• Title/Summary/Keyword: an algebraic approach

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A study on the analytic geometric characteristics of Archimedes' 《The Method》 and its educational implications (아르키메데스의 《The Method》의 해석기하학적 특성과 그 교육적 시사점에 대한 연구)

  • Park, Sun-Yong
    • Journal for History of Mathematics
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    • v.27 no.4
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    • pp.271-283
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    • 2014
  • This study takes a look at Polya's analysis on Archimedes' "The Method" from a math-historical perspective. We, based on the elaboration of Polya's analysis, investigate the analytic geometric characteristics of Archimedes' "The Method" and discuss the way of using the characteristics in education of school calculus. So this study brings up the educational need of approach of teaching the definite integral by clearly disclosing the transition from length, area, volume etc into the length as an area function under a curve. And this study suggests the approach of teaching both merit and deficiency of the indivisibles method, and the educational necessity of making students realizing that the strength of analytic geometry lies in overcoming deficiency of the indivisibles method by dealing with the relation of variation and rate of change by means of algebraic expression and graph.

A simple and rapid approach to modeling chromium breakthrough in fixed bed adsorber

  • Chu, Khim Hoong
    • Advances in environmental research
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    • v.7 no.1
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    • pp.29-37
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    • 2018
  • A simple mathematical model for predicting fixed bed adsorption dynamics is described. The model is characterized by a linear adsorption isotherm and a linear driving force expression for mass transfer. Its analytic solution can be approximated with an algebraic equation in closed form which is easily evaluated by spreadsheet computation. To demonstrate one application of the fixed bed model, a previously published adsorption system is used as a case study in this work. The adsorption system examined here describes chromium breakthrough in a fixed bed adsorber packed with imidazole functionalized adsorbent particles and is characterized by a nonlinear adsorption isotherm. However, the equilibrium behavior of the fixed bed adsorber is in essence governed by a linear adsorption isotherm due to the use of a low influent chromium concentration. It is shown that chromium breakthrough is predicted reasonably well by the fixed bed model. The model's parameters can be easily extracted from independent batch experiments. The proposed modeling approach is very simple and rapid, and only Excel is used for computation.

Robust $L_2$Optimization for Uncertain Systems

  • Kim, Kyung-Soo;Park, Youngjin
    • 제어로봇시스템학회:학술대회논문집
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    • 1995.10a
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    • pp.348-351
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    • 1995
  • This note proposes a robust LQR method for systems with structured real parameter uncertainty based on Riccati equation approach. Emphasis is on the reduction of design conservatism in the sense of quadratic performance by utilizing the uncertainty structure. The class of uncertainty treated includes all the form of additive real parameter uncertainty, which has the multiple rank structure. To handle the structure of uncertainty, the scaling matrix with block diagonal structure is introduced. By changing the scaling matrix, all the possible set of uncertainty structures can be represented. Modified algebraic Riccati equation (MARE) is newly proposed to obtain a robust feedback control law, which makes the quadratic cost finite for an arbitrary scaling matrix. The remaining design freedom, that is, the scaling matrix is used for minimizing the upper bound of the quadratic cost for all possible set of uncertainties within the given bounds. A design example is shown to demonstrate the simplicity and the effectiveness of proposed method.

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The Iterated Ritz Method: Basis, implementation and further development

  • Dvornik, Josip;Lazarevic, Damir;Uros, Mario;Novak, Marta Savor
    • Coupled systems mechanics
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    • v.7 no.6
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    • pp.755-774
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    • 2018
  • The Ritz method is known as very successful strategy for discretizing continuous problems, but it has never been used for solving systems of algebraic equations. The Iterated Ritz Method (IRM) is a novel iterative solver based on the discretized Ritz procedure applied at each iteration step. With an appropriate choice of coordinate vectors, the method may be efficient in linear, nonlinear and optimization problems. Additionally, some iterative methods can be explained as special cases of this approach, which helps to understand advantages and limitations of these methods and gives motivation for their improvement in sense of IRM. In this paper, some ideas for generation of efficient coordinate vectors are presented. The algorithm was developed and tested independently and then implemented into the open source program FEAP. Method has been successfully applied to displacement based (even ill-conditioned) models of structural engineering practice. With this original approach, a new iterative solution strategy has been opened.

Word problem solving of simultaneous equations by 5th and 6th grade students (5.6학년 학생들의 이원일차연립방정식 형태의 문장제 해결 과정 분석)

  • Yun, Min-Ji;Pang, Jeong-Suk
    • Communications of Mathematical Education
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    • v.23 no.3
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    • pp.761-783
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    • 2009
  • Problem solving ability can be fostered by dealing with many different types of problems. We investigated how $5^{th}$ and $6^{th}$ graders who did not learn traditional algebraic methods might approach the word problems of simultaneous equations. This result reveals that the strategy of guess-and-check serves as a basis for elementary school students in solving simultaneous equations. A noticeable remark is that students used the guess-and-check strategy in various ways. Whereas some students changed a variable given in the problem step by step, others did in a sophisticated way focusing on the relation between two variables. Moreover, some students were able to write an equation which was not typical but meaningful and correct. This paper emphasizes the need of connections between pre-algebraic and algebraic solutions.

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Elastic local buckling of thin-walled elliptical tubes containing elastic infill material

  • Bradford, M.A.;Roufegarinejad, A.
    • Interaction and multiscale mechanics
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    • v.1 no.1
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    • pp.143-156
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    • 2008
  • Elliptical tubes may buckle in an elastic local buckling failure mode under uniform compression. Previous analyses of the local buckling of these members have assumed that the cross-section is hollow, but it is well-known that the local buckling capacity of thin-walled closed sections may be increased by filling them with a rigid medium such as concrete. In many applications, the medium many not necessarily be rigid, and the infill can be considered to be an elastic material which interacts with the buckling of the elliptical tube that surrounds it. This paper uses an energy-based technique to model the buckling of a thin-walled elliptical tube containing an elastic infill, which elucidates the physics of the buckling phenomenon from an engineering mechanics basis, in deference to a less generic finite element approach to the buckling problem. It makes use of the observation that the local buckling in an elliptical tube is localised with respect to the contour of the ellipse in its cross-section, with the localisation being at the region of lowest curvature. The formulation in the paper is algebraic and it leads to solutions that can be determined by implementing simple numerical solution techniques. A further extension of this formulation to a stiffness approach with multiple degrees of buckling freedom is described, and it is shown that using the simple one degree of freedom representation is sufficiently accurate for determining the elastic local buckling coefficient.

Derivation Algorithm of State-Space Equation for Production Systems Based on Max-Plus Algebra

  • Goto, Hiroyuki;Masuda, Shiro
    • Industrial Engineering and Management Systems
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    • v.3 no.1
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    • pp.1-11
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    • 2004
  • This paper proposes a new algorithm for determining an optimal control input for production systems. In many production systems, completion time should be planned within the due dates by taking into account precedence constraints and processing times. To solve this problem, the max-plus algebra is an effective approach. The max-plus algebra is an algebraic system in which the max operation is addition and the plus operation is multiplication, and similar operation rules to conventional algebra are followed. Utilizing the max-plus algebra, constraints of the system are expressed in an analogous way to the state-space description in modern control theory. Nevertheless, the formulation of a system is currently performed manually, which is very inefficient when applied to practical systems. Hence, in this paper, we propose a new algorithm for deriving a state-space description and determining an optimal control input with several constraint matrices and parameter vectors. Furthermore, the effectiveness of this proposed algorithm is verified through execution examples.

Mutual Detectability and System Enlargement of Detection Filters: An Invariant Zero Approach

  • Kim, Yong-Min;Park, Jae-Hong
    • International Journal of Control, Automation, and Systems
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    • v.4 no.6
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    • pp.769-774
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    • 2006
  • In this paper, we discuss the problem of non-mutual detectability using the invariant zero. We propose a representation method for excess spaces by linear equation based on the Rosenbrock system matrix. As an alternative to the system enlargement method proposed by White[1], we propose an appropriate form of an enlarged system to make a set of faults mutually detectable by assigning sufficient geometric multiplicity of invariant zeros. We show the equivalence between the two methods and a necessary condition for the system enlargement in terms of the geometric and algebraic multiplicities of invariant zeros.

Design of a Low-Order Sensorless Controller by Robust H∞ Control for Boost Converters

  • Li, Xutao;Chen, Minjie;Shinohara, Hirofumi;Yoshihara, Tsutomu
    • Journal of Power Electronics
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    • v.16 no.3
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    • pp.1025-1035
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    • 2016
  • Luenberger observer (LO)-based sensorless multi-loop control of a converter requires an iterative trial-and-error design process, considering that many parameters should be determined, and loop gains are indirectly related to the closed-loop characteristics. Robust H∞ control adopts a compact sensorless controller. The algebraic Riccati equation (ARE)-based and linear matrix inequality (LMI)-based H∞ approaches need an exhaustive procedure, particularly for a low-order controller. Therefore, in this study, a novel robust H∞ synthesis approach is proposed to design a low-order sensorless controller for boost converters, which need not solve any ARE or LMI, and to parameterize the controller by an adjustable parameter behaving like a "knob" on the closed-loop characteristics. Simulation results show the straightforward closed-loop characteristics evaluation and better dynamic performance by the proposed H∞ approach, compared with the LO-based sensorless multi-loop control. Practical experiments on a digital processor confirmed the simulation results.

Self-tuning optimal control of an active suspension using a neural network

  • Lee, Byung-Yun;Kim, Wan-Il;Won, Sangchul
    • 제어로봇시스템학회:학술대회논문집
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    • 1996.10b
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    • pp.295-298
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    • 1996
  • In this paper, a self-tuning optimal control algorithm is proposed to retain the optimal performance of an active suspension system, when the vehicle has some time varying parameters and parameter uncertainties. We consider a 2 DOF time-varying quarter car model which has the parameter variation of sprung mass, suspension spring constant and suspension damping constant. Instead of solving algebraic riccati equation on line, we propose a neural network approach as an alternative. The optimal feedback gains obtained from the off line computation, according to parameter variations, are used as the neural network training data. When the active suspension system is on, the parameters are identified by the recursive least square method and the trained neural network controller designer finds the proper optimal feedback gains. The simulation results are represented and discussed.

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