Thesis (M.A.)--Boston University / It has been noted that the writing of a system of linear equations in matrix form quite naturally suggests the computation of an inverse matrix which may then be used to compute solutions to a system of equations in which the coefficients remain the same but the constants have been changed. The elementary properties of matrices and determinants have been reviewed and a general method of matrix inversion based on these properties was considered. The very considerable practical deficiencies of this theoretically attractive method suggested that the method would be almost completely worthless for a system larger than two by two [TRUNCATED]

Orientador: Eduardo Kokubun / Banca: Fábio Yuzo Nakamura / Banca: Cláudio Alexandre Gobatto / Resumo: Manchado et al. (2002) determinaram os parâmetros do modelo da PCrit através dos valores submáximos da FC. Foi reportado a existência de uma relação linear da PCritFC e AWCFC submáxima com seus respectivos valores máximos. Diante da correlação entre a FC e o VO2, essas equações lineares foram re-inseridas no modelo hiperbólico da PCrit e deduzidas para predizer a cinética do VO2. O objetivo do presente estudo foi verificar a possibilidade de determinar os parâmetros da PCrit através da cinética do VO2 de apenas uma sessão de exercício supramáximo. Para isso, foram analisadas a resposta do VO2 de jovens universitários, 13 homens e 2 mulheres, submetidos a três ou quatro testes preditivos da PCrit e CTA. O VO2 foi coletado a cada 3 ciclos respiratórios, interpolado a cada um segundo e alisado com médias móveis de 15 segundos. Foram determinados os tempos para se atingir, 60, 65, 70, 75, 80, 85, 90, 95 e 100% do VO2max. Esses tempos apresentaram uma relação hiperbólica com a intensidade de trabalho e possibilitaram determinar os valores de PCrit e CTA correspondentes aos valores abaixo e no VO2max. Ancova demonstrou que esses valores de PCrit e CTA sub-VO2max se relacionavam de maneira linear com os valores máximos correspondentes. As equações lineares foram deduzidas ao modelo hiperbólico 2 2 2 1 * TVO k E P VO VO rep + = +, o qual apresentou bom ajuste à cinética do VO2 (0.90 < r2 <0.99). Os valores encontrados, para os termos da equação, possibilitaram predizer os valores de PCrit (PCrit_pred) e CTA (CTA_pred). Esses valores foram comparados, através do intervalo de confiança (IC), com seus respectivos parâmetros reais, mas não foi encontrada diferença para a PCrit. Entretanto, as grandes amplitudes do IC para a CTA e CTA_pred não permitiram realizar determinações acuradas para este parâmetro / Abstract: Manchado et al. (2002) determined CP model parameters through heart rate (HR) submaximal values. The authors reported a linear relationship between submaximal CPHR and AWCHR to its respectives maximal values. This work was based on the assumption that HR and VO2 have a qualitative correlation. Thus, those linear equations were re-inserted in CP hyperbolic model and deduced to predict VO2 kinetics. The main purpose of this study was test the use of VO2 kinetics, of one supramaximal exercise session, to determine the CP parameters. Were submitted to the experiment 13 men and 2 women. They realized 3 or 4 CP and AWC prediction tests and, during the trials, the VO2 response was collected every 3-breath cycle. It was interpolated to one-second values and smoothed by 15 s rolling average. The time to achieve 60, 65, 70, 75, 80, 85, 90, 95 and 100% of VO2max was recorded. Those times fitted to a hyperbolic relationship with exercise intensity, thus the CP and AWC related to sub- and VO2max were determined. ANCOVA revealed that each parameter sub- VO2max value was linear related to its maximal value. These two linear equations were deduced to the hyperbolic model 2 2 2 1 * TVO k E P VO VO rep + = + . The VO2 response was good fitted (0.90 < r2 < 0.99) and the equation term’s values were used to determine predicted values of CP (CP_pred) and AWC (AWC_pred). These values were compared, by confidence intervals (CI), with its respective measured real values and no differences were found to CP. However, AWC and AWC_pred presented elevated CI ranges and does not allowed an accurate prediction of this parameter. In Summary, the kinetics model adopted to VO2 makes possible determine CP values using just one rectangular test / Mestre

Made available in DSpace on 2014-06-11T19:22:51Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-11-27Bitstream added on 2014-06-13T19:28:18Z : No. of bitstreams: 1 moura_rf_me_rcla.pdf: 364424 bytes, checksum: 2ba8adc0206af1d0845327bf8bf6f7e8 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Manchado et al. (2002) determinaram os parâmetros do modelo da PCrit através dos valores submáximos da FC. Foi reportado a existência de uma relação linear da PCritFC e AWCFC submáxima com seus respectivos valores máximos. Diante da correlação entre a FC e o VO2, essas equações lineares foram re-inseridas no modelo hiperbólico da PCrit e deduzidas para predizer a cinética do VO2. O objetivo do presente estudo foi verificar a possibilidade de determinar os parâmetros da PCrit através da cinética do VO2 de apenas uma sessão de exercício supramáximo. Para isso, foram analisadas a resposta do VO2 de jovens universitários, 13 homens e 2 mulheres, submetidos a três ou quatro testes preditivos da PCrit e CTA. O VO2 foi coletado a cada 3 ciclos respiratórios, interpolado a cada um segundo e alisado com médias móveis de 15 segundos. Foram determinados os tempos para se atingir, 60, 65, 70, 75, 80, 85, 90, 95 e 100% do VO2max. Esses tempos apresentaram uma relação hiperbólica com a intensidade de trabalho e possibilitaram determinar os valores de PCrit e CTA correspondentes aos valores abaixo e no VO2max. Ancova demonstrou que esses valores de PCrit e CTA sub-VO2max se relacionavam de maneira linear com os valores máximos correspondentes. As equações lineares foram deduzidas ao modelo hiperbólico 2 2 2 1 * TVO k E P VO VO rep + = + , o qual apresentou bom ajuste à cinética do VO2 (0.90 < r2 <0.99). Os valores encontrados, para os termos da equação, possibilitaram predizer os valores de PCrit (PCrit_pred) e CTA (CTA_pred). Esses valores foram comparados, através do intervalo de confiança (IC), com seus respectivos parâmetros reais, mas não foi encontrada diferença para a PCrit. Entretanto, as grandes amplitudes do IC para a CTA e CTA_pred não permitiram realizar determinações acuradas para este parâmetro. / Manchado et al. (2002) determined CP model parameters through heart rate (HR) submaximal values. The authors reported a linear relationship between submaximal CPHR and AWCHR to its respectives maximal values. This work was based on the assumption that HR and VO2 have a qualitative correlation. Thus, those linear equations were re-inserted in CP hyperbolic model and deduced to predict VO2 kinetics. The main purpose of this study was test the use of VO2 kinetics, of one supramaximal exercise session, to determine the CP parameters. Were submitted to the experiment 13 men and 2 women. They realized 3 or 4 CP and AWC prediction tests and, during the trials, the VO2 response was collected every 3-breath cycle. It was interpolated to one-second values and smoothed by 15 s rolling average. The time to achieve 60, 65, 70, 75, 80, 85, 90, 95 and 100% of VO2max was recorded. Those times fitted to a hyperbolic relationship with exercise intensity, thus the CP and AWC related to sub- and VO2max were determined. ANCOVA revealed that each parameter sub- VO2max value was linear related to its maximal value. These two linear equations were deduced to the hyperbolic model 2 2 2 1 * TVO k E P VO VO rep + = + . The VO2 response was good fitted (0.90 < r2 < 0.99) and the equation term s values were used to determine predicted values of CP (CP_pred) and AWC (AWC_pred). These values were compared, by confidence intervals (CI), with its respective measured real values and no differences were found to CP. However, AWC and AWC_pred presented elevated CI ranges and does not allowed an accurate prediction of this parameter. In Summary, the kinetics model adopted to VO2 makes possible determine CP values using just one rectangular test.

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The thesis involved the design, implementation and testing of a second order neuro-fuzzy controller for the speed control of an AC induction motor, and a comparison of the neuro-fuzzy controller's performance with that of the PI algorithm. It was found experimentally, that the operating temperature of the AC induction motor affected the ability of the PI controller to maintain the set speed. The linear PI algorithm approximation was observed to produce transient speed responses when sudden changes in load occurred. The neuro-fuzzy design was found to be quite involved in the initial design stages. However, after the initial design, it was a simple matter of fine-tuning the algorithm, to optimize performance for any parameter variations of the motor due to temperature or due to sudden changes in load. The neuro-fuzzy algorithm can be developed utilising one of two methods. The first method utilises sensor-less control by detailed modeling of the induction motor; where all varying parameters of the motor are modeled mathematically. This involves using differential equations, and representing them in the form of system response block diagrams. When the overall plant transfer function is known, a fuzzy PI algorithm can be utilised to control the processes of the plant. The second method involves modeling the overall output response as a second order system. Raw data can then be generated in a text file format, providing control data according to the modeled second order system. Using the raw data, development software such as FuzzyTECH is utilised to perform supervised learning, so to produce the knowledge base for the overall system. This method was utilised in this thesis and compared to the conventional PI algorithm. The neuro-fuzzy algorithm implemented on a PLC was found to provide better performance than the PI algorithm implemented on the same PLC. It provided also in the added flexibility for further fine-tuning and avoided the need for rigorous mathematical manipulation of linear equations / Master of Engineering (Hons)

This thesis investigates the computation of piecewise polynomial solutions to ill- conditioned linear systems of equations when noise on the linear measurements is observed. Specifically, we enhance our understanding of and provide qualifications on when such ill-conditioned systems of equations can be solved to a satisfactory accuracy. We show that the conventional condition number of the coefficient matrix is not sufficiently informative in this regard and propose a more relevant conditioning measure that takes into account the decay rate of singular values. We also discuss interactions of several factors affecting the solvability of such systems, including the number of discontinuities in solutions, as well as the distribution of nonzero entries in sparse matrices. In addition, we construct and test an approach for computing piecewise polynomial solutions of highly ill-conditioned linear systems using a randomized, SVD-based truncation, and L1-norm regularization. The randomized truncation is a stabilization technique that helps reduce the cost of the traditional SVD truncation for large and severely ill-conditioned matrices. For L1-minimization, we apply a solver based on the Alternating Direction Method. Numerical results are presented to compare our approach that is faster and can solve larger problems, called RTL1 (randomized truncation L1-minimization), with a well-known solver PP-TSVD.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Historicamente o estudo de sistemas de equaÃÃes lineares precede o estudo de matrizes e determinantes. Seguiremos esta ordem e mostraremos que seguir o curso histÃrico à viÃvel e e dar significado preciso ao conceito de matriz e a definiÃÃo de determinante, os quais tÃm sidos imprecisos e incompreendido pela ordem estudada. Atualmente quase todos ou totalmente os livros didÃticos de MatemÃtica para o ensino mÃdio que abordam o tema tem adotado a seguinte ordem de estudo: matrizes, determinantes e sistemas de equaÃÃes lineares. Esta ordem tem tornado inviÃvel a compreensÃo dos conceitos de matriz e determinante por parte dos alunos e tambÃm dos professores que os ensinam. Essa prÃtica tem deixado prejuÃzos enormes na formaÃÃo dos alunos. Com uma abordagem simples e objetiva, introduzimos de maneira natural e sucessiva, os conceitos de: equaÃÃes lineares, sistema de equaÃÃes lineares, resoluÃÃo de sistemas de equaÃÃes lineares, determinante do sistema linear, matrizes e determinante de matrizes, sendo matrizes e determinantes considerados como ferramentas essenciais para o estudo e resoluÃÃo de sistema de equaÃÃes lineares. Acreditamos que essa ordem na abordagem pode servir para a melhoria do processo ensino-aprendizagem. / The study of linear equation systems has historically preceded the study of matrixes and determinants. Following this order, we will show that the historical course is viable,and we will give precise significance to the concept of matrix and to the definition of determinant, which have been imprecise and misunderstood due to the order of study. Nowadays almost all of the mathematics high school textbooks which approach this theme have adopted the following order of study: matrixes, determinants, and linear equation systems. This order has made it impracticable for students and teachers to comprehend the concepts of matrix and determinant. This kind of practice has caused enormous damage in the education process of the students. With a simple and objective approach, we introduce in a natural and successive way the concepts of: linear equations, linear equation systems, resolution of linear equation systems, linear system determinant, matrixes, and determinant of matrixes, keeping in mind that matrixes and detetrminants are essential tools for the study and resolution of linear equation systems.We believe that this order of approach can improve the teaching-learning process.

This research referred to Basic Competency Test from 2001 to 2006 to construct test and analyzed 207 ninth-graders¡¦ problem-solving strategies as well as errors in solving linear equations with two unknowns. Furthermore, the investigator referred to the contents of interview, to investigate the factors that cause students¡¦ mistakes. Results shows that the main strategy for solving equations is 'to add and subtract the elimination approach', while for solving application problems is 'organizing side by side'. The errors for solving equations are mistaking concepts including Equality Axiom, etc. The errors for solving application problems are mostly concerned about the translation and holistic mistake. Through analyzing data from interviews, the reasons for mistakes in solving equations are: mutual interference of experience; mixed up different operation rules; or, solving a problem with the wrong concept built by themselves. The reasons for mistakes in solving application problems are: insufficient language ability; the lack of the self-monitoring; and limitation in strategies for solving problems. Finally, based on the results of this research, the researcher gave suggestions in three aspects. Hopefully, this research can assist teachers to have more variety in teaching methods heading towards an aim to benefit in students¡¦ learning.

Abstract The fundamental purpose of this research is to discuss the solution strategies and error types when seventh-grade students in middle school solved simple equations with one unknown and application problems; we hope results can provide reference for teachers in remedial teaching and in improving instructional. The items in the examination paper used in this research were taken from 12 past papers of basic competency test in Taiwan (2001 to 2006, twice a year). The subjects were students from Kaohsiung City Y Middle School (City region), Kaohsiung County T Middle School (Ordinary town) and Pingtung County Z Middle School (Near seacoast) forming a convenience sample. The investigators selected 2 classes from each school by random, with a total of 188 students. The main results of this research are as follows: 1. There are ten questions and the average proportion of correct answers is 49.10%; it shows that, at least 51% students have difficulties in solving symbolic operation and linear equations with one unknown. 2. The solution strategy student used most frequently is Removal of Term. Equal Axiom and Enumeration Evaluation Method were used occasionally. No student used Substitution method, Reduction method and Hidden method. 3. The main error types are: missing knowledge of using symbol; do not understand meanings of symbols; combine dissimilar terms freely; do not understand the meaning of bracket; cannot do calculation involving bracket; and, cannot simplify equation etc. 4. The main reasons for failures are: do not understand the meaning of equation; do not understand combination rules of like terms, position of symbol, unknown variables; whether containing fractions in reducing equation will also cause an error. Keywords: linear equations with one unknown, solution strategies, error types.

We study three different problems in non linear PDE. The first problem relates to finding equilibrium configurations in periodic media, more precisely, given an Area-Dirichlet functional J, which is periodic under integer translations and given three planes in R[superscript d], we proof there exists at least one minimizer such that it’s positive part, negative part and zero set remain at a uniform bounded distance of each plane. The second and third problem are related to non local diffusion, in the elliptic non symmetric case and parabolic case. In both cases we are interested in proving interior regularity for solutions of the aforementioned equations. / text