been quantitatively characterized by using the ideas of fractal and multifractal distributions, which correspond in a unique way to the geometrical shape and dynamical properties of the systems under study. In this thesis we present the Space of Fractals and the methods of Hausdorff-Besicovitch, box-counting and Scaling to calculate the fractal dimension of a set. In this Thesis we investigate also percolation phenomena in multifractal objects that are built in a simple way. The central object of our analysis is a multifractal object that we call Qmf . In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, c, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability p. Depending on a parameter, ρ, characterizing the multifractal and the lattice size, L, the histogram can have two peaks. We observe that the probability of occupation at the percolation threshold, pc, for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent β. Despite the topological differences, we find that the percolation in a multifractal support is in the same universality class as standard percolation. The area and the number of neighbors of the blocks of Qmf show a non-trivial behavior. A general view of the object Qmf shows an anisotropy. The value of pc is a function of ρ which is related to its anisotropy. We investigate the relation between pc and the average number of neighbors of the blocks as well as the anisotropy of Qmf. In this Thesis we study likewise the distribution of shortest paths in percolation systems at the percolation threshold in two dimensions (2D). We study paths from one given point to multiple other points. In oil recovery terminology, the given single point can be mapped to an injection well (injector) and the multiple other points to production wells (producers). In the previously standard case of one injection well and one production well separated by Euclidean distance r, the distribution of shortest paths l, P(l|r), shows a power-law behavior with exponent gl = 2.14 in 2D. Here we analyze the situation of one injector and an array A of producers. Symmetric arrays of producers lead to one peak in the distribution P(l|A), the probability that the shortest path between the injector and any of the producers is l, while the asymmetric configurations lead to several peaks in the distribution. We analyze configurations in which the injector is outside and inside the set of producers. The peak in P(l|A) for the symmetric arrays decays faster than for the standard case. For very long paths all the studied arrays exhibit a power-law behavior with exponent g ∼= gl.

Condições de energia de hawking e ellis e a equação de raychaudhuri

In the Einstein s theory of General Relativity the field equations relate the geometry of space-time with the content of matter and energy, sources of the gravitational field. This content is described by a second order tensor, known as energy-momentum tensor. On the other hand, the energy-momentum tensors that have physical meaning are not specified by this theory. In the 700s, Hawking and Ellis set a couple of conditions, considered feasible from a physical point of view, in order to limit the arbitrariness of these tensors. These conditions, which became known as Hawking-Ellis energy conditions, play important roles in the gravitation scenario. They are widely used as powerful tools for analysis; from the demonstration of important theorems concerning to the behavior of gravitational fields and geometries associated, the gravity quantum behavior, to the analysis of cosmological models. In this dissertation we present a rigorous deduction of the several energy conditions currently in vogue in the scientific literature, such as: the Null Energy Condition (NEC), Weak Energy Condition (WEC), the Strong Energy Condition (SEC), the Dominant Energy Condition (DEC) and Null Dominant Energy Condition (NDEC). Bearing in mind the most trivial applications in Cosmology and Gravitation, the deductions were initially made for an energy-momentum tensor of a generalized perfect fluid and then extended to scalar fields with minimal and non-minimal coupling to the gravitational field. We also present a study about the possible violations of some of these energy conditions. Aiming the study of the single nature of some exact solutions of Einstein s General Relativity, in 1955 the Indian physicist Raychaudhuri derived an equation that is today considered fundamental to the study of the gravitational attraction of matter, which became known as the Raychaudhuri equation. This famous equation is fundamental for to understanding of gravitational attraction in Astrophysics and Cosmology and for the comprehension of the singularity theorems, such as, the Hawking and Penrose theorem about the singularity of the gravitational collapse. In this dissertation we derive the Raychaudhuri equation, the Frobenius theorem and the Focusing theorem for congruences time-like and null congruences of a pseudo-riemannian manifold. We discuss the geometric and physical meaning of this equation, its connections with the energy conditions, and some of its several aplications.

Criação da matéria em cosmologias com velocidade da luz variável e eletrônico não-linear

In this work we obtain the cosmological solutions and investigate the thermodynamics of matter creation in two diferent contexts. In the first we propose a cosmological model with a time varying speed of light c. We consider two diferent time dependence of c for a at Friedmann-Robertson- Walker (FRW) universe. We write the energy conservation law arising from Einstein equations and study how particles are created as c decreases with cosmic epoch. The variation of c is coupled to a cosmological Λ term and both singular and non-singular solutions are possible. We calculate the "adiabatic" particle creation rate and the total number of particles as a function of time and find the constrains imposed by the second law of thermodynamics upon the models. In the second scenario, we study the nonlinearity of the electrodynamics as a source of matter creation in the cosmological models with at FRW geometry. We write the energy conservation law arising from Einstein field equations with cosmological term Λ, solve the field equations and study how particles are created as the magnetic field B changes with cosmic epoch. We obtain solutions for the adiabatic particle creation rate, the total number of particles and the scale factor as a function of time in three cases: Λ = 0, Λ = constant and Λ α H2 (cosmological term proportional to the Hubble parameter). In all cases, the second law of thermodynamics demands that the universe is not contracting (H ≥ 0). The first two solutions are non-singular and exhibit in ationary periods. The third case studied allows an always in ationary universe for a suficiently large cosmological term

Cálculos AB initio de propriedades estruturais, eletrônicas e ópticas de colmpostos CaXo3 (X=Si, Ge, Sn)

In this work we present a study of structural, electronic and optical properties, at ambient conditions, of CaSiO3, CaGeO3 and CaSnO3 crystals, all of them a member of Ca-perovskite class. To each one, we have performed density functional theory ab initio calculations within LDA and GGA approximations of the structural parameters, geometry optimization, unit cell volume, density, angles and interatomic length, band structure, carriers effective masses, total and partial density of states, dielectric function, refractive index, optical absorption, reflectivity, optical conductivity and loss function. A result comparative procedure was done between LDA and GGA calculations, a exception to CaSiO3 where only LDA calculation was performed, due high computational cost that its low symmetry crystalline structure imposed. The Ca-perovskite bibliography have shown the absence of electronic structure calculations about this materials, justifying the present work

Sistemas Nanoestruturados: Heteroestruturas Quasi-Periódicas de Nitretos e Cálculos Ab initio em Polimorfos CaCO3

The physical properties and the excitations spectrum in oxides and semiconductors materials are presented in this work, whose the first part presents a study on the confinement of optical phonons in artificial systems based on III-V nitrides, grown in periodic and quasiperiodic forms. The second part of this work describes the Ab initio calculations which were carried out to obtain the optoeletronic properties of Calcium Oxide (CaO) and Calcium Carbonate (CaCO3) crystals. For periodic and quasi-periodic superlattices, we present some dynamical properties related to confined optical phonons (bulk and surface), obtained through simple theories, such as the dielectric continuous model, and using techniques such as the transfer-matrix method. The localization character of confined optical phonon modes, the magnitude of the bands in the spectrum and the power laws of these structures are presented as functions of the generation number of sequence. The ab initio calculations have been carried out using the CASTEP software (Cambridge Total Sequential Energy Package), and they were based on ultrasoft-like pseudopotentials and Density Functional Theory (DFT). Two di®erent geometry optimizations have been e®ectuated for CaO crystals and CaCO3 polymorphs, according to LDA (local density approximation) and GGA (generalized gradient approximation) approaches, determining several properties, e. g. lattice parameters, bond length, electrons density, energy band structures, electrons density of states, e®ective masses and optical properties, such as dielectric constant, absorption, re°ectivity, conductivity and refractive index. Those results were employed to investigate the confinement of excitons in spherical Si@CaCO3 and CaCO3@SiO2 quantum dots and in calcium carbonate nanoparticles, and were also employed in investigations of the photoluminescence spectra of CaCO3 crystal

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

Estudo da digestão de sedimento de curtume visando a extração do cromo por microemulsão

Leather tanneries generate effluents with high content of heavy metals, especially chromium, which is used in the mineral tanning process. Microemulsions have been studied in the extraction of heavy metals from aqueous solutions. Considering the problems related with the sediment resulting from the tanning process, due to its high content in chromium, in this work this sediment was characterized and microemulsion systems were applied for chromium removal. The extraction process consists in the removal of heavy metal ions present in an aqueous feeding solution (acid digestion solution) by a microemulsion system. First three different solid sludge digestion methods were evaluated, being chosen the method with higher digestion capacity. For this digestion method, seeking its optimization, was evaluated the influence of granule size, temperature and digestion time. Experimental results showed that the method proposed by USEPA (Method A) was the most efficient one, being obtained 95.77% of sample digestion. Regarding to the evaluated parameters, the best results were achieved at 95°C, 14 Mesh granule size, and 60 minutes digestion time. For chromium removal, three microemulsion extraction methods were evaluated: Method 1, in a Winsor II region, using as aqueous phase the acid digestion solution; Method 2, in a Winsor IV region, being obtained by the addition of the acid digestion solution to a microemulsion phase, whose aqueous phase is distilled water, until the formation of Winsor II system; and Method 3, in a Winsor III region, consisting in the formation of a Winsor III region using as aqueous phase the acid digestion solution, diluted in NaOH 0.01N. Seeking to optimize the extraction process only Method 1 (Systems I, II, and VIII) and Method 2 (System IX) were evaluated, being chosen points inside the interest regions (studied domains) to study the influence of contact time and pH in the extraction percentiles. The studied systems present the following compositions: System I: Surfactant Saponified coconut oil, Cosurfactant 1-Butanol, Oil phase Kerosene, Aqueous phase 2% NaCl solution; System II: Aqueous phase Acid digestion solution with pH adjusted using KOH (pH 3.5); System VIII: Aqueous phase - Acid digestion solution (pH 0.06); and System IX Aqueous phase Distilled water (pH 10.24), the other phases of Systems II, VIII and IX are similar to System I. Method 2 showed to be the more efficient one regarding chromium extraction percentile (up to 96.59% - pH 3.5). Considering that with Method 2 the microemulsion region only appears in the Winsor II region, it was studied Method 3 (System X) for the evaluation and characterization of a triphasic system, seeking to compare with a biphases system. System X is composed by: Surfactant Saponified coconut oil, Cosurfactant 1-Butanol, Oil phase Kerosene, Aqueous phase Acid digestion solution diluted with water and with its pH adjusted using 0.01N NaOH solution. The biphasic and triphasic microemulsion systems were analyzed regarding its viscosity, extraction efficiency and drop effective diameter. The experimental results showed that for viscosity studies the obtained values were low for all studied systems, the diameter of the drop is smaller in the Winsor II region, with 15.5 nm, reaching 46.0 nm in Winsor III region, being this difference attributed to variations in system compositions and micelle geometry. In chromium extraction, these points showed similar results, being achieved 99.76% for Winsor II system and 99.62% for Winsor III system. Winsor III system showed to be more efficient due to the obtaining of a icroemulsion with smaller volume, with the possibility to recover the oil phase in excess, and the use of a smaller proportion of surfactant and cosurfactant (C/S)

Estudo do desempenho de catalisadores tipo Ni/CexM1-xO2 (M = Zr ou Mn) na reação de oxidação parcial do metano

One of the main applications of methane is in the production of syngas, a mixture of hydrogen and carbon monoxide. Procedures used in this process are steam reforming, CO2 reforming, partial oxidation and autothermal reforming. The present study evaluated and compared the behavior of nickel catalysts supported on mixed oxides of cerium and manganese in the partial oxidation of methane with that of nickel catalysts supported on mixed oxides of cerium and zirconium. Mixed oxides of cerium and zirconium or cerium and manganese were synthesized using two different preparation methods, the polymeric precursor based on Pechini method and combustion reaction using a microwave. This was followed by impregnation with nickel content of 15 %. Samples were calcined at 300, 800 and 900 °C and characterized by specific surface area (SSA), X-ray fluorescence (XRF), X-ray diffraction (XRD), scanning electron microscopy (SEM), temperature programmed reduction (TPR) and the reaction of partial oxidation of methane. The specific areas of samples decrease with the rise in calcination temperature and after nickel impregnation. Metal-cerium solid solution was formed and the presence of other manganese species outside the solid solution structure was confirmed in the compound with the highest amounts of manganese oxides showed. With regard to scanning electron microscopy, supports based on cerium and zirconium prepared by Pechini method exhibited agglomerated particles without uniform geometry or visible pores on the surface. However, compounds containing manganese presented empty spaces in its structure. Through synthesis by combustion reaction, morphology acquired independently of the proposed composition demonstrated greater porosity in relation to Pechini synthesis. Although catalysts were prepared using different synthesis methods, the insertion of nickel showed very similar reduction profiles (TPR). In relation to nickel catalysts supported on mixed oxide of cerium and zirconium, there is an initial reduction of NiO species that present certain interaction with the support. This is followed by the reduction of Ce4+ in Ce3+ surface, with subsequent bulk reduction. For catalysts containing manganese, a reduction of nickel oxide species occurs, followed by two stages of reduction for species Mn2O3 in Mn3O4 and Mn3O4 in MnO, with subsequent reduction of bulk. With respect to partial oxidation reactions, the nickel catalyst supported on mixed oxide of cerium and zirconium, prepared using the Pechini method, exhibited CH4 conversion of approximately 80 %, with conversion of 81 % when prepared by combustion. This behavior continued for 10 hours of reaction. Manganese content was also found to directly influence catalytic activity of materials; the greater the manganese oxide content, the faster deactivation and destabilization occurred in the catalyst. In both synthesis methods, the nickel catalyst supported on mixed oxide of cerium and zirconium maintained an H2/CO ratio very close to 2 during the 10 hours of partial oxidation reaction. Samples containing manganese displayed smaller H2/CO ratios and lower performance in partial oxidation.

Sistema microemulsionado: caracterização e aplicação na indústria de petróleo

Alkyl polyethoxylates are surfactants widely used in vastly different fields, from oil exploitation to pharmaceutical applications. One of the most interesting characteristics of these surfactants is their ability to form micellar systems with specific geometry, the so-called wormlike micelle. In this work, microemulsions with three distinct compositions (C/T = 40 %, 30 % and 25 %) was used with contain UNITOL / butanol / water / xylene, cosurfactant / surfactante (C/S) ratio equal to 0,5. The microemulsion was characterized by dynamic light scattering (DLS), capillary viscometry, torque rheometry and surface tensiometry experiments carried out with systems based on xylene, water, butanol (cosurfactant) and nonaethyleneglycolmonododecyl ether (surfactant), with fixed surfactant:cosurfactant:oil composition (with and without oil phase) and varying the overall concentration of the microemulsion. The results showed that a transition from wormlike micelles to nanodrops was characterized by maximum relative viscosity (depending on how relative viscosity was defined), which was connected to maximum effective diameter, determined by DLS. Surface tension suggested that adsorption at the air water interface had a Langmuir character and that the limiting value of the surfactant surface excess was independent of the presence of cosurfactant and xylene. The results of the solubilization of oil sludge and oil recovery with the microemulsion: C/S = 40%, 30% and 25% proved to be quite effective in solubilization of oil sludge, with the percentage of solubilization (%solubilization) as high as 92.37% and enhanced oil recovery rates up to 90.22% for the point with the highest concentration of active material (surfactant), that is, 40%.

Síntese e caracterização de compostos de coordenação homolépticos e heterolépticos com o ligante ácido kójico

The present work has as main objective to contribute to the coordination chemistry of the ligand kojic acid, with the synthesis and characterization of the homoleptic compounds [Al(kj)3], [Fe(kj)3], [Fe(kj)2], [Cu(kj)2] e [Ru(kj)3], and the new heteroleptic complexes, trans- K2[Fe(kj)2(CN)2] and trans-Na2[Ru(kj)2(CN)2]. The obtained compounds were characterized by vibrational spectroscopy in the infrared region (IV) and Electronic spectroscopy in the ultraviolet and visible region (Uv-Vis). The infrared results indicated the coordination of the bidentate ligand kojic acid, due to reductions in the values of the stretching frequencies of the carbonyl and double bonds, compared to the free ligand for all complexes obtained. The presence of new vibrational modes indicated the change of symmetry of the molecules in the new compounds synthesized. Additionally, the presence of vibrational modes assigned to metal-oxygen also contributed to confirm the ligand coordinating to the metal ions. Through this technique, was also possible to perform correlations of the numbers of vibrational modes, in the region 1400-900 cm-1 and the compounds geometry. The heteroleptic compounds exhibited υC≡N in 2065 and 2053 cm-1, respectively, for the trans-K2[Fe(kj)2(CN)2] and trans-Na2[Ru(kj)2(CN)2], indicating coordination of the cyano ligand to metal ions FeII e RuII. Comparing the obtained values with literature data was possible to identify the complex isomerism as trans. In relation to the results of electronic spectroscopy, studies of pH variation of kojic acid provided information on the distribution of electron density in the molecule, showing characteristic spectral profile of kojic ion and its protonated form (Hkj, kojic acid), with two bands at 215 and 269 nm, or deprotonated (kj-), with bands at 226 and 315 nm. The electronic spectra obtained for all complexes in aqueous medium, in the ultraviolet region, exhibited variations of the energies assigned to kojic acid intraligand transitions while in the visible region, only transitions assigned to charge transfer of iron and ruthenium complex have been identified