Uma fundamentação para sinais e sistemas intervalares

In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given

Estudo numérico de escoamento turbulento em padrão anular gás-líquido em dutos verticais

Multiphase flows in ducts can adopt several morphologies depending on the mass fluxes and the fluids properties. Annular flow is one of the most frequently encountered flow patterns in industrial applications. For gas liquid systems, it consists of a liquid film flowing adjacent to the wall and a gas core flowing in the center of the duct. This work presents a numerical study of this flow pattern in gas liquid systems in vertical ducts. For this, a solution algorithm was developed and implemented in FORTRAN 90 to numerically solve the governing transport equations. The mass and momentum conservation equations are solved simultaneously from the wall to the center of the duct, using the Finite Volumes Technique. Momentum conservation in the gas liquid interface is enforced using an equivalent effective viscosity, which also allows for the solution of both velocity fields in a single system of equations. In this way, the velocity distributions across the gas core and the liquid film are obtained iteratively, together with the global pressure gradient and the liquid film thickness. Convergence criteria are based upon satisfaction of mass balance within the liquid film and the gas core. For system closure, two different approaches are presented for the calculation of the radial turbulent viscosity distribution within the liquid film and the gas core. The first one combines a k- Ɛ one-equation model and a low Reynolds k-Ɛ model. The second one uses a low Reynolds k- Ɛ model to compute the eddy viscosity profile from the center of the duct right to the wall. Appropriate interfacial values for k e Ɛ are proposed, based on concepts and ideas previously used, with success, in stratified gas liquid flow. The proposed approaches are compared with an algebraic model found in the literature, specifically devised for annular gas liquid flow, using available experimental results. This also serves as a validation of the solution algorithm

Implementação de um modelo computacional para estudo do processo Fischer-Tropsch em reator de leito de lama

This work aims at the implementation and adaptation of a computational model for the study of the Fischer-Tropsch reaction in a slurry bed reactor from synthesis gas (CO+H2) for the selective production of hydrocarbons (CnHm), with emphasis on evaluation of the influence of operating conditions on the distribution of products formed during the reaction.The present model takes into account effects of rigorous phase equilibrium in a reactive flash drum, a detailed kinetic model able of predicting the formation of each chemical species of the reaction system, as well as control loops of the process variables for pressure and level of slurry phase. As a result, a system of Differential Algebraic Equations was solved using the computational code DASSL (Petzold, 1982). The consistent initialization for the problem was based on phase equilibrium formed by the existing components in the reactor. In addition, the index of the system was reduced to 1 by the introduction of control laws that govern the output of the reactor products. The results were compared qualitatively with experimental data collected in the Fischer-Tropsch Synthesis plant installed at Laboratório de Processamento de Gás - CTGÁS-ER-Natal/RN

Avaliacao de dados de secagem de suspensoes de polpas de frutas em leito de jorro com alimentacao intermitente

The objective of this work was the development and improvement of the mathematical models based on mass and heat balances, representing the drying transient process fruit pulp in spouted bed dryer with intermittent feeding. Mass and energy balance for drying, represented by a system of differential equations, were developed in Fortran language and adapted to the condition of intermittent feeding and mass accumulation. Were used the DASSL routine (Differential Algebraic System Solver) for solving the differential equation system and used a heuristic optimization algorithm in parameter estimation, the Particle Swarm algorithm. From the experimental data food drying, the differential models were used to determine the quantity of water and the drying air temperature at the exit of a spouted bed and accumulated mass of powder in the dryer. The models were validated using the experimental data of drying whose operating conditions, air temperature, flow rate and time intermittency, varied within the limits studied. In reviewing the results predicted, it was found that these models represent the experimental data of the kinetics of production and accumulation of powder and humidity and air temperature at the outlet of the dryer

Clássicos versus Keynes: a abordagem formal de David Champernowne

Este texto tem por objetivo ressaltar um aspecto que não tem sido tratado com a devida profundidade na literatura que estuda a formalização da Teoria Geral do Emprego, dos Juros e da Moeda de John Maynard Keynes (1936). Mais precisamente, o texto destaca a estratégia de formalização adotada por David G. Champernowne em seu artigo intitulado Unemployment, Basic and Monetary: the classical analysis and the keynesian, publicado em 1935-36 na Review of Economic Studies. Chamamos a atenção para o fato dele distinguir a teoria clássica da teoria de Keynes não apenas pelos pressupostos adotados por cada teoria, mas principalmente pela construção de subsistemas a partir de um sistema geral, com características recursivas (relações de causalidade) distintas. As explicações em prosa, a descrição algébrica das funções comportamentais e condições de equilíbrio e a ilustração por meio de diagramas, além da escolha de conjuntos específicos de variáveis para representar cada uma das teorias e suas diferentes versões são aspectos deste artigo de Champernowne que merecem uma análise mais minuciosa.

Números complexos: uma proposta de mudança metodológica para uma aprendizagem significativa no ensino médio

This work presents a proposal of a methodological change to the teaching and learning of the complex numbers in the Secondary education. It is based on the inquiries and difficulties of students detected in the classrooms about the teaching of complex numbers and a questioning of the context of the mathematics teaching - that is the reason of the inquiry of this dissertation. In the searching for an efficient learning and placing the work as a research, it is presented a historical reflection of the evolution of the concept of complex numbers pointing out their more relevant focuses, such as: symbolic, numeric, geometrical and algebraic ones. Then, it shows the description of the ways of the research based on the methodology of the didactic engineering. This one is developed from the utilization of its four stages, where in the preliminary analysis stage, two data surveys are presented: the first one is concerning with the way of presenting the contents of the complex numbers in math textbooks, and the second one is concerning to the interview carried out with High school teachers who work with complex numbers in the practice of their professions. At first, in the analysis stage, it is presented the prepared and organized material to be used in the following stage. In the experimentation one, it is presented the carrying out process that was made with the second year High school students in the Centro Federal de Educação tecnológica do Rio Grande do Norte CEFET-RN. At the end, it presents, in the subsequent and validation stages, the revelation of the obtained results from the observations made in classrooms in the carrying out of the didactic sequence, the students talking and the data collection

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

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

The Quasi-lattice of Indiscernible Elements

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

Structural aspects of the fermion-boson mapping in two-dimensional gauge and anomalous gauge theories with massive fermions

Using a synthesis of the functional integral and operator approaches we discuss the fermion-buson mapping and the role played by the Bose field algebra in the Hilbert space of two-dimensional gauge and anomalous gauge field theories with massive fermions. In QED, with quartic self-interaction among massive fermions, the use of an auxiliary vector field introduces a redundant Bose field algebra that should not be considered as an element of the intrinsic algebraic structure defining the model. In anomalous chiral QED, with massive fermions the effect of the chiral anomaly leads to the appearance in the mass operator of a spurious Bose field combination. This phase factor carries no fermion selection rule and the expected absence of Theta-vacuum in the anomalous model is displayed from the operator solution. Even in the anomalous model with massive Fermi fields, the introduction of the Wess-Zumino field replicates the theory, changing neither its algebraic content nor its physical content. (C) 2002 Elsevier B.V. (USA).

Scattering and bound states of spinless particles in a mixed vector-scalar smooth step potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Exotic dark spinor fields

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Optimal attitude corrections for cylindrical spacecraft

A first order analytical model for optimal small amplitude attitude maneuvers of spacecraft with cylindrical symmetry in an elliptical orbits is presented. The optimization problem is formulated as a Mayer problem with the control torques provided by a power limited propulsion system. The state is defined by Seffet-Andoyer's variables and the control by the components of the propulsive torques. The Pontryagin Maximum Principle is applied to the problem and the optimal torques are given explicitly in Serret-Andoyer's variables and their adjoints. For small amplitude attitude maneuvers, the optimal Hamiltonian function is linearized around a reference attitude. A complete first order analytical solution is obtained by simple quadrature and is expressed through a linear algebraic system involving the initial values of the adjoint variables. A numerical solution is obtained by taking the Euler angles formulation of the problem, solving the two-point boundary problem through the shooting method, and, then, determining the Serret-Andoyer variables through Serret-Andoyer transformation. Numerical results show that the first order solution provides a good approximation to the optimal control law and also that is possible to establish an optimal control law for the artificial satellite's attitude. (C) 2003 COSPAR. Published by Elsevier B.V. Ltd. All rights reserved.

A geometric interpretation for transmission real losses minimization through the optimal power flow and its influence on voltage collapse

This work presents an approach for geometric solution of an optimal power flow (OPF) problem for a two bus system (a slack and a PV busses). Additionally, the geometric relationship between the losses minimization and the increase of the reactive margin and, therefore, the maximum loading point, is shown. The algebraic equations for the calculation of the Lagrange multipliers and for the minimum losses value are obtained. These equations are used to validate the results obtained using an OPF program. (C) 2002 Elsevier B.V. B.V. All rights reserved.

Systèmes de numération et automates

Let beta be an hyperbolic algebraic integer of modulus greater than 1. Lot A be a finite set of Q[beta] and D-beta = {(a(i), b(i))(igreater than or equal to0) is an element of (A x A)(N) \ Sigma(i=0)(infinity) a(i)beta(-i)}. We give a necessary and sufficient condition for D-beta to be sofic. As a consequence, we obtain a result due to Thurston (see Corollary 1). We also treat the case where the set of digits A is given by the greedy algorithm and study the connection with the beta-shift. (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.