Uma investigação histórica sobre os logaritmos com sugestões didáticas para a sala de aula

This study was conducted from a preliminary research to identify the conceptual and didactic approach to the logarithms given in the main textbooks adopted by the Mathematics teachers in state schools in the School of Natal, in Rio Grande do Norte. I carried out an historical investigation of the logarithms in order to reorient the math teacher to improve its educational approach this subject in the classroom. Based on the research approach I adopted a model of the log based on three concepts: the arithmetic, the geometric and algebraic-functional. The main objective of this work is to redirect the teacher for a broad and significant understanding of the content in order to overcome their difficulties in the classroom and thus realize an education that can reach the students learning. The investigative study indicated the possibility of addressing the logarithms in the classroom so transversalizante and interdisciplinary. In this regard, I point to some practical applications of this matter are fundamental in the study of natural phenomena as earthquakes, population growth, among others. These practical applications are connected, approximately, Basic Problematization Units (BPUs) to be used in the classroom. In closing, I offer some activities that helped teachers to understand and clarify the meaningful study of this topic in their teaching practice

Application of NMR techniques for the evaluation of fruit internal quality and physical properties of food

Los alimentos son sistemas complejos, formados por diversas estructuras a diferentes escalas: macroscópica y microscópica. Muchas propiedades de los alimentos, que son importantes para su procesamiento, calidad y tratamiento postcosecha, están relacionados con su microestructura. La presente tesis doctoral propone una metodología completa para la determinación de la estructura de alimentos desde un punto de vista multi-escala, basándose en métodos de Resonancia Magnética Nuclear (NMR). Las técnicas de NMR son no invasivas y no destructivas y permiten el estudio tanto de macro- como de microestructura. Se han utilizado distintos procedimientos de NMR dependiendo del nivel que se desea estudiar. Para el nivel macroestructural, la Imagen de Resonancia Magnética (MRI) ha resultado ser muy útil para la caracterización de alimentos. Para el estudio microestructural, la MRI requiere altos tiempos de adquisición, lo que hace muy difícil la transferencia de esta técnica a aplicaciones en industria. Por tanto, la optimización de procedimientos de NMR basados en secuencias relaxometría 2D T1/T2 ha resultado ser una estrategia primordial en esta tesis. Estos protocolos de NMR se han implementado satisfactoriamente por primera vez en alto campo magnético. Se ha caracterizado la microestructura de productos alimentarios enteros por primera vez utilizando este tipo de protocolos. Como muestras, se han utilizado dos tipos de productos: modelos de alimentos y alimentos reales (manzanas). Además, como primer paso para su posterior implementación en la industria agroalimentaria, se ha mejorado una línea transportadora, especialmente diseñada para trabajar bajo condiciones de NMR en trabajos anteriores del grupo LPF-TAGRALIA. Se han estudiado y seleccionado las secuencias más rápidas y óptimas para la detección de dos tipos de desórdenes internos en manzanas: vitrescencia y roturas internas. La corrección de las imágenes en movimiento se realiza en tiempo real. Asimismo, se han utilizado protocolos de visión artificial para la clasificación automática de manzanas potencialmente afectadas por vitrescencia. El presente documento está dividido en diferentes capítulos: el Capítulo 2 explica los antecedentes de la presente tesis y el marco del proyecto en el que se ha desarrollado. El Capítulo 3 recoge el estado del arte. El Capítulo 4 establece los objetivos de esta tesis doctoral. Los resultados se dividen en cinco sub-secciones (dentro del Capítulo 5) que corresponden con los trabajos publicados bien en revistas revisadas por pares, bien en congresos internacionales o bien como capítulos de libros revisados por pares. La Sección 5.1. es un estudio del desarrollo de la vitrescencia en manzanas mediante MRI y lo relaciona con la posición de la fruta dentro de la copa del árbol. La Sección 5.2 presenta un trabajo sobre macro- y microestructura en modelos de alimentos. La Sección 5.3 es un artículo en revisión en una revista revisada por pares, en el que se hace un estudio microestrcutural no destructivo mediante relaxometría 2D T1/T2. la Sección 5.4, hace una comparación entre manzanas afectadas por vitrescencia mediante dos técnicas: tomografía de rayos X e MRI, en manzana. Por último, en la Sección 5.5 se muestra un trabajo en el que se hace un estudio de secuencias de MRI en línea para la evaluación de calidad interna en manzanas. Los siguientes capítulos ofrecen una discusión y conclusiones (Capítulo 6 y 7 respectivamente) de todos los capítulos de esta tesis doctoral. Finalmente, se han añadido tres apéndices: el primero con una introducción de los principios básicos de resonancia magnética nuclear (NMR) y en los otros dos, se presentan sendos estudios sobre el efecto de las fibras en la rehidratación de cereales de desayuno extrusionados, mediante diversas técnicas. Ambos trabajos se presentaron en un congreso internacional. Los resultados más relevantes de la presente tesis doctoral, se pueden dividir en tres grandes bloques: resultados sobre macroestructura, resultados sobre microestructura y resultados sobre MRI en línea. Resultados sobre macroestructura: - La imagen de resonancia magnética (MRI) se aplicó satisfactoriamente para la caracterización de macroestructura. En particular, la reconstrucción 3D de imágenes de resonancia magnética permitió identificar y caracterizar dos tipos distintos de vitrescencia en manzanas: central y radial, que se caracterizan por el porcentaje de daño y la conectividad (número de Euler). - La MRI proveía un mejor contraste para manzanas afectadas por vitrescencia que las imágenes de tomografía de rayos X (X-Ray CT), como se pudo verificar en muestras idénticas de manzana. Además, el tiempo de adquisición de la tomografía de rayos X fue alrededor de 12 veces mayor (25 minutos) que la adquisición de las imágenes de resonancia magnética (2 minutos 2 segundos). Resultados sobre microestructura: - Para el estudio de microestructura (nivel subcelular) se utilizaron con éxito secuencias de relaxometría 2D T1/T2. Estas secuencias se usaron por primera vez en alto campo y sobre piezas de alimento completo, convirtiéndose en una forma no destructiva de llevar a cabo estudios de microestructura. - El uso de MRI junto con relaxometría 2D T1/T2 permite realizar estudios multiescala en alimentos de forma no destructiva. Resultados sobre MRI en línea: - El uso de imagen de resonancia magnética en línea fue factible para la identificación de dos tipos de desórdenes internos en manzanas: vitrescencia y podredumbre interna. Las secuencias de imagen tipo FLASH resultaron adecuadas para la identificación en línea de vitrescencia en manzanas. Se realizó sin selección de corte, debido a que la vitrescencia puede desarrollarse en cualquier punto del volumen de la manzana. Se consiguió reducir el tiempo de adquisición, de modo que se llegaron a adquirir 1.3 frutos por segundos (758 ms por fruto). Las secuencias de imagen tipo UFLARE fueron adecuadas para la detección en línea de la podredumbre interna en manzanas. En este caso, se utilizó selección de corte, ya que se trata de un desorden que se suele localizar en la parte central del volumen de la manzana. Se consiguió reducir el tiempo de adquisicón hasta 0.67 frutos por segundo (1475 ms por fruto). En ambos casos (FLASH y UFLARE) fueron necesarios algoritmos para la corrección del movimiento de las imágenes en tiempo real. ABSTRACT Food is a complex system formed by several structures at different scales: macroscopic and microscopic. Many properties of foods that are relevant to process engineering or quality and postharvest treatments are related to their microstructure. This Ph.D Thesis proposes a complete methodology for food structure determination, in a multiscale way, based on the Nuclear Magnetic Resonance (NMR) phenomenon since NMR techniques are non-invasive and non-destructive, and allow both, macro- and micro-structure study. Different NMR procedures are used depending on the structure level under study. For the macrostructure level, Magnetic Resonance Imaging (MRI) revealed its usefulness for food characterization. For microstructure insight, MRI required high acquisition times, which is a hindrance for transference to industry applications. Therefore, optimization of NMR procedures based on T1/T2 relaxometry sequences was a key strategy in this Thesis. These NMR relaxometry protocols, are successfully implemented in high magnetic field. Microstructure of entire food products have been characterized for the first time using these protocols. Two different types of food products have been studied: food models and actual food (apples). Furthermore, as a first step for the food industry implementation, a grading line system, specially designed for working under NMR conditions in previous works of the LPF-TAGRALIA group, is improved. The study and selection of the most suitable rapid sequence to detect two different types of disorders in apples (watercore and internal breakdown) is performed and the real time image motion correction is applied. In addition, artificial vision protocols for the automatic classification of apples potentially affected by watercore are applied. This document is divided into seven different chapters: Chapter 2 explains the thesis background and the framework of the project in which it has been worked. Chapter 3 comprises the state of the art. Chapter 4 establishes de objectives of this Ph.D thesis. The results are divided into five different sections (in Chapter 5) that correspond to published peered reviewed works. Section 5.1 assesses the watercore development in apples with MRI and studies the effect of fruit location in the canopy. Section 5.2 is an MRI and 2D relaxometry study for macro- and microstructure assessment in food models. Section 5.3 is a non-destructive microstructural study using 2D T1/T2 relaxometry on watercore affected apples. Section 5.4 makes a comparison of X-ray CT and MRI on watercore disorder of different apple cultivars. Section 5.5, that is a study of online MRI sequences for the evaluation of apple internal quality. The subsequent chapters offer a general discussion and conclusions (Chapter 6 and Chapter 7 respectively) of all the works performed in the frame of this Ph.D thesis (two peer reviewed journals, one book chapter and one international congress).Finally, three appendices are included in which an introduction to NMR principles is offered and two published proceedings regarding the effect of fiber on the rehydration of extruded breakfast cereal are displayed. The most relevant results can be summarized into three sections: results on macrostructure, results on microstructure and results on on-line MRI. Results on macrostructure: - MRI was successfully used for macrostructure characterization. Indeed, 3D reconstruction of MRI in apples allows to identify two different types of watercore (radial and block), which are characterized by the percentage of damage and the connectivity (Euler number). - MRI provides better contrast for watercore than X-Ray CT as verified on identical samples. Furthermore, X-Ray CT images acquisition time was around 12 times higher (25 minutes) than MRI acquisition time (2 minutes 2 seconds). Results on microstructure: - 2D T1/T2 relaxometry were successfully applied for microstructure (subcellular level) characterization. 2D T1/T2 relaxometry sequences have been applied for the first time on high field for entire food pieces, being a non-destructive way to achieve microstructure study. - The use of MRI together with 2D T1/T2 relaxometry sequences allows a non-destructive multiscale study of food. Results on on-line MRI: - The use of on-line MRI was successful for the identification of two different internal disorders in apples: watercore and internal breakdown. FLASH imaging was a suitable technique for the on-line detection of watercore disorder in apples, with no slice selection, since watercore is a physiological disorder that may be developed anywhere in the apple volume. 1.3 fruits were imaged per second (768 ms per fruit). UFLARE imaging is a suitable sequence for the on-line detection of internal breakdown disorder in apples. Slice selection was used, as internal breakdown is usually located in the central slice of the apple volume. 0.67 fruits were imaged per second (1475 ms per fruit). In both cases (FLASH and UFLARE) motion correction was performed in real time, during the acquisition of the images.

Efeito da concentração de sólidos e de variáveis operacionais no desempenho de um hidrociclone concentrador

The hydrocycloning operation has a goal to separate solid-liquid suspensions and liquid-liquid emulsions through the centrifugal force action. Hydrocyclones are equipment with reduced size and used in both clarification and thickening. This device is used in many areas, like petrochemical and minerals process, and accumulate advantages like versatility and low cost of maintenance. However, the demand to improve the process and to reduce the costs has motivated several studies of equipment optimization. The filtering hydrocyclone is a non-conventional equipment developed at FEQUI/UFU with objective to improve the hydrocycloning separation efficiency. The purpose of this study is to evaluate the operating conditions of feed concentration and underflow diameter on the performance of a filtering geometry optimized to minimization of energy costs. The filtration effect was investigated through the comparison between the performance of the Optimized Filtering Hydrocyclone (HCOF) and the Optimized Concentrator Hydrocyclone (HCO). Because of the resemblance of hydrocyclones performance, the filtration did not represent significant effect on the performance of the HCOF. It was found that in this geometry the decrease of the variable underflow diameter was very favorable to thickening operation. The suspension concentration of quartzite at 1.0% of solids in volume was increased about 42 times when the 3 mm underflow diameter was used. The increase on the feed solid percentage was good for decreasing the energy spent, so that a minimum number of Euler of 730 was achieved at CVA = 10.0%v. However, a greater amount of solids in suspension leads to a lower efficiency of the equipment. Therefore, to minimize the underflow-to-throughput ratio and keep a high efficiency level, it is indicated to work with dilute suspension (CVA = 1.0%) and 3 mm underflow diameter (η = 67%). But if it is necessary to work with high feed concentration, the use of 5 mm underflow diameter provides a rise in the efficiency. The HCO hydrocyclone was compared to the traditional family of hydrocyclones Rietema and presented advantages like higher efficiency (34% higher in average) and lower energy costs (20% lower in average). Finally, the efficiency curves and project equation have been raised for the HCO hydrocyclone each with satisfactory adjust.

Euler y el número π

Uno de los más prolíficos matemáticos, sino el que más, que han existido a lo largo de toda la historia, ha sido el suizo Leonhard Euler (1707-1783). Además de, en nuestro caso, introducir el uso de la letra griega π (inicial de perímetro) para nuestro número, dió numerosas aproximaciones mediante desarrollos en serie de la relación existente entre la circunferencia y su diámetro. El presente articulo trata de explicar el ingenioso procedimiento de dos de dichas series: la serie de Jacques Bernouilli y, la serie de Leibniz.

La música y el número siete. Historia de una relación controvertida.

Se estudia la relación del número siete con las matemáticas. Las notas musicales, los sonidos o las harmonías son algunos de los conceptos analizados. Además se realiza una aproximación a la figura de Leibniz, Bach y Euler, matemáticos que tambien tuvieron cierta relación con la música.

Sobre el número e. : algunas ideas para la Secundaria y el Bachillerato.

A partir de un caso práctico se explica el número matemático e. Leonhard Euler fue el matemático que hizo más descubrimientos relativos a este número, aunque el primero en estudiar el límite fue Jacob Bernoulli. Este número debería figurar en los libros de texto de Matemáticas por su interés didáctico. Leonhard Euler calculó el número e con mucha exactitud, para lo que desarrolló las herramientas adecuadas y supo ver su utilidad. Una ventaja de la nueva expresión para el número e es la rapidez en el cálculo. Por otro lado, se puede utilizar para dar una demostración asequible de la irracionalidad del número. Por último, se da una bibliografía donde encontrar ideas interesantes para ilustrar cuestiones relativas al número e..

Euler e os números pentagonais

The present investigation includes a study of Leonhard Euler and the pentagonal numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524. After a brief review of the life and work of Euler, we analyze the mathematical concepts covered in that article as well as its historical context. For this purpose, we explain the concept of figurate numbers, showing its mode of generation, as well as its geometric and algebraic representations. Then, we present a brief history of the search for the Eulerian pentagonal number theorem, based on his correspondence on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t know to prove it. Finally, in a letter to Goldbach in 1750, he presents a demonstration, which is published in E541, along with an alternative proof. The expansion of the concept of pentagonal number is then explained and justified by compare the geometric and algebraic representations of the new pentagonal numbers pentagonal numbers with those of traditional pentagonal numbers. Then we explain to the pentagonal number theorem, that is, the fact that the infinite product(1 x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7 x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers (expanded) and the sign is determined by whether as more or less as the exponent is pentagonal number (traditional or expanded). We also mention that Euler relates the pentagonal number theorem to other parts of mathematics, such as the concept of partitions, generating functions, the theory of infinite products and the sum of divisors. We end with an explanation of Euler s demonstration pentagonal number theorem

Deconvolução de Euler: passado, presente e futuro – um tutorial

Neste tutorial apresentamos uma revisão da deconvolução de Euler que consiste de três partes. Na primeira parte, recordamos o papel da clássica formulação da deconvolução de Euler 2D e 3D como um método para localizar automaticamente fontes de campos potenciais anômalas e apontamos as dificuldades desta formulação: a presença de uma indesejável nuvem de soluções, o critério empírico usado para determinar o índice estrutural (um parâmetro relacionado com a natureza da fonte anômala), a exeqüibilidade da aplicação da deconvolução de Euler a levantamentos magnéticos terrestres, e a determinação do mergulho e do contraste de susceptibilidade magnética de contatos geológicos (ou o produto do contraste de susceptibilidade e a espessura quando aplicado a dique fino). Na segunda parte, apresentamos as recentes melhorias objetivando minimizar algumas dificuldades apresentadas na primeira parte deste tutorial. Entre estas melhorias incluem-se: i) a seleção das soluções essencialmente associadas com observações apresentando alta razão sinal-ruído; ii) o uso da correlação entre a estimativa do nível de base da anomalia e a própria anomalia observada ou a combinação da deconvolução de Euler com o sinal analítico para determinação do índice estrutural; iii) a combinação dos resultados de (i) e (ii), permitindo estimar o índice estrutural independentemente do número de soluções; desta forma, um menor número de observações (tal como em levantamentos terrestres) pode ser usado; iv) a introdução de equações adicionais independentes da equação de Euler que permitem estimar o mergulho e o contraste de susceptibilidade das fontes magnéticas 2D. Na terceira parte apresentaremos um prognóstico sobre futuros desenvolvimentos a curto e médio prazo envolvendo a deconvolução de Euler. As principais perspectivas são: i) novos ataques aos problemas selecionados na segunda parte deste tutorial; ii) desenvolvimento de métodos que permitam considerar interferências de fontes localizadas ao lado ou acima da fonte principal, e iii) uso das estimativas de localização da fonte anômala produzidas pela deconvolução de Euler como vínculos em métodos de inversão para obter a delineação das fontes em um ambiente computacional amigável.

História do número e: gênese e aplicações

The aim of this paper is to present some fundamental aspects of the history of the number e, in particular those related to its origin, a little uncertain, and their unavoidable presence in the most diverse applications in various branches of science. We will highlight the importance of this number in compound interest problems, in the Napier’s logarithms, in the quadrature of the hyperbola, in the catenary problem and mostly in the lush Euler’s contribution to the subject.

Elementos finitos con acciones repartidas equivalentes de cualquier orden. Aplicación a los modelos de vigas de Timoshenko y Bernoulli-Euler

En este trabajo se introducen, en el contexto del Método de Elementos Finitos, dos alternativas posibles en relación con el concepto de acción repartida equivalente. La primera consiste en emplear pocos elementos, elevando el orden de dicha acción, mientras que la segunda se basa en emplear un mayor número de elementos dejando la acción en el orden más bajo posible. Se ilustran ambas situaciones mediante aplicaciones a los modelos de vigas de Timoshenko y Bernoulli-Euler, empleando estas acciones con diferentes órdenes, las cuales aproximan a la acción original, mediante polinomios ortogonales de Legendre en cada elemento. Como conclusión destacable, se indica que cuando se considera el menor número posible de elementos, es decir uno, para los casos de carga poco regular, ha bastado con utilizar acciones repartidas equivalentes de orden ligeramente superior al mínimo (orden cuatro), para obtener una excelente aproximación en los desplazamientos, giros y esfuerzos en el interior de los elementos.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems

In order to simulate stiff biochemical reaction systems, an explicit exponential Euler scheme is derived for multidimensional, non-commutative stochastic differential equations with a semilinear drift term. The scheme is of strong order one half and A-stable in mean square. The combination with this and the projection method shows good performance in numerical experiments dealing with an alternative formulation of the chemical Langevin equation for a human ether a-go-go related gene ion channel mode

Generalized Burgers equations and Euler-Painleve transcendents. II

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler Painlevé equation yy[script `]+ay[script ']2+ f(x)yy[script ']+g(x) y2+by[script ']+c=0 represents the generalized Burgers equations (GBE's) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE ut+uaux+Ju/2t =(gd/2)uxx (the nonplanar Burgers equation) is considered. It is found that its self-similar form is again governed by the Euler Painlevé equation. The ranges of the parameter alpha for which solutions of the connection problem to the self-similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE's. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well-known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler Painlevé equation with respect to GBE's. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.