Iterative methods are considered for the numerical solution of large, sparse, nonsingular, and nonsymmetric systems of linear equations Ax=b, where it is also required that A is p-cyclic (p≥2). Firstly, it is shown that the SOR method applied to the system with A as p-cyclic, if p > 2, has a slower rate of convergence than the SOR method applied to the same system with A considered as 2-cyclic under some conditions. Therefore, the p-cyclic matrix A should be partitioned into 2-cyclic form when the SOR method is applied.

Abstract The objective of this study was to explore the feasibility of using the One-Variable Linear Equation Situational Test to investigate the development of the concept of one-variable linear equation for middle school students. The conduct of the first stage of this study was as follows: first, a thorough literature review was made; which was followed by interviews with middle and high school math teachers; finally a survey of the eighth-grade students was made using the ¡§One-Variable Linear Equation Concept Unstructured Questionnaire¡¨. The second stage of this study was to devise the One-Variable Linear Equation Situational Test. First, a detailed concept map of the one-variable linear equation was made based on the results obtained in the first stage of this study. Then, a situational test of one-variable linear equation was constructed according to the map. This test was to be used later in the one-to-one interviews with twelve seventh-grade students from a middle school in Kaohsiung, who did not learn the one-variable linear equation before. These twelve subjects were randomly devided into two groups: with guidance and without guidance. The data of the math achievement tests were also collected for these subjects. The results of the test interviews were analyzed and the feasibility of using the One-Variable Linear Equation Situational Test to investigate the development of the concept of one-variable linear equation was discussed. The analysis results of individual questions of the situational test of one-variable linear equation indicated that the concepts of one-variable linear equation for middle school students were detectable. This suggested that it was feasible to use the One-Variable Linear Equation Situational Test to investigate the development of the concept of one-variable linear equation for middle school students. All subjects who participated in this study already had some preliminary ideas of the one-variable linear equation. The factor of providing guidance would enhance the development of the concept of one-variable linear equation and it would reduce the differences in numbers of various concepts of one-variable linear equation developed among the high, medium and low achievement students. Therefore, the variable of with- or without-guidance would have some effects on the detectable concepts of one-variable linear equation by the One-Variable Linear Equation Situational Test.

The purpose of this study is to investigate students¡¦ understanding of the equal sign, problem-solving strategies of equations with one unknown, and the strategies of solving equations with one unknown under different understanding types of the equal sign. To achieve this purpose, the investigator did a survey and development instruments. The participants were 203 seventh-grade students in a convenient sample. Descriptive statistics were used to analyze data in frequency and percentages. The main results was that participants with a relational definition of the equal sign were the most (close to 50%), and an operational definition of the equal sign was approximately 1/4. There was a higher successful performance associated with a relational definition than an operational definition. The primary strategy of operations on the left-hand side of equal sign is the mathematical operations; the main strategy of an unknown quantity on the right-hand side of the equal sign was by going to the parenthesis-reverse and bringing different denominators into a common denominator; the principal strategies of one number on the right-hand side of the equal sign, equations with operations on the right side of the equal sign and equations with operations on both sides of the equal sign are cover-up and transposing. To use the strategies of trial and error substitution and undoing is minority in a linear equation with one unknown. The strategy of an operational definition participant in five equal sign topics is similar to the strategy of one with a relational definition. However, those with a relational definition apply multiple strategies and exhibited varying particular and algebraic property. On the other hand, participants with an operational definition used arithmetic strategies more frequently than participants with a relational definition. From the above results, the researcher suggested instruction to include strategies with algebraic property to help learners to develop stable understanding of the equal sign in Algebra. In addition, the recommendation is to have teachers to encourage students to apply multi-dimensional thinking and different strategies in algebraic problem-solving.

This study reanalyzed a part of the national data of the responses of 288 students in grades 6 to 8 on the ¡§One-Variable Linear Equation Virtual Situational Test¡¨ collected by Professor Pao-Kuei Wu from August 1, 2001 through July 31, 2003. The analyses were based on the ¡§One-Variable Linear Equation Conceptual Tables¡¨. The results of the analyses are the following. I. The use of variables A. Compared to 7th and 8th graders, 6th graders would first solve the numerical arithmetic and solve the unknown parts next. But if the students could not handle the unknown parts, the 6th graders tended to ignore or even not list the unknown variable in the equations. B. When encountering the unknown situations, most 6th graders are not accustomed to using symbols to represent unknown variables. Instead, they would observe the numerical components first to try to deduce what the unknown variable would be, and proceed from there. Some students would even set up some constants to represent those unknown variables. These results indicate that the 6th graders¡¦ ability to use symbolic representation is still in the beginning stages. C. In the unknown virtual situations, the majority of 7th graders were able to use symbolic representations. However, most of them would use pictorial representations such as ¡¼, instead of alphabetical representations such as x, y and z. Moreover, many students use the same symbols to represent different variables; this shows that although the 7th graders know to use symbols to represent unknown variables, they still are not able to fully comprehend unknown variables. Hence, the 7th graders¡¦ ability to use symbolic representation is in the transitional stage. D. When encountering unknown virtual situations, the majority of the 8th graders would able to use the numerical symbols such as x, y and z to represent the unknown variables. The frequency of using pictorial representations such as ¡¼ becomes less and less, and the tendency to use the same symbols to represent different variables is decreasing. All these indicate that the 8th graders¡¦ development of the concept of unknown variables is maturing. II. The concept of problem solving A. The 6th graders¡¦ ability to use symbolic representation is still in the beginning stages: 1. They only deal with the simple part; for the more complicated part, they chose to ignore. 2. Due to their immature development of symbol representation, when encountering the two variable linear equation problems, they even do not have the ability to write the ¡¥complete¡¦ equation, not to mention to solve the equations. B. The 7th graders¡¦ ability to use symbolic representation is in the transitional stage: 1. Compared to the 6th graders, the 7th graders are more able to draw relationships among the different components of the problem. 2. The fact that the substantially decreasing proportion of 7th graders conceiving the unknown variable as a certain numeric compared with 6th graders means that the 7th graders have deeper recognition of unknown variables. 3. When encountering ¡¥simple¡¦ two-variable linear equation virtual situations, some 7th graders can translate at least one condition into an equation. This result shows that the 7th graders have developed some ability to translate the conditions embedded in the virtual situation into some equations. But when the situation gets more complicated, due to conception immaturity of solving two equations simultaneously, the 7th graders either solve each equation independently, or mess up and tangle the clues of all the conditions together. Moreover, they would use the same symbol to stand for different variables. C. The 8th graders¡¦ development of the concept of unknown variables is maturing: 1. Most of the 8th graders can use the clues of all the conditions in the virtual situation in a sufficient way. 2. Only a few 8th graders would use the same symbol to stand for different variables during their problem-solving procedure. This result indicates that the ability to use the symbolic way to represent unknown variables is more mature among the 8th grade students. 3. When encountering two-variable linear equation virtual situations, the 8th graders can formulate two independent equations and solve them simultaneously. This result shows that the 8th grade students possess more profound skills to solve two-variable linear equations. III. Proportion of answering questions correctly: In general, for simpler virtual problems, there does not exist many differences among grades. Whereas, for the more difficult virtual problems, the 8th graders outperform the 7th graders, and the 7th graders, in turn, outdo the 6th grade students.

No description available.

<p> A new computer executive system for the steady state simulation of chemical processes has been developed which combines modular (GEMCS) approach with the simultaneous linear equation (SYMBØL) approach to simulation. In the combined system, a GEMCS simulation, using non-linear models, is used to generate the coefficients for the set of linear equations describing the process. This linear system of equations may also include the constraints on the process which dictate the operating conditions for the actual process. The solution of the linear equations then provide new operating conditions (feed flowrates together with the component flowrates in the recycle streams) for the modular simulation, which in turn provides new coefficients; etc. This iterative procedure is automatically continued until the system is converged to the desired point. </p> <p> A modular simulation for an actual Naphtha Reforming Plant has also been achieved and it was used as a test case to demonstrate the use and effectiveness of this new executive system. In the course of developing this simulation, the application of a method for correcting plant data was demonstrated. This is the first real application of this method to be reported in the current literature. </p> / Thesis / Master of Engineering (ME)

Title: Optimization Problems under (max, min)-Linear Constraints and Some Related Topics. Author: Mahmoud Gad Department/Institue: Department of Probability and Mathematical Statis- tics Supervisor of the doctoral thesis: 1. Prof. RNDr. Karel Zimmermann,DrSc 2. Prof. Dr. Assem Tharwat, Cairo University, Egypt Abstract: Problems on algebraic structures, in which pairs of operations such as (max, +) or (max, min) replace addition and multiplication of the classical linear algebra have appeared in the literature approximately since the sixties of the last century. The first publications on these algebraic structures ap- peared by Shimbel [37] who applied these ideas to communication networks, Cunninghame-Green [12, 13], Vorobjov [40] and Gidffer [18] applied these alge- braic structures to problems of machine-time scheduling. A systematic theory of such algebraic structures was published probable for the first time in [14]. In recently appeared book [4] the readers can find latest results concerning theory and algorithms for (max, +)-linear systems of equations and inequalities. Since operation max replacing addition in no more a group, but a semigroup oppera- tion, it is a substantial difference between solving systems with variables on one side and systems with variables occuring on both sides of the equations....

Este trabalho apresenta um estudo qualitativo das equações diferenciais nãolineares que descrevem o sincronismo de fase nos PLLs de 3ª ordem que compõem redes OWMS de topologia mista, Estrela Simples e Cadeia Simples. O objetivo é determinar, através da Teoria de Bifurcações, os valores ou relações entre os parâmetros constitutivos da rede que permitam a existência e a estabilidade do estado síncrono, quando são aplicadas, no oscilador mestre, duas funções de excitação muito comuns na prática: o degrau e a rampa de fase. Na determinação da estabilidade dos pontos de equilíbrio, sob o ponto de vista de Lyapunov, a existência de pontos de equilíbrio não-hiperbólicos não permite uma aproximação linear e, nesses casos, é aplicado o Teorema da Variedade Central. Essa técnica de simplificação de sistemas dinâmicos permite fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases e possibilitando determinar localmente suas estabilidades. / This work presents a qualitative study of the non-linear differential equations that describe the synchronous state in 3rd order PLLs that compose One-way masterslave time distribution networks with Single Star and Single Chain topologies. Using bifurcation theory, the dynamical behavior of third-order phase-locked loops employed to extract the syncronous state in each node is analyzed depending on constitutive node parameters when two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node. When parameter combinations result in non hyperbolic synchronous states, from Lyapunov point of view, the linear approximation does not provide any information about the local behavior of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behavior of the original system in the neighborhood of these points. Thus, the local stability can be determined.

Submitted by Franciéli Pereira Fernandes null (fran_pefer_@hotmail.com) on 2016-03-01T18:54:35Z No. of bitstreams: 1 DissertacaoFrancieliPereiraFernandes_versao-final.pdf: 2523902 bytes, checksum: 0bec87e8e15f5fe4320867983b94456c (MD5) / Approved for entry into archive by Sandra Manzano de Almeida (smanzano@marilia.unesp.br) on 2016-03-02T13:14:53Z (GMT) No. of bitstreams: 1 fernandes_fp_me_sjrp.pdf: 2523902 bytes, checksum: 0bec87e8e15f5fe4320867983b94456c (MD5) / Made available in DSpace on 2016-03-02T13:14:53Z (GMT). No. of bitstreams: 1 fernandes_fp_me_sjrp.pdf: 2523902 bytes, checksum: 0bec87e8e15f5fe4320867983b94456c (MD5) Previous issue date: 2016-02-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / No presente trabalho é apresentado um estudo analítico sobre retas no plano, tendo em vista a sua representatividade no significado da Geometria Analítica como um método de abordagem de problemas geométricos. Durante o desenvolvimento desse estudo, é visto que, a partir da escolha de um sistema de coordenadas de um plano, as retas do mesmo podem ser representadas por equações lineares com duas incógnitas e coeficientes reais, e vice-versa. Também são abordados os tópicos: equação reduzida, inclinação, coeficientes angular e linear, e as posições relativas de duas retas. Nas orientações dadas pelo Currículo de Matemática da Secretaria da Educação do Estado de São Paulo (SEE), um estudo das retas, com suas equações, propriedades e aplicações deve ser introduzido no ensino básico, mais precisamente, no início da 3ª Série do Ensino Médio. Baseados nessas orientações e objetivando sempre tornar o processo de ensino-aprendizagem de matemática mais interessante e prazeroso, são propostas e desenvolvidas atividades em sala de informática, para se explorar retas no plano com o auxílio das ferramentas do software GeoGebra. / In the present work, it is presented an analytical study on lines in the plane, bearing in mind its representativeness in the Analytic Geometry meaning as a method of geometrical problems approach. During the development of this study it is seen that, from the selection of a coordinate system of a plane, the lines of the same plane can be represented by linear equations with two unknowns and real coefficients, and vice versa. The following topics are also discussed: reduced equation, inclination, angular and linear coefficients, and the relative positions of two lines. At the guidelines given by the Mathematics Curriculum of São Paulo State Education (SEE), a study of the lines, with their equations, properties and applications should be introduced in basic education, more precisely, at the beginning of the 3rd grade of high school. Based on the guidelines and always aiming to become the mathematic's teaching-learning process more interesting and enjoyable, activities in computer room are proposed and developed, to explore lines in the plane with the support of the GeoGebra software tools.

Este trabalho apresenta um estudo qualitativo das equações diferenciais nãolineares que descrevem o sincronismo de fase nos PLLs de 3ª ordem que compõem redes OWMS de topologia mista, Estrela Simples e Cadeia Simples. O objetivo é determinar, através da Teoria de Bifurcações, os valores ou relações entre os parâmetros constitutivos da rede que permitam a existência e a estabilidade do estado síncrono, quando são aplicadas, no oscilador mestre, duas funções de excitação muito comuns na prática: o degrau e a rampa de fase. Na determinação da estabilidade dos pontos de equilíbrio, sob o ponto de vista de Lyapunov, a existência de pontos de equilíbrio não-hiperbólicos não permite uma aproximação linear e, nesses casos, é aplicado o Teorema da Variedade Central. Essa técnica de simplificação de sistemas dinâmicos permite fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases e possibilitando determinar localmente suas estabilidades. / This work presents a qualitative study of the non-linear differential equations that describe the synchronous state in 3rd order PLLs that compose One-way masterslave time distribution networks with Single Star and Single Chain topologies. Using bifurcation theory, the dynamical behavior of third-order phase-locked loops employed to extract the syncronous state in each node is analyzed depending on constitutive node parameters when two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node. When parameter combinations result in non hyperbolic synchronous states, from Lyapunov point of view, the linear approximation does not provide any information about the local behavior of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behavior of the original system in the neighborhood of these points. Thus, the local stability can be determined.