Relevant bibliographies by topics / Second degree equation

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Second degree equation'

Author: Grafiati
Published: 4 June 2021
Last updated: 13 February 2022

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Journal articles on the topic "Second degree equation"

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Kesavan, S. "On the general equation of the second degree." Resonance 20, no. 7 (July 2015): 643–62. http://dx.doi.org/10.1007/s12045-015-0222-3.

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Oskolkov, K. "Schrödinger equation and oscillatory Hilbert transforms of second degree." Journal of Fourier Analysis and Applications 4, no. 3 (May 1998): 341–56. http://dx.doi.org/10.1007/bf02476032.

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Berezkina, N. S., I. P. Martynov, and V. A. Pron’ko. "Classes of Painlevé type second-order differential equation of the second degree." Differential Equations 36, no. 1 (January 2000): 39–46. http://dx.doi.org/10.1007/bf02754161.

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hadj ali, B. Ben, and M. Mejri. "Algebraic equation and symmetric second degree forms of class two." Integral Transforms and Special Functions 28, no. 9 (July 5, 2017): 682–701. http://dx.doi.org/10.1080/10652469.2017.1345903.

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Roitenberg, V. Sh. "ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 12, no. 4 (2020): 33–40. http://dx.doi.org/10.14529/mmph200404.

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In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.
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Guan‐quan, Zhang, Zhang Shu‐lun, Wang Ying‐xiang, and Liu Chau‐ying. "A new algorithm for finite‐difference migration of steep dips." GEOPHYSICS 53, no. 2 (February 1988): 167–75. http://dx.doi.org/10.1190/1.1442451.

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One‐way wave propagation is formulated in the (x, t) domain as an integro‐differential equation and is applied to the migration of stacked data. A key step is to replace the phase‐shift square root in the frequency‐domain representation by an integral of a rational function; the resulting expression is interpreted in the space‐time domain. Approximating the integral by a finite sum leads to a number of practical approximations, the lowest order being Claerbout’s 15 degree equation and others being various high‐order equations in the literature. Optimal mth‐order quadrature formulas based upon Chebychev criteria suggest a second‐order approximation which takes 20 percent more time than the 15 degree equation but is accurate to over 50 degrees.
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Zhao, Mi, Huifang Li, Shengtao Cao, and Xiuli Du. "An explicit time integration algorithm for linear and non-linear finite element analyses of dynamic and wave problems." Engineering Computations 36, no. 1 (December 19, 2018): 161–77. http://dx.doi.org/10.1108/ec-07-2018-0312.

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Purpose The purpose of this paper is to propose a new explicit time integration algorithm for solution to the linear and non-linear finite element equations of structural dynamic and wave propagation problems. Design/methodology/approach The algorithm is completely explicit so that no linear equation system requires solving, if the mass matrix of the finite element equation is diagonal and whether the damping matrix does or not. The algorithm is a single-step method that has the simple starting and is applicable to the analysis with the variable time step size. The algorithm is second-order accurate and conditionally stable. Its numerical stability, dissipation and dispersion are analyzed for the dynamic single-degree-of-freedom equation. The stability of the multi-degrees-of-freedom non-proportional damping system can be evaluated directly by the stability theory on ordinary differential equation. Findings The performance of the proposed algorithm is demonstrated by several numerical examples including the linear single-degree-of-freedom problem, non-linear two-degree-of-freedom problem, wave propagation problem in two-dimensional layer and seismic elastoplastic analysis of high-rise structure. Originality/value A new single-step second-order accurate explicit time integration algorithm is proposed to solve the linear and non-linear dynamic finite element equations. The algorithm has advantages on the numerical stability and accuracy over the existing modified central difference method and Chung-Lee method though the theory and numerical analyses.
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Kebede, Tesfaye. "Second Degree Refinement Jacobi Iteration Method for Solving System of Linear Equation." International Journal of Computing Science and Applied Mathematics 3, no. 1 (March 1, 2017): 5. http://dx.doi.org/10.12962/j24775401.v3i1.2114.

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Halburd, R. G. "Elementary exact calculations of degree growth and entropy for discrete equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2201 (May 2017): 20160831. http://dx.doi.org/10.1098/rspa.2016.0831.

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Second-order discrete equations are studied over the field of rational functions C ( z ) , where z is a variable not appearing in the equation. The exact degree of each iterate as a function of z can be calculated easily using the standard calculations that arise in singularity confinement analysis, even when the singularities are not confined. This produces elementary yet rigorous entropy calculations.
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He, Tieshan, Yimin Lu, Youfa Lei, and Fengjian Yang. "Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations." Discrete Dynamics in Nature and Society 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/153082.

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This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories.
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Dissertations / Theses on the topic "Second degree equation"

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Silva, Fabiano Luiz da. "As diferentes estratÃgias de resoluÃÃo das equaÃÃes algÃbricas atà o terceiro grau." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15501.

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O objetivo desse trabalho à apresentar explanaÃÃes e estratÃgias de resoluÃÃo das equaÃÃes algÃbricas do primeiro, segundo e terceiro graus, uma vez que o ensino relativo à resoluÃÃes dessas equaÃÃes tem se restringido praticamente a apresentaÃÃo da fÃrmula resolutiva e as relaÃÃes entre seus coeficientes e suas raÃzes. Desta maneira procuramos demonstrar e atà mesmo justificar todas as formas apresentadas para se resolver equaÃÃes atà o terceiro grau atravÃs de mÃtodos puramente algÃbricos ou geomÃtricos, como tambÃm, exemplificar todos os mÃtodos que foram exibidos no intuito de satisfazer as expectativas dos leitores, por isso, o texto foi produzido em uma linguagem simples, acessÃvel à professores e alunos. Nesse contexto, espera-se que essa proposta de trabalho estimule os professores de MatemÃtica do Ensino BÃsico a realizarem essa abordagem diferenciada das equaÃÃes algÃbricas em questÃo, pois acredita-se que com essa abordagem ocorram reflexos positivos no processo de ensino e aprendizagem das equaÃÃes e da MatemÃtica.
The aim of this paper is to present explanations and solving strategies of algebraic equations of the first, second and third degrees, since the relative teaching on the resolutions of these equations has been restricted practically the presentation of solving formula and the relationships between its coefficients and its roots. In this way we try to demonstrate and even justify all forms presented to solve equations to the third degree by purely algebraic or geometric methods, but also exemplify all methods that were displayed in order to meet the expectations of readers, so the text was produced in simple language, accessible to teachers and students. In this context, it is expected that this work proposal stimulate the mathematics teachers of Basic Education to perform this differentiated approach to algebraic equations in question, since it is believed that with this approach occur positive reflexes in the teaching and learning of equations and of Mathematics.
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Kuroiwa, Elisabete Tiyoko Nishimura [UNESP]. "Uma abordagem peculiar da equação do segundo grau no ensino fundamental e médio." Universidade Estadual Paulista (UNESP), 2016. http://hdl.handle.net/11449/145534.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Há cerca de 2000 a.C, com o desenvolvimento do conhecimento matemático, a equação do 2º grau tem mostrado sua aplicabilidade principalmente em problemas de medidas, áreas e repartição de herança, pois, o Alcorão prescrevia uma divisão de herança de acordo com a idade. Além das circunstancias descritas acima, há uma infinidade de situações problema que envolve a resolução de uma equação deste tipo. Ao longo de minha jornada como docente, observei as dificuldades encontradas pelos discentes, que não conseguem aplicá-las e resolvê-las. Procuramos uma abordagem diferenciada para priorizar a aprendizagem, sanar suas dificuldades e efetivar a resolução. Assim, o objetivo deste trabalho consiste no auxílio aos professores na importante tarefa de ensinar, esclarecer, fundamentar e sedimentar os conceitos e as resoluções da equação do 2º grau, promovendo aprendizagem efetiva, com maior interação e cumplicidade ajudando-os em seus questionamentos, providenciando a retomada e a aquisição de saberes com suas intervenções propícias e auxiliando-os a encontrar as respostas às suas indagações. Para isto utilizamos a Metodologia de Resolução de Problemas, como ferramenta viável para a demonstração da fórmula de Bháskara, proporcionando uma motivação em sua construção, possibilitando ao aluno, maior envolvimento, participação e interação com manejo de material concreto. Essa metodologia proporciona maior interação entre o professor e os alunos auxiliando-os e estimulando-os com questionamento e intervenções convenientes. Iniciamos com uma abordagem histórica em ordem cronológica, de acordo com os conhecimentos desenvolvidos, bem como sua contextualização teórica, envolvendo inclusive os diversos métodos de resolução, através dos tempos, bem como seus exemplos. Apresentamos uma aplicação em sala de aula, com levantamento diagnóstico para verificar os conhecimentos dos estudantes, aplicabilidade e resoluções desta equação, sendo posteriormente feita uma reavaliação para verificar o nível de aprendizado, mostrando de maneira geral o método e a metodologia proposta.
There are about 2000 B.C with the development of mathematical knowledge, the 2nd degree equation has shown its applicability especially in problems of measures, areas and division of inheritance, as the Qur'an prescribed a division of inheritance according to age. In addition to the circumstances described above, there are plenty of situations that involve problem solving an equation of this kind. Along my journey as a teacher, I noted the difficulties encountered by students who can not apply them and solve them. We seek a differentiated approach to prioritize learning, solve their problems and carry out the resolution. The objective of this work is to assist teachers in the important task of teaching, clarify, justify and consolidate the concepts and resolutions of the 2nd degree equation, promoting effective learning, with greater interaction and complicity helping them in their inquiries, arranging the recovery and the acquisition of knowledge with its favorable interventions and helping them find answers to their questions. For this we use the Troubleshooting Methodology as a viable tool for demonstrating the formula Bháskara, providing a motivation in their construction, allowing the student to greater involvement, participation and interaction with management of concrete material. This methodology provides greater interaction between teacher and students helping them and encouraging them to questioning and appropriate interventions. We begin with a historical approach in chronological order, according to the developed knowledge and its theoretical context, including the various methods of resolution, through the ages, as well as their examples. We present an application in the classroom, with diagnosis survey to check students' knowledge, applicability and resolutions of this equation, a reassessment later being done to check the level of learning, showing generally the method and the proposed methodology.
3107510001F5
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Kuroiwa, Elisabete Tiyoko Nishimura. "Uma abordagem peculiar da equação do segundo grau no ensino fundamental e médio /." São José do Rio Preto, 2016. http://hdl.handle.net/11449/145534.

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Orientador: José Carlos Rodrigues
Resumo: Há cerca de 2000 a.C, com o desenvolvimento do conhecimento matemático, a equação do 2º grau tem mostrado sua aplicabilidade principalmente em problemas de medidas, áreas e repartição de herança, pois, o Alcorão prescrevia uma divisão de herança de acordo com a idade. Além das circunstancias descritas acima, há uma infinidade de situações problema que envolve a resolução de uma equação deste tipo. Ao longo de minha jornada como docente, observei as dificuldades encontradas pelos discentes, que não conseguem aplicá-las e resolvê-las. Procuramos uma abordagem diferenciada para priorizar a aprendizagem, sanar suas dificuldades e efetivar a resolução. Assim, o objetivo deste trabalho consiste no auxílio aos professores na importante tarefa de ensinar, esclarecer, fundamentar e sedimentar os conceitos e as resoluções da equação do 2º grau, promovendo aprendizagem efetiva, com maior interação e cumplicidade ajudando-os em seus questionamentos, providenciando a retomada e a aquisição de saberes com suas intervenções propícias e auxiliando-os a encontrar as respostas às suas indagações. Para isto utilizamos a Metodologia de Resolução de Problemas, como ferramenta viável para a demonstração da fórmula de Bháskara, proporcionando uma motivação em sua construção, possibilitando ao aluno, maior envolvimento, participação e interação com manejo de material concreto. Essa metodologia proporciona maior interação entre o professor e os alunos auxiliando-os e estimulando-os com questionamento... (Resumo completo, clicar acesso eletrônico abaixo)
Mestre
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Fernandes, Marcos Vinicius Ferreira. "Métodos históricos utilizados para a resolução de uma equação do segundo grau." Universidade Federal de São Carlos, 2014. https://repositorio.ufscar.br/handle/ufscar/4468.

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In this work we attempt to answer the question: Can the methods that great mathematicians have used for almost 4 thousand years of history be helpful in the learning process? With a few years of experience in 2nd degree equations, we realize that as soon as the students learn the popular and well-known Bháskara formula (continue...).
Neste trabalho procuramos responder ao questionamento: Os métodos que grandes Matemáticos usaram no decorrer de quase 4 mil anos de história podem auxiliar no aprendizado? Com alguns anos de experiência no ensino de equações do 2º grau, percebemos que assim que os alunos aprendem a popularmente conhecida fórmula de Bháskara, , qualquer outro esforço no sentido de justificar ou discutir uma equação da forma , é feito em vão (continua...).
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Oliveira, Josà Adriano dos Santos. "Sobre seÃÃes cÃnicas." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=14712.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
O estudo realizado nesta dissertaÃÃo, busca apresentar as seccÃes cÃnicas, dando Ãnfase a uma abordagem por meio de uma geometria sintÃtica e elementar, onde o trabalho à desenvolvido da seguinte forma: inicia-se com uma abordagem histÃrica, assim como a sua relaÃÃo com o cone circular; em seguida, à feito um estudo sintÃtico sobre as cÃnicas, exclusivamente, no plano; apresenta-se algumas superfÃcies quÃdricas; a equaÃÃo geral do segundo grau à apresentada como uma representaÃÃo algÃbrica de uma cÃnica e sÃo mostradas diversas situaÃÃes, onde as cÃnicas surgem de forma, curiosamente, natural, alÃm das inÃmeras aplicaÃÃes prÃticas em diversas Ãreas do conhecimento.
The study in this dissertation, seeks to present the conic sections, emphasizing an approach by means of a synthetic and elementary geometry, where the work is carried out as follows: begins with a historical approach, as well as their relationship with the circular cone; then itâs done a synthetic study on the conical exclusively on the plan; It presents some quadric surfaces; the general equation of the second degree is presented as an algebraic representation of a conic and are shown several situations where the conical arise so, curiously, natural, in addition to numerous practical applications in various fields of knowledge.
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Santos, Leonardo Silva. "Uma abordagem histórica e metodológica dos métodos de resolução de equação do 2º grau desenvolvidos por Al-Khwarizmi." Universidade Estadual da Paraíba, 2017. http://tede.bc.uepb.edu.br/jspui/handle/tede/2978.

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Faced with all problems related to the teaching and learning of Algebra, this work proposes the elaboration and application of a methodological didactic alternative that seeks to give meaning to the teaching and learning process of the methods of resolution of equations of the second degree. Thus, the present work presents the results of the application in a class of 9th year of elementary school of a sequence of activities mediated by the History of Mathematics, which approaches the methods of solving the equation of the second degree developed by Al - Khwarizmi . In order to do so, we sought to investigate the effects of the applied activities, so that the students perceived the construction of the methods while interacting wit h the activities, making the study of this topic more significant. As a methodological approach was used the qualitative research, being characterized as an action type research, with interventions in the classroom. The instruments used for data collection were: questionnaire, sequence of activities in light of the History of Mathematics, observation, logbook, final evaluation and interview. The analysis of the results confirm the objectives of the study, indicating that a methodology based on the development of teaching activities supported in constructivism and mediated by the History of Mathematics provides meaningful learning for the student, since most of the students performed and obtained good results in the applied activities.
Diante de toda problemática relativa ao ensino e aprendizagem da Álgebra, este trabalho propõe a elaboração e aplicação de uma alternativa didática metodológica que busca dar significado ao processo de ensino e aprendizagem dos métodos de resolução de equações do 2º grau. Sendo assim, o presente trabalho, apresenta os resultados da aplicação em uma turma de 9º ano do ensino fundamental, de uma sequência de atividades mediadas pela História da Matemática, a qual aborda os métodos de resolução da equação do 2º grau desenvolvidos por Al-Khwarizmi. Para tanto, buscou-se investigar, os efeitos das atividades aplicadas, de modo que os estudantes percebessem a construção dos métodos ao passo que interagiam com as atividades, tornando o estudo deste tópico mais significativo. Como abordagem metodológica foi utilizada a pesquisa qualitativa, sendo caracterizada como uma pesquisa do tipo ação, com intervenções na sala de aula. Os instrumentos utilizados para a coleta de dados foram: questionário, sequência de atividades à luz da História da Matemática, observação, diário de bordo, avaliação final e entrevista. A análise dos resultados confirmam os objetivos do estudo, indicando que uma metodologia baseada no desenvolvimento de atividades de ensino apoiadas no construtivismo e mediada pela História da Matemática proporciona aprendizado significativo para o aluno, pois a maioria dos alunos realizou e obteve bons resultados nas atividades aplicadas.
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BAPOUNGUE-LISSOUCK, LIONEL. "Sur la resolubilite de certaines equations diophantiennes du second degre." Caen, 1989. http://www.theses.fr/1989CAEN2005.

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Ce travail a son origine dans un article de kenneth hardy et kenneth s. Williams qui etudie la resolution complete de certaines equations diophantiennes du second degre. Nous considerons des familles plus generales d'equations de meme type. Nous caracterisons la resolubilite de ces equations en termes de la solution minimale d'une equation de pell associee. Nous discutons egalement la description de toutes les solutions et la maniere de les obtenir
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Noubiagain, Chomchie Fanny Larissa. "Contributions to second order reflected backward stochastic differentials equations." Thesis, Le Mans, 2017. http://www.theses.fr/2017LEMA1016/document.

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Cette thèse traite des équations différentielles stochastiques rétrogrades réfléchies du second ordre dans une filtration générale . Nous avons traité tout d'abord la réflexion à une barrière inférieure puis nous avons étendu le résultat dans le cas d'une barrière supérieure. Notre contribution consiste à démontrer l'existence et l'unicité de la solution de ces équations dans le cadre d'une filtration générale sous des hypothèses faibles. Nous remplaçons la régularité uniforme par la régularité de type Borel. Le principe de programmation dynamique pour le problème de contrôle stochastique robuste est donc démontré sous les hypothèses faibles c'est à dire sans régularité sur le générateur, la condition terminal et la barrière. Dans le cadre des Équations Différentielles Stochastiques Rétrogrades (EDSRs ) standard, les problèmes de réflexions à barrières inférieures et supérieures sont symétriques. Par contre dans le cadre des EDSRs de second ordre, cette symétrie n'est plus valable à cause des la non linéarité de l'espérance sous laquelle est définie notre problème de contrôle stochastique robuste non dominé. Ensuite nous un schéma d'approximation numérique d'une classe d'EDSR de second ordre réfléchies. En particulier nous montrons la convergence de schéma et nous testons numériquement les résultats obtenus
This thesis deals with the second-order reflected backward stochastic differential equations (2RBSDEs) in general filtration. In the first part , we consider the reflection with a lower obstacle and then extended the result in the case of an upper obstacle . Our main contribution consists in demonstrating the existence and the uniqueness of the solution of these equations defined in the general filtration under weak assumptions. We replace the uniform regularity by the Borel regularity(through analytic measurability). The dynamic programming principle for the robust stochastic control problem is thus demonstrated under weak assumptions, that is to say without regularity on the generator, the terminal condition and the obstacle. In the standard Backward Stochastic Differential Equations (BSDEs) framework, there is a symmetry between lower and upper obstacles reflection problem. On the contrary, in the context of second order BSDEs, this symmetry is no longer satisfy because of the nonlinearity of the expectation under which our robust stochastic non-dominated stochastic control problem is defined. In the second part , we get a numerical approximation scheme of a class of second-order reflected BSDEs. In particular we show the convergence of our scheme and we test numerically the results
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Olteanu, Constanta. "”Vad skulle x kunna vara?” : andragradsekvation och andragradsfunktion som objekt för lärande." Doctoral thesis, Umeå University, Mathematics, Technology and Science Education, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-1363.

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Algebraic equations and functions play an important role in various mathematical topics, including algebra, trigonometry, linear programming and calculus. Accordingly, various documents, such as the most recent Swedish curriculum (Lpf 94) for upper secondary school and the course syllabi in mathematics, specify what the students should learn in Mathematics Course B. They should be able to solve quadratic equations and apply this knowledge in solving problems, explain the properties of a function, as well as be able to set up, interpret and use some nonlinear functions as models for real processes. To implement these recommendations, it is crucial to understand the students’ way of experiencing quadratic equations and functions, and describe the meaning these have for the students in relation to the possibility they have to their experience of them.

The aim of this thesis is to analyse, understand and explain the relation between the handled and learned content, which consists of second-degree equations and quadratic functions, in classroom practice. This means that content is the research object and not the teacher’s conceptions or knowledge of, or about this content. This restriction implies that the handled and learned contents are central in this study and will be analysed from different perspectives.

The study includes two teachers and 45 students in two different classes. The data consist of video-recordings of lessons, individual sessions, interviews and the teachers’/researcher’s review of the individual sessions. The students’ tests also constituted an important part of the data collection.

When analysing the data, concepts relating to variation theory have been used as analytical tools. Data have been analysed in respect of the teachers’ focus on the lesson content, which aspects are ignored and which patterns of dimensions of variations are constituted when the contents are handled by the teachers in the classroom. Also, data have been analysed in respect of the students’ focus when they solve different exercises in a test situation. It can be shown that the meaning of parameters, the unknown quantity in an equation and the function’s argument change several times when the teacher presents the content in the classroom and when the students solve different exercises. It can also be shown that the teachers and the students develop complicated patterns of variation during the lessons and that the ways in which the teachers open up dimensions of variation play an important role in the learning process. The results indicate that there is a convergent variation leading the students to improve their learning. By focusing on some aspects of the objects of learning and create convergent variations, it is possible for the students to understand the difference between various interpretations of these aspects and thereafter focus on the interpretation that fits in a certain context. Furthermore, this variation leads the students to make generalisations in each object of learning (equations and functions) and between these objects of learning. These generalisations remain over time, despite working with new objects of learning. An important result in this study is that the implicit or explicit arguments of a function can make it possible to discern an equation from a function despite the fact that they are constituted by the same algebraic expression.

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Silva, M?rcio Vieira da. "Equa??es de segundo grau e mudan?a de vari?veis." Universidade Federal do Rio Grande do Norte, 2014. http://repositorio.ufrn.br:8080/jspui/handle/123456789/18669.

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In this work are presented, as a review and in a historical context, the most used methods to solve quadratic equations. It is also shown the simplest type of change of variables, namely: x = Ay + B where A;B 2 R, and some changes of variables that were used to solve quadratic equations throughout history. Finally, a change of variable, which has been used by the author in the classroom as an alternative method, is presented and the result of this methodoly is illustrated by the responses of a test that was done by the students in classroom
Neste trabalho s?o apresentados, como revis?o e num contexto hist ?rico, os m?todos mais utilizados de resolver equa??es de 2? grau. E apresentado tamb ?m o tipo maissimples de mudan ?a de vari ?veis, a saber: x = Ay + B onde A;B 2 R, e mostrado como algumas mudan ?as de vari ?veis foram utilizadas na resolu ??o de equa ??ess do segundo grau ao longo da hist ?ria. Finalmente, uma mudan ?a de vari ?vel, que tem sido utilizada pelo autor em sala de aula como um m et?do alternativo, e apresentada e o resultado da aplica ??o de tal m ?todo e ilustrado atr?v es das respostas de um teste
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Books on the topic "Second degree equation"

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Sherwood, Dennis, and Paul Dalby. Order, information and time. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198782957.003.0011.

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This chapter broadens the reader’s appreciation, and understanding, of entropy. Starting with the reader’s intrinsic recognition of the difference between an ‘ordered’ and a ‘disordered’ state, this chapter introduces the concepts of microstates, macrostates and thermodynamic probability, leading firstly to the Boltzmann equation, and then to the relationships between entropy and the flow of time, entropy and information, and Maxwell’s demon. Finally, this chapter introduces the less familiar topics of ‘thermoeconomics’ - the application of the principles of thermodynamics to economic systems – and ‘organodynamics’ – the idea that high-performing teams are systems which maintain a high degree of order, and low entropy, over time, without breaking the Second Law.
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Book chapters on the topic "Second degree equation"

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Wille, Katrin. "Equations of the Second Degree." In George Spencer Brown, 172–91. Wiesbaden: VS Verlag für Sozialwissenschaften, 2004. http://dx.doi.org/10.1007/978-3-322-95679-8_15.

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Wille, Katrin. "Equations of the second degree." In George Spencer Brown, 174–93. Wiesbaden: VS Verlag für Sozialwissenschaften, 2009. http://dx.doi.org/10.1007/978-3-531-91964-5_15.

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Gauss, Carl Friedrich. "Forms and Indeterminate Equations of the Second Degree." In Disquisitiones Arithmeticae, 108–374. New York, NY: Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4939-7560-0_5.

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Magnaghi-Delfino, Paola, and Tullia Norando. "How to Solve Second Degree Algebraic Equations Using Geometry." In Lecture Notes in Networks and Systems, 121–30. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-29796-1_11.

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Egorov, Yu V., and M. A. Shubin. "The Function P+λ for Polynomials of Second-degree and its Application in the Construction of Fundamental Solutions." In Partial Differential Equations II, 185–212. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-642-57876-2_12.

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Takagi, Teiji. "On the theory of indeterminate equations of the second degree in two variables." In Springer Collected Works in Mathematics, 256–61. Tokyo: Springer Japan, 1990. http://dx.doi.org/10.1007/978-4-431-54995-6_24.

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Harris, Michael. "How to Explain Number Theory at a Dinner Party." In Mathematics without Apologies. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691175836.003.0006.

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This chapter continues the discussion began in Chapter α‎. It presents the second part of author's response to the question, “What is it you do in number theory, anyway?” Working his way up to the guiding problem of the Birch–Swinnerton–Dyer conjecture, here he deals with equations of degree 3 (or 4) in one variable or degree 2 in two variables. He says that if we are willing to allow square roots into our arithmetic, we can consider the quadratic equation a problem whose solution has been long understood (in some cases by the ancient Babylonians). Equations of degree 3 and 4, such as x3 − 2x2 + 14x + 9 and x4 + 5x3 + 11x2 + 17x − 29, were first solved in Renaissance Italy to great acclaim; the solutions are given by formulas involving cube roots and fourth roots.
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K. Makarios, Triantafyllos. "Identification of Eigen-Frequencies and Mode-Shapes of Beams with Continuous Distribution of Mass and Elasticity and for Various Conditions at Supports." In Number Theory and Its Applications. IntechOpen, 2020. http://dx.doi.org/10.5772/intechopen.92185.

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In the present article, an equivalent three degrees of freedom (DoF) system of two different cases of inverted pendulums is presented for each separated case. The first case of inverted pendulum refers to an amphi-hinge pendulum that possesses distributed mass and stiffness along its height, while the second case of inverted pendulum refers to an inverted pendulum with distributed mass and stiffness along its height. These vertical pendulums have infinity number of degree of freedoms. Based on the free vibration of the above-mentioned pendulums according to partial differential equation, a mathematically equivalent three-degree of freedom system is given for each case, where its equivalent mass matrix is analytically formulated with reference on specific mass locations along the pendulum height. Using the three DoF model, the first three fundamental frequencies of the real pendulum can be identified with very good accuracy. Furthermore, taking account the 3 × 3 mass matrix, it is possible to estimate the possible pendulum damages using a known technique of identification mode-shapes via records of response accelerations. Moreover, the way of instrumentation with a local network by three accelerometers is given via the above-mentioned three degrees of freedom.
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Bandle, Catherine, and Wolfgang Reichel. "Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory." In Handbook of Differential Equations: Stationary Partial Differential Equations, 1–70. Elsevier, 2004. http://dx.doi.org/10.1016/s1874-5733(04)80003-2.

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Nitzan, Abraham. "Introduction To Quantum Relaxation Processes." In Chemical Dynamics in Condensed Phases. Oxford University Press, 2006. http://dx.doi.org/10.1093/oso/9780198529798.003.0015.

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The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.
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Conference papers on the topic "Second degree equation"

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Riley, Peregrine E. J. "Alternative Inverse Kinematic Equations for General RRP and RRR Regional Structures of Manipulators." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/mech-1009.

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Abstract Many manipulators with six degrees of freedom are constructed with two distinct sections, a regional structure for spatial positioning, and an orientational structure having a common intersection point for the joint axes. With this arrangement, inverse kinematic solutions for position and orientation may be found separately. While solutions for general three link manipulators have been available since the work of Pieper in 1969, this paper presents new forms of the inverse kinematic equations for general RRP and RRR regional structures. Cartesian coordinates of the F-surface (generated by movement of the outer two joints) together with the outer joint angle are used as the equation variables. In addition, a second degree polynomial approxiamation of the equation may be used for quick iteration to a solution. It is hoped that these new equations will be useful by themselves and in workspace regions where solutions using equations in terms of the joint variables are numerically inaccurate or impossible.
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Khanuja, Sukhwant S., Andi I. Mahyuddin, and Ashok Midha. "Analytical and Experimental Investigations of a Multi-Degree-of-Freedom Flexible Cam-Follower Mechanisma." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/mech-1182.

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Abstract A multi-degree-of-freedom model is developed herein for prediction of response of an experimental cam-follower mechanism. Both transverse and axial flexibility of the follower rod and return spring, as well as transverse and torsional flexibility of the camshaft are included. The camshaft is assigned two rotational degrees of freedom, one at the cam and the other at the flywheel. The follower mass motion is also described by two degrees of freedom, one each in the axial and the transverse directions. The model takes into account the fluctuating camshaft angular speed and treats it as an input excitation. The governing second-order, nonlinear, nondimensionalized ordinary differential equations of motion, with time-periodic coefficients, are developed. In doing so, a comprehensive modeling of the kinematics of deformation of the flexible camshaft and follower system is considered, with the first inclusion of the transverse flexibility of the follower rod and return spring. With axial deflection, the transverse flexibility of the return spring gives rise to a phenomenon defined as moment stiffening. The transverse degree of freedom of the follower mass significantly influences the equivalent axial stiffness of the system. Its inclusion in the equation of motion yields a more accurate prediction of the experimental system behavior.
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Huang, Qinghua, and Wei-Chau Xie. "Stability of SDOF Linear Viscoelastic System Under the Excitation of Narrow-Band Noise." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34366.

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The moment Lyapunov exponents of a single degree-of-freedom (SDOF) viscoelastic system under the excitation of a narrow-band noise, which is described as a bounded noise, is studied in this paper. An example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The equation of motion is an integro-differential equation with parametric excitation. The method of stochastic averaging for integro-differential equations, both first-order and second-order, is applied and the eigenvalue problems governing the moment Lyapunov exponents are established. Numerical results from Monte Carlo simulation are compared with the approximate analytical results, and the variations of the moment Lyapunov exponents with the change of different parameters are discussed.
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Li, Like, Chen Chen, Renwei Mei, and James F. Klausner. "Conjugate Interface Heat and Mass Transfer Simulation With the Lattice Boltzmann Equation Method." In ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels collocated with the ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/icnmm2014-21864.

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An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation (LBE) method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including: (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, and (iii) steady heat conduction inside a circular domain with two different solid materials. The accuracy and order-of-convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration) and heat (mass) flux are examined in detail and compared with those obtained from the “half lattice division” treatment in the literature. The present analysis and numerical results show that the half lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved interfaces, the present treatment yields second-order accurate interior and interfacial temperatures (concentrations) and first-order accurate interfacial heat (mass) flux. An increase of order-of-convergence by one degree is obtained for each of these three quantities compared with the half lattice division scheme.
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Venkataraman, P. "Solving Inverse ODE Using Bezier Functions." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86331.

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The simplest inverse boundary value problem is to identify the differential equation and the boundary conditions from a given set of discrete data points. For an ordinary differential equation, it would involve finding a function, which when expressed through some function of itself and its derivatives, and integrated using particular boundary conditions would generate the given data. Parametric Bezier functions are excellent candidates for these functions. They allow efficient approximation of data and its derivative content. The Bezier function is smooth and continuous to a high degree. In this paper the best Bezier function to fit the data represents this function which is being sought. This Bezier approximation also determines the boundary conditions. Next, a generic form of the differential equation is assumed. The Bezier function and its derivatives are then used in this generic form to establish the exponents and coefficients of the various terms in the actual differential equation. The paper looks at homogeneous ordinary differential equations and shows it can recover the exact form of both linear and nonlinear differential equations. Two examples are presented. The first example uses data from the Bessel equation, representing a linear equation. The second example uses the data from the Blassius equation which is nonlinear. In both cases the exact form of the equation is identified from the given discrete data.
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Liu, Jian, David T. Martin, Karthik Kadirvel, Toshikazu Nishida, Louis N. Cattafesta, Mark Sheplak, and Brian P. Mann. "Nonlinear System Identification of a MEMS Dual-Backplate Capacitance Microphone by Harmonic Balance Method." In ASME 2005 International Mechanical Engineering Congress and Exposition. ASMEDC, 2005. http://dx.doi.org/10.1115/imece2005-82880.

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This paper presents the nonlinear identification of system parameters for a capacitive dual-backplate MEMS microphone. First, the microphone is modeled by a single-degree-of-freedom (SDOF) second order differential equation with electrostatic and cubic mechanical nonlinearities. A harmonic balance nonlinear identification approach is then applied to the governing equation to obtain a set of algebraic equations that relate the unknown system parameters to the steady-state response of the microphone under the harmonic excitation. The microphone is experimentally characterized and a nonlinear least-squares technique is implemented to identify the system parameters from experimental data. The experimentally extracted bandwidth of the microphone is over 218 kHz. Finally, numerical simulations of the governing equation are performed, using the identified system parameters, to validate the accuracy of the approximate solution. The differences between the properties of the integrated measured center velocity and simulated center displacement responses in the steady state are less than 1%.
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Duan, Zhipeng. "Second-Order Gaseous Flow Models in Long Circular and Noncircular Microchannels and Nanochannels." In ASME 2011 9th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2011. http://dx.doi.org/10.1115/icnmm2011-58040.

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Gaseous flow in circular and noncircular microchannels has been examined and a simple analytical model with second-order slip boundary conditions for normalized Poiseuille number is proposed. The model is applicable to arbitrary length scale. It extends previous studies to the transition regime by employing the second-order slip boundary conditions. The effects of the second-order slip boundary conditions are analyzed. As in slip and transition regimes, no solutions or graphical and tabulated data exist for most geometries, the developed simple model can be used to predict friction factor, mass flow rate, tangential momentum accommodation coefficient, pressure distribution of gaseous flow in noncircular microchannels by the research community for the practical engineering design of microchannels such as rectangular, trapezoidal, double-trapezoidal, triangular, rhombic, hexagonal, octagonal, elliptical, semielliptical, parabolic, circular sector, circular segment, annular sector, rectangular duct with unilateral elliptical or circular end, annular, and even comparatively complex doubly-connected microducts. The developed second-order models are preferable since the difficulty and “investment” is negligible compared with the cost of alternative methods such as molecular simulations or solutions of Boltzmann equation. Navier-Stokes equations with second-order slip models can be used to predict quantities of engineering interest such as Poiseuille number, tangential momentum accommodation coefficient, mass flow rate, pressure distribution, and pressure drop beyond its typically acknowledged limit of application. The appropriate or effective second-order slip coefficients include the contribution of the Knudsen layers in order to capture the complete solution of the Boltzmann equation for the Poiseuille number, mass flow rate, and pressure distribution. It could be reasonable that various researchers proposed different second-order slip coefficients because the values are naturally different in different Knudsen number regimes. The transition regime is a varying mixture of different transport mechanisms and the mixed degree relies on the magnitude of the Knudsen number. It is analytically shown that the Knudsen’s minimum can be predicted with the second-order model and the Knudsen value of the occurrence of Knudsen’s minimum depends on inlet and outlet pressure ratio. The compressibility and rarefaction effects on mass flow rate and the curvature of the pressure distribution by employing first-order and second-order slip flow models are analyzed and compared. The condition of linear pressure distribution is given. This paper demonstrates that with some relatively simple ideas from knowledge, observation, and intuition, one can predict some fairly complex flows.
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Manickam, Vijaymaran, Ismail B. Celik, Jerry Mason, and Yuxin Liu. "Assessment of the Degree of Mixing in Microchannel Two-Fluid Flows." In ASME 2012 Fluids Engineering Division Summer Meeting collocated with the ASME 2012 Heat Transfer Summer Conference and the ASME 2012 10th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/fedsm2012-72268.

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Microchannels are used for delivery of two-or more fluids for multiple purposes, such as drug delivery, where good mixing is desired in a very short time (or distance). For this purpose, many design options are being proposed in the literature. For example herringbone baffles at the bottom of a rectangular channel are proposed to enhance mixing in a drug delivery device [6]. To assess the effectiveness of such devices many experiments need to be performed thus reducing the design cycle time and the cost involved. Computational fluid dynamics (CFD) can and is being used to shorten this design cycle by performing parametric analysis; however, due to numerical errors it is also necessary to verify and then validate the numerical models to ensure that the predictions are indeed accurate. In this study a recently developed microchannel presently in use is analyzed using CFD to determine its mixing efficiency. There are two common ways of assessing the degree of mixing: (a) via calculation of a passive scalar transport equation, (b) by following fluid particle trajectories and calculating the statistics. The first approach suffers from high degree of numerical diffusion. The second approach is usually used to only obtain qualitative information rather than quantitative assessment. In this study we explore both approaches and reconcile both of these in terms of extracting quantitative information. Furthermore we assess the results of simulations using both approaches to determine a mixing index that provides directly a measure of the extent of mixing.
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Reiss, Robert, Bo Qian, and Win Aung. "Eigenvalues for Moderately Damped Linear Systems Determined by Eigensensitivity Analysis." In ASME 1990 Design Technical Conferences. American Society of Mechanical Engineers, 1990. http://dx.doi.org/10.1115/detc1990-0079.

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Abstract A new method is presented to determine approximate closed-form solutions for the complex-valued frequencies of moderately damped linearly elastic structures. The approach is equally applicable to finite degree of freedom systems and distributed parameter systems. The damping operator is split into two components, the first of which uncouples the quadratic eigenvalue equation, and the eigenvalues are expressed as a power series in the second component of the damping operator. Specific numerical examples include both finite degree of freedom and distributed parameter systems. It is shown that for moderate damping, that is, when the second component of the damping operator is small, but not negligible, the series solution truncated after quadratic terms provides an excellent approximation to the true eigenvalues.
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Hassanpour, Pezhman A. "Approximate Response of Beam-Type Resonant Biosensors." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-88535.

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In this paper, the effect of absorption of antigens to the functionalized surface of a biosensor is modeled using a single degree-of-freedom mass-spring-damper system. The change in the mass of the system due to absorption is modeled with an exponential function. The governing equations of motion is derived considering the change in the mass of the system as well as the impact force due to absorption. It has been demonstrated that this equation is a linear second-order ordinary differential equation with time-varying coefficients. The solution of this differential equation is approximated by expanding the exponential function with a Taylor series and applying the method of multiple scales. The advantage of using the method of multiple scales to derive an approximate solution is in the insight it provides on the effect of each parameter on the response of the system. The free vibration response of the biosensor is derived using the approximate solution under different conditions, namely, with and without viscous damping, with and without considering the impact force, and for different binding rates.
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