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

Os Polímeros como Sistemas Complexos

Neste trabalho, através de simulações computacionais, identificamos os fenômenos físicos associados ao crescimento e a dinâmica de polímeros como sistemas complexos exibindo comportamentos não linearidades, caos, criticalidade auto-organizada, entre outros. No primeiro capítulo, iniciamos com uma breve introdução onde descrevemos alguns conceitos básicos importantes ao entendimento do nosso trabalho. O capítulo 2 consiste na descrição do nosso estudo da distribuição de segmentos num polímero ramificado. Baseado em cálculos semelhantes aos usados em cadeias poliméricas lineares, utilizamos o modelo de crescimento para polímeros ramificados (Branched Polymer Growth Model - BPGM) proposto por Lucena et al., e analisamos a distribuição de probabilidade dos monômeros num polímero ramificado em 2 dimensões, até então desconhecida. No capítulo seguinte estudamos a classe de universalidade dos polímeros ramificados gerados pelo BPGM. Utilizando simulações computacionais em 3 dimensões do modelo proposto por Lucena et al., calculamos algumas dimensões críticas (dimensões fractal, mínima e química) para tentar elucidar a questão da classe de universalidade. Ainda neste Capítulo, descrevemos um novo modelo para a simulação de polímeros ramificados que foi por nós desenvolvido de modo a poupar esforço computacional. Em seguida, no capítulo 4 estudamos o comportamento caótico do crescimento de polímeros gerados pelo BPGM. Partimos de polímeros criticamente organizados e utilizamos uma técnica muito semelhante aquela usada em transições de fase em Modelos de Ising para estudar propagação de danos chamada de Distância de Hamming. Vimos que a distância de Hamming para o caso dos polímeros ramificados se comporta como uma lei de potência, indicando um caráter não-extensivo na dinâmica de crescimento. No Capítulo 5 analisamos o movimento molecular de cadeias poliméricas na presença de obstáculos e de gradientes de potenciais. Usamos um modelo generalizado de reptação para estudar a difusão de polímeros lineares em meios desordenados. Investigamos a evolução temporal destas cadeias em redes quadradas e medimos os tempos característicos de transporte t. Finalizamos esta dissertação com um capítulo contendo a conclusão geral denoss o trabalho (Capítulo 6), mais dois apêndices (Apêndices A e B) contendo a fenomenologia básica para alguns conceitos que utilizaremos ao longo desta tese (Fractais e Percolação respectivamente) e um terceiro e ´ultimo apêndice (Apêndice C) contendo uma descrição de um programa de computador para simular o crescimentos de polímeros ramificados em uma rede quadrada

Estudo de sistemas complexos com interações de longo alcance : percolação, redes e tráfego

In this thesis we investigate physical problems which present a high degree of complexity using tools and models of Statistical Mechanics. We give a special attention to systems with long-range interactions, such as one-dimensional long-range bondpercolation, complex networks without metric and vehicular traffic. The flux in linear chain (percolation) with bond between first neighbor only happens if pc = 1, but when we consider long-range interactions , the situation is completely different, i.e., the transitions between the percolating phase and non-percolating phase happens for pc < 1. This kind of transition happens even when the system is diluted ( dilution of sites ). Some of these effects are investigated in this work, for example, the extensivity of the system, the relation between critical properties and the dilution, etc. In particular we show that the dilution does not change the universality of the system. In another work, we analyze the implications of using a power law quality distribution for vertices in the growth dynamics of a network studied by Bianconi and Barabási. It incorporates in the preferential attachment the different ability (fitness) of the nodes to compete for links. Finally, we study the vehicular traffic on road networks when it is submitted to an increasing flux of cars. In this way, we develop two models which enable the analysis of the total flux on each road as well as the flux leaving the system and the behavior of the total number of congested roads

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

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

Otimização da síntese de nanoferritas de nizn dopada com cobre e cobalto

Different compositions of Ni0,5-xCuxZn0,5Fe2O4 and Ni0,5-xCoxZn0,5Fe2O4 0 ≤ x ≤ 0.3 were synthesized ferrite y the citrate precursor method. The stoichiometric compositions were calcined in air at 350°C and then pressed into pellets and toroids. The pressed samples were sintered at temperatures of 1000, 1050 and 1100°C/3h in air control at the speed of heating and cooling. The calcined powders were characterized by XRD, TGA / DTG, FTIR, SEM and vibrating sample magnetometry (VSM) and the sintered samples by XRD, SEM, MAV, density and measurements of permeability and magnetic losses. There was pure phase formation ferrimagnetism applied at all temperatures except for A-I composition at all sintering temperatures and A-II only at a temperature of 1100°C. Crystallite sizes were obtained by Rietveld analysis, nanometer size from 11 to 20 nm for the calcined powders. For SEM, the sintered samples showed grain size between 1 and 10 micrometers. Bulk density (ρ) of sintered material presented to the Families almost linear behavior with increasing temperature and a tendency to decrease with increasing concentration of copper, different behavior of the B Family, where the increase in temperature decreased the density. The magnetic measurements revealed the powder characteristics of a soft ferrimagnetic material. Two processes of magnetization were considered, the superparamagnetism at low temperatures (350°C) and the formation of magnetic domains at higher temperatures. Obtaining the best parameters for P and B-II magnetic ferrites at high temperatures. The sintered material at 1000°C showed a relative permeability (μ) from 50 to 800 for the A Family and from 10 to 600 for the B Family. The samples sintered at 1100°C, B Family showed a variation from 10 to 1000 and the magnetic loss (tan δ) of A and B Families, around of 1. The frequency response of the toroidal core is in the range of 0.3 kHz. Several factors contribute to the behavior of microstructure considering the quantities μ and tan δ, such as the grain size, inter-and intragranular porosity, amount of grain boundary and the aspects of the dynamics of domain walls at high frequencies.

Eficiência de extração de cobre e níquel utilizando sistemas microemulsionados bifásicos e trifásicos

The environmental impact due to the improper disposal of metal-bearing industrial effluents imposes the need of wastewater treatment, since heavy metals are nonbiodegradable and hazardous substances that may cause undesirable effects to humans and the environment. The use of microemulsion systems for the extraction of metal ions from wastewaters is effective when it occurs in a Winsor II (WII) domain, where a microemulsion phase is in equilibrium with an aqueous phase in excess. However, the microemulsion phase formed in this system has a higher amount of active matter when compared to a WIII system (microemulsion in equilibrium with aqueous and oil phases both in excess). This was the reason to develop a comparative study to evaluate the efficiency of two-phases and three-phases microemulsion systems (WII and WIII) in the extraction of Cu+2 and Ni+2 from aqueous solutions. The systems were composed by: saponified coconut oil (SCO) as surfactant, n-Butanol as cosurfactant, kerosene as oil phase, and synthetic solutions of CuSO4.5H2O and NiSO4.6H2O, with 2 wt.% NaCl, as aqueous phase. Pseudoternary phase diagrams were obtained and the systems were characterized by using surface tension measurements, particle size determination and scanning electron microscopy (SEM). The concentrations of metal ions before and after extraction were determined by atomic absorption spectrometry. The extraction study of Cu+2 and Ni+2 in the WIII domain contributed to a better understanding of microemulsion extraction, elucidating the various behaviors presented in the literature for these systems. Furthermore, since WIII systems presented high extraction efficiencies, similar to the ones presented by Winsor II systems, they represented an economic and technological advantage in heavy metal extraction due to a small amount of surfactant and cosurfactant used in the process and also due to the formation of a reduced volume of aqueous phase, with high concentration of metal. Considering the reextraction process, it was observed that WIII system is more effective because it is performed in the oil phase, unlike reextraction in WII, which is performed in the aqueous phase. The presence of the metalsurfactant complex in the oil phase makes possible to regenerate only the surfactant present in the organic phase, and not all the surfactant in the process, as in WII system. This fact allows the reuse of the microemulsion phase in a new extraction process, reducing the costs with surfactant regeneration

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

Estudo de Fractalidade e Evolução Dinâmica de Sistemas Complexos

In this work, the study of some complex systems is done with use of two distinct procedures. In the first part, we have studied the usage of Wavelet transform on analysis and characterization of (multi)fractal time series. We have test the reliability of Wavelet Transform Modulus Maxima method (WTMM) in respect to the multifractal formalism, trough the calculation of the singularity spectrum of time series whose fractality is well known a priori. Next, we have use the Wavelet Transform Modulus Maxima method to study the fractality of lungs crackles sounds, a biological time series. Since the crackles sounds are due to the opening of a pulmonary airway bronchi, bronchioles and alveoli which was initially closed, we can get information on the phenomenon of the airway opening cascade of the whole lung. Once this phenomenon is associated with the pulmonar tree architecture, which displays fractal geometry, the analysis and fractal characterization of this noise may provide us with important parameters for comparison between healthy lungs and those affected by disorders that affect the geometry of the tree lung, such as the obstructive and parenchymal degenerative diseases, which occurs, for example, in pulmonary emphysema. In the second part, we study a site percolation model for square lattices, where the percolating cluster grows governed by a control rule, corresponding to a method of automatic search. In this model of percolation, which have characteristics of self-organized criticality, the method does not use the automated search on Leaths algorithm. It uses the following control rule: pt+1 = pt + k(Rc − Rt), where p is the probability of percolation, k is a kinetic parameter where 0 < k < 1 and R is the fraction of percolating finite square lattices with side L, LxL. This rule provides a time series corresponding to the dynamical evolution of the system, in particular the likelihood of percolation p. We proceed an analysis of scaling of the signal obtained in this way. The model used here enables the study of the automatic search method used for site percolation in square lattices, evaluating the dynamics of their parameters when the system goes to the critical point. It shows that the scaling of , the time elapsed until the system reaches the critical point, and tcor, the time required for the system loses its correlations, are both inversely proportional to k, the kinetic parameter of the control rule. We verify yet that the system has two different time scales after: one in which the system shows noise of type 1 f , indicating to be strongly correlated. Another in which it shows white noise, indicating that the correlation is lost. For large intervals of time the dynamics of the system shows ergodicity

Obtenção e caracterização de complexos polieletrolíticos de quitosana e poli (metacrilato de sódio)

Polyelectrolyte complexes (PECs) nanoparticles were prepared using chitosan and sodium polymethacrylate. The complex formation was investigated using turbidimetry, conductometry, viscometry, and dynamic light scattering. The presence of excess positive charges was evidenced by zeta potential measurements. The particle diameter was characterized by dynamic light scattering and the morphology by atomic force microscopy. In all experiments an abrupt change in behavior was observed at a carboxyl:amino molar ratio around 0.7−0.8. Those changes in behavior were related to a proposed mechanism of complex formation based on the decrease of macromolecular dimensions of soluble polyelectrolyte complex clusters, followed by phase segregation

Densificação e microestrutura de um compósito Mo-Cu preparado a partir de um pó de heptamolibidato de amônia e cobre

The Cu-Mo system is a composite used in the electrical industry as material for electrical contact and resistance welding electrode as well as the heat sink and microwave absorber in microelectronic devices. The use of this material in such applications is due to the excellent properties of thermal and electrical conductivity and the possibility of adjustment of its coefficient of thermal expansion to meet those of materials used as substrates in the semiconductor micoreletrônic industry. Powder metallurgy through the processes of milling, pressing shaping and sintering is a viable technique for consolidation of such material. However, the mutual insolubility of both phases and the low wettability of liquid Cu on Mo impede its densification. However, the mutual insolubility of both phases and the low wettability of liquid Cu on Mo impede its densification. The mechanical alloying is a technique for preparation of powders used to produce nanocrystalline composite powder with amorphous phase or extended solid solution, which increases the sinterability immiscible systems such as the Mo-Cu. This paper investigates the influence of ammonium heptamolybdate (HMA) and the mechanical alloying in the preparation of a composite powder HMA-20% Cu and the effect of this preparation on densification and structure of MoCu composite produced. HMA and Cu powders in the proportion of 20% by weight of Cu were prepared by the techniques of mechanical mixing and mechanical alloying in a planetary mill. These were milled for 50 hours. To observe the evolution of the characteristics of the particles, powder samples were taken after 2, 10, 15, 20, 30 and 40 hours of milling. Cylindrical samples 5 to 8 mm in diameter and 3 to 4 mm thickness were obtained by pressing at 200 MPa to the mixed powders so as to ground. These samples were sintered at 1200 ° C for 60 minutes under an atmosphere of H2. To determine the effect of heating rate on the structure of the material during the decomposition and reduction of HMA, rates of 2, 5 and 10 ° C / min were used .. The post and the structures of the sintered samples were characterized by SEM and EDS. The density of the green and sintered bodies was measured using the geometric method (weight / volume). Vickers microhardness with a load of 1 N for 15 s were performed on sintered structures. The density of the sintered structures 10 ° C / min. reached 99% of theoretical density, how the density of sintered structures to 2 ° C / min. reached only 90% of the theoretical density

Transportadores sólidos de oxigênio à base de níquel e cobre suportados em aluminatos para combustão do metano pela tecnologia de recirculação química

In recent years, solid carriers suitable oxygen have been developed for use in different chemical processes recirculation. The success of this technology is directly related to the chemical reactivity and the oxygen storage capacity of the carrier. Thus, research into the development of new materials that can be applied to the process becomes extremely important. Possible candidates are the carriers based on nickel and copper for presenting favorable thermodynamic properties. In this work, aluminates type MAl2O4 (M = Mg and Ca) and M0,9B0,1Al2O4 (B = Ni and Cu) that are used as supports were synthesized by combustion reactions assisted by microwave and calcined at 900°C/2h. Then, the carriers were impregnated with 10% (m/m) of nickel or copper, and subsequently calcined at 600°C/2h to obtain the solid oxygen carriers, which were characterized by X-ray diffraction (XRD) Microscopy scanning electron microscopy (SEM) and temperature programmed reduction (TPR). Reactions simulating the combustion process by chemical recirculation were performed by cycles reduction/oxidation, in order to evaluate the reactivity of carriers. XRD analysis revealed diffraction peaks of the spinel type structures. In the doped substrates were verified the presence of secondary phases, suggesting that all the metal was incorporated into the spinel structure. In solid oxygen carriers, the NiO and CuO phases were observed after impregnation of active phases on different media. The results of evaluations of chemical cycles reduction/oxidation revealed that TSO's impregnated with nickel in various media were more active and are potential candidates for use in the chemical recirculation technology

Síntese e caracterização de complexos de Co(II) e Ni(II) com l-dopa, carbidopa e benzimidazol

Were synthesized in this work in the following aqueous solution coordination compounds: [Ni(LDP)(H2O)2Cl2].2H2O, [Co(LDP)Cl2].3H2O, [Ni(CDP)Cl2].4H2O, [Co(CDP)Cl2].4H2O, [Ni(BDZ)2Cl2].4H2O and [Co(BDZ)2Cl2(H2O)2]. These complexes were synthesized by stoichiometric addition of the binder in the respective metal chloride solutions. Precipitation occurred after drying the solvent at room temperature. The characterization and proposed structures were made using conventional analysis methods such as elemental analysis (CHN), absorption spectroscopy in the infrared Fourier transform spectroscopy (FTIR), X-ray diffraction by the powder method and Technical thermoanalytical TG / DTG (thermogravimetry / derivative thermogravimetry) and DSC (differential scanning calorimetry). These techniques provided information on dehydration, coordination modes, thermal performance, composition and structure of the synthesized compounds. The results of the TG curve, it was possible to establish the general formula of each compound synthesized. The analysis of X-ray diffraction was observed that four of the synthesized complex crystal structure which does not exhibit the complex was obtained from Ldopa and carbidopa and the complex obtained from benzimidazole was obtained crystal structures. The observations of the spectra in the infrared region suggested a monodentate ligand coordination to metal centers through its amine group for all complexes. The TG-DTG and DSC curves provide important information and on the behavior and thermal decomposition of the synthesized compounds. The molar conductivity data indicated that the solutions of the complexes formed behave as a nonelectrolyte, which implies that chlorine is coordinated to the central atom in the complex.