Uma análise dos questionamentos dos alunos nas aulas de números complexos

This work presents a contribution for the studies reffering to the use of the History of Mathematics focusing on the improvement of the Teaching and Learning Process. It considers that the History of Matematics, as a way of giving meaning to the discipline and improve the quality of the Teaching and Learning Process. This research focuses on the questions of the students, classified in three categories of whys: the chronological, the logical and the pedagogical ones. Therefore, it is investigated the teaching of the Complex Numbers, from the questions of the students of the Centro Federal de Educação Tecnológica do Rio Grande do Norte (Educational Institution of Professional and Technology Education from Rio Grande do Norte). The work has the following goals: To classify and to analyse the questions of the students about the Complex Numbers in the classes of second grade of the High School, and to collate with the pointed categories used by Jones; To disccus what are the possible guidings that teachers of Mathematics can give to these questions; To present the resources needed to give support to the teacher in all things involving the History of Mathematics. Finally, to present a bibliographic research, trying to reveal supporting material to the teacher, with contents that articulate the Teaching of Mathematics with the History of Mathematics. It was found that the questionings of the pupils reffers more to the pedagogical whys, and the didatic books little contemplate other aspects of the history and little say about the sprouting and the evolution of methods of calculations used by us as well

O uso pedagógico de uma seqüência didática para a aquisição de algumas idéias relacionadas ao conceito de números complexos

The aim of the present work is to contribute to the teaching-learning process in Mathematics through an alternative which tries to motivate the student so that he/she will learn the basic concepts of Complex Numbers and realize that they are not pointless. Therefore, this work s general objective is to construct a didactic sequence which contains structured activities that intends to build up, in each student s thought, the concept of Complex Numbers. The didactic sequence is initially based on a review of the main historical aspects which begot the construction of those numbers. Based on these aspects, and the theories of Richard Skemp, was elaborated a sequence of structured activities linked with Maths history, having the solution of quadratic equations as a main starting point. This should make learning more accessible, because this concept permeates the students previous work and, thus, they should be more familiar with it. The methodological intervention began with the application of that sequence of activities with grade students in public schools who did not yet know the concept of Complex Numbers. It was performed in three phases: a draft study, a draft study II and the final study. Each phase was applied in a different institution, where the classes were randomly divided into groups and each group would discuss and write down the concepts they had developed about Complex Numbers. We also use of another instrument of analysis which consisted of a recorded interview of a semi-structured type, trying to find out the ways the students thought in order to construct their own concepts, i.e. the solutions of the previous activity. Their ideas about Complex Numbers were categorized according to their similarities and then analyzed. The results of the analysis show that the concepts constructed by the students were pertinent and that they complemented each other this supports the conclusion that the use of structured activities is an efficient alternative for the teaching of mathematics

Extração automatica de segmentos lineares de rodovias em imagens de resoluções variadas

In this paper a methodology for automatic extraction of road segments from images with different resolutions (low, middle and high resolution) is presented. It is based on a generalized concept of lines in digital images, by which lines can be described by the centerlines of two parallel edges. In the specific case of low resolution images, where roads are manifested as entities of 1 or 2 pixels wide, the proposed methodology combines an automatic image enhancement operation with the following strategies: automatic selection of the hysteresis thresholds and the Gaussian scale factor; line length thresholding; and polygonization. In medium and high resolution images roads manifest as narrow and elongated ribbons and, consequently, the extraction goal becomes the road centerlines. In this case, it is not necessary to apply the previous enhancement step used to enhance roads in low resolution images. The results obtained in the experimental evaluation satisfied all criteria established for the efficient extraction of road segments from different resolution images, providing satisfactory results in a completely automatic way.

Estudo de propriedades críticas em sistemas complexos: método de procura automática em sistemas tipo Ising e percolação de longo alcance

The new technique for automatic search of the order parameters and critical properties is applied to several well-know physical systems, testing the efficiency of such a procedure, in order to apply it for complex systems in general. The automatic-search method is combined with Monte Carlo simulations, which makes use of a given dynamical rule for the time evolution of the system. In the problems inves¬tigated, the Metropolis and Glauber dynamics produced essentially equivalent results. We present a brief introduction to critical phenomena and phase transitions. We describe the automatic-search method and discuss some previous works, where the method has been applied successfully. We apply the method for the ferromagnetic fsing model, computing the critical fron¬tiers and the magnetization exponent (3 for several geometric lattices. We also apply the method for the site-diluted ferromagnetic Ising model on a square lattice, computing its critical frontier, as well as the magnetization exponent f3 and the susceptibility exponent 7. We verify that the universality class of the system remains unchanged when the site dilution is introduced. We study the problem of long-range bond percolation in a diluted linear chain and discuss the non-extensivity questions inherent to long-range-interaction systems. Finally we present our conclusions and possible extensions of this work

Espectro das ondas de spin lineares e não-lineares em multicamadas magnéticas anisotrópicas

Conselho Nacional de Desenvolvimento Científico e Tecnológico

Representações espectrais de sistemas complexos: aplicações à síntese de superfícies brownianas fracionárias anisotrópicas, filtragem de sinais e identificação de correlações

In this thesis, we study the application of spectral representations to the solution of problems in seismic exploration, the synthesis of fractal surfaces and the identification of correlations between one-dimensional signals. We apply a new approach, called Wavelet Coherency, to the study of stratigraphic correlation in well log signals, as an attempt to identify layers from the same geological formation, showing that the representation in wavelet space, with introduction of scale domain, can facilitate the process of comparing patterns in geophysical signals. We have introduced a new model for the generation of anisotropic fractional brownian surfaces based on curvelet transform, a new multiscale tool which can be seen as a generalization of the wavelet transform to include the direction component in multidimensional spaces. We have tested our model with a modified version of the Directional Average Method (DAM) to evaluate the anisotropy of fractional brownian surfaces. We also used the directional behavior of the curvelets to attack an important problem in seismic exploration: the atenuation of the ground roll, present in seismograms as a result of surface Rayleigh waves. The techniques employed are effective, leading to sparse representation of the signals, and, consequently, to good resolutions

Métodos numéricos para resolução de equações diferenciais ordinárias lineares baseados em interpolação por spline

In this work we have elaborated a spline-based method of solution of inicial value problems involving ordinary differential equations, with emphasis on linear equations. The method can be seen as an alternative for the traditional solvers such as Runge-Kutta, and avoids root calculations in the linear time invariant case. The method is then applied on a central problem of control theory, namely, the step response problem for linear EDOs with possibly varying coefficients, where root calculations do not apply. We have implemented an efficient algorithm which uses exclusively matrix-vector operations. The working interval (till the settling time) was determined through a calculation of the least stable mode using a modified power method. Several variants of the method have been compared by simulation. For general linear problems with fine grid, the proposed method compares favorably with the Euler method. In the time invariant case, where the alternative is root calculation, we have indications that the proposed method is competitive for equations of sifficiently high order.

Realimentação da derivada dos estados em sistemas multivariáveis lineares usando LMIs

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Controle ótimo H∞ de sistemas não-lineares com modelos fuzzy takagi-sugeno

Neste trabalho é proposta uma metodologia de rastreamento de sinais e rejeição de distúrbios aplicada a sistemas não-lineares. Para o projeto do sistema de rastreamento, projeta-se os controladores fuzzy M(a) e N(a) que minimizam o limitante superior da norma H∞ entre o sinal de referência r(t) e o sinal de erro de rastreamento e(t), sendo e(t) a diferença entre a entrada de referência e a saída do sistema z(t). No método de rejeição de distúrbio utiliza-se a realimentação dinâmica da saída através de um controlador fuzzy Kc(a) que minimiza o limitante superior da norma H∞ entre o sinal de entrada exógena w(t) e o sinal de saída z(t). O procedimento de projeto proposto considera as não-linearidades da planta através dos modelos fuzzy Takagi-Sugeno. Os métodos são equacionados utilizando-se inequações matriciais lineares (LMIs), que quando factíveis, podem ser facilmente solucionados por algoritmos de convergência polinomial. Por fim, um exemplo ilustra a viabilidade da metodologia proposta.

Espectroscopia no infravermelho próximo e métodos de calibração multivariada aplicados à determinação simultânea de parâmetros bioquímicos em plasma sanguíneo

In this work, the quantitative analysis of glucose, triglycerides and cholesterol (total and HDL) in both rat and human blood plasma was performed without any kind of pretreatment of samples, by using near infrared spectroscopy (NIR) combined with multivariate methods. For this purpose, different techniques and algorithms used to pre-process data, to select variables and to build multivariate regression models were compared between each other, such as partial least squares regression (PLS), non linear regression by artificial neural networks, interval partial least squares regression (iPLS), genetic algorithm (GA), successive projections algorithm (SPA), amongst others. Related to the determinations of rat blood plasma samples, the variables selection algorithms showed satisfactory results both for the correlation coefficients (R²) and for the values of root mean square error of prediction (RMSEP) for the three analytes, especially for triglycerides and cholesterol-HDL. The RMSEP values for glucose, triglycerides and cholesterol-HDL obtained through the best PLS model were 6.08, 16.07 e 2.03 mg dL-1, respectively. In the other case, for the determinations in human blood plasma, the predictions obtained by the PLS models provided unsatisfactory results with non linear tendency and presence of bias. Then, the ANN regression was applied as an alternative to PLS, considering its ability of modeling data from non linear systems. The root mean square error of monitoring (RMSEM) for glucose, triglycerides and total cholesterol, for the best ANN models, were 13.20, 10.31 e 12.35 mg dL-1, respectively. Statistical tests (F and t) suggest that NIR spectroscopy combined with multivariate regression methods (PLS and ANN) are capable to quantify the analytes (glucose, triglycerides and cholesterol) even when they are present in highly complex biological fluids, such as blood plasma

Complexos metálicos de cobalto, níquel e cobre com a Pirazina-2-carboxamida e 4- hidrazida ácida piridincarboxílica: síntese e caracterização

This work involved the synthesis, characterization and proposing the molecular structure of coordination compounds involving ligands pyrazine-2-carboxamide (PZA) and 4- hydrazide acidic pyridine carboxylic (INH) and metals of the first transition series (M = Co2+, Ni2+ and Cu2+). For the characterization of the compounds used were analytical techniques such as infrared absorption spectroscopy average (FT-IR) molar conductivity measurements, CHN elemental analysis, EDTA Complexometric, measurement of melting point, X-ray diffraction by powder method, Thermogravimetry (TG) and Differential Thermal Analysis (DTA) and Simultaneous Differential Scanning Calorimetry (DSC). The absorption spectra in the infrared region suggested that the ligand coordination to the metal center occurs through the carbonyl oxygen atom and nitrogen alpha pyrazine ring to those complexes formed with PZA. For INH complexes with metal-ligand coordination is through the carbonyl oxygen and nitrogen of the terminal hydrazide grouping. The conductivity measurements of the complexes in aqueous solution they suggest to all behavior of the type 1:2 electrolytes, and conduct of non-electrolytes in acetonitrile. The results obtained by CHN elemental analysis and EDTA Complexometric allowed to infer the stoichiometry of the compounds synthesized. For all of the complexes obtained was possible to record the melting points, neither of which melted near the melting temperature of the free ligands. The X-ray diffraction showed that the complexes of pyrazinamide exhibited diffraction lines, suggesting that these compounds are crystalline, while compounds of isoniazid, with the exception of cobalt, exhibited diffraction lines, indicating that they are crystalline. The results from the TG-DTA and DSC allowed information regarding the dehydration and thermal decomposition of these complexes

Síntese, caracterização e estudo fotofísico de complexos de 4,4- diaminoestilbeno-2,2-dissulfonato com íons lantanídeos

This is a work involving fundamental studies of chemistry where the synthesis and structural characterization, as well as a possible future application of these new compounds as luminescent sensors or sunscreen agents, complexes with 4,4 diaminostilbene-2,2-disulfonic (DSD) and trivalent lanthanide ions La3+, Nd3+, Eu3+, Gd3+ and Yb3+, were synthesized in the ratio of 3 mmol: 1 mmol (DSD: lanthanides). The complexes obtained with these ions were present in powder form and were characterized by complexometric titration with EDTA CHN Elemental analysis, molecular absorption spectroscopy in the ultraviolet region, the absorption spectroscopy in the infrared, thermal analysis (TG / DTG), Nuclear Magnetic Resonance - NMR 1H and Luminescence Spectroscopy. The complexometric titration and CHN analysis, confirmed the TG / DTG which suggest that these complexes have the following general chemical formulas: [La2(C14H12S2O6N2)2(H2O)2Cl2].7H2O,[Nd2(C14H12S2O6N2)2(H2O)2Cl2].6H2O,[Eu2(C14 H12S2O6N2)2(H2O)2Cl2].7H2O,[Gd2(C14H12S2O6N2)2(H2O)2Cl2].4H2O e [Yb2(C14H12S2O6N2)2(H2O)2].6H2O. The disappearance of the bands in the infrared spectrum at 2921 cm-1 and 2623 cm-1 and the displacement of the bands in the spectra of the amine complex indicate that the lanthanide ion is coordinated to the oxygen atoms and the sulfonate groups of the nitrogens amines, suggesting the formation of the dimer. The disappearance of the signal and the displacement signal SO3H amines in the 1H NMR spectrum of this complex are also indicative coordination and dimer formation. The Thermogravimetry indicates that the DSD is thermally stable in the range of 40º to 385°C and their complexes with lanthanide ions exhibit weight loss between 4 and 5 stages. The Uv-visible spectra indicated that the DSD and complexes exhibit cis isomers. The analysis of luminescence indicates that the complexes do not exhibit emission in the region of the lanthanides but an intense emission part of the binder. This is related to the triplet states of the ligand, which are in the lowest energy state emitting lanthanide ions, and also the formation of the dimer that suppress the luminescence of ion Eu3+. The formation of dimer was also confirmed by calculating the europium complex structure using the model Hamiltonian PM6 and Sparkle

Obtenção de dispersões de complexos polieletrolíticos à base de quitosana e poli(ácido metacrílico) e análise de adsorção de albumina bovina sérica

Dispersions composed of polyelectrolyte complexes based on chitosan and poly(methacrylic acid), PMAA, were obtained by the dropping method and template polymerization. The effect of molecular weight of PMAA and ionic strength on the formation of chitosan/poly(methacrylic acid), CS/PMAA, complexes was evaluated using the dropping method. The increase in molecular weight of PMAA inhibited the formation of insoluble complexes, while the increase in ionic strength first favored the formation of the complex followed by inhibiting it at higher concentrations. The polyelectrolyte complexation was strongly dependent on macromolecular dimensions, both in terms of molecular weight and of coil expansion/contraction driven by polyelectrolyte effect. The resultant particles from dropping method and template polymerization were characterized as having regions with different charge densities: chitosan predominating in the core and poly(methacrylic acid) at the surface, the particles being negatively charged, as a consequence. Albumin was adsorbed on templatepolymerized CS/PMAA complexes (after crosslinking with glutardialdehyde) and pH was controlled in order to obtain two conditions: (i) adsorption of positively charged albumin, and (ii) adsorption of albumin at its isoelectric point. Adsorption isotherms and zeta potential measurements showed that albumin adsorption was controlled by hydrogen bonding/van der Waals interactions and that brushlike structures may enhance adsorption of albumin on these particles

Controlador adaptativo backstepping a estrutura variavel com Observadores de estado

This research aims at developing a variable structure adaptive backstepping controller (VS-ABC) by using state observers for SISO (Single Input Single Output), linear and time invariant systems with relative degree one. Therefore, the lters were replaced by a Luenberger Adaptive Observer and the control algorithm uses switching laws. The presented simulations compare the controller performance, considering when the state variables are estimated by an observer, with the case that the variables are available for measurement. Even with numerous performance advantages, adaptive backstepping controllers still have very complex algorithms, especially when the system state variables are not measured, since the use of lters on the plant input and output is not something trivial. As an attempt to make the controller design more intuitive, an adaptive observer as an alternative to commonly used K lters can be used. Furthermore, since the states variables are considered known, the controller has a reduction on the dependence of the unknown plant parameters on the design. Also, switching laws could be used in the controller instead of the traditional integral adaptive laws because they improve the system transient performance and increase the robustness against external disturbances in the plant input

Termodinâmica de um gás de fótons no contexto de eletrodinâmicas não-lineares

The objective of this dissertation is the development of a general formalism to analyze the thermodynamical properties of a photon gas under the context of nonlinear electrodynamics (NLED). To this end it is obtained, through the systematic analysis of Maxwell s electromagnetism (EM) properties, the general dependence of the Lagrangian that describes this kind of theories. From this Lagrangian and in the background of classical field theory, we derive the general dispersion relation that photons must obey in terms of a background field and the NLED properties. It is important to note that, in order to achieve this result, an aproximation has been made in order to allow the separation of the total electromagnetic field into a strong background electromagnetic field and a perturbation. Once the dispersion relation is in hand, the usual Bose-Einstein statistical procedure is followed through which the thermodynamical properties, energy density and pressure relations are obtained. An important result of this work is the fact that equation of state remains identical to the one obtained under EM. Then, two examples are made where the thermodynamic properties are explicitly derived in the context of two NLED, Born-Infelds and a quadratic approximation. The choice of the first one is due to the vast appearance in literature and, the second one, because it is a first order approximation of a large class of NLED. Ultimately, both are chosen because of their simplicity. Finally, the results are compared to EM and interpreted, suggesting possible tests to verify the internal consistency of NLED and motivating further developement into the formalism s quantum case