• Communications of Mathematical Education
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• v.23 no.3
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• pp.625-642
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• 2009

In the school mathematics the concept of tangent is introduced in several steps in suitable contexts. Students are required to reflect and revise their concepts of tangent in order to apply the improved concept to wider range of contexts. In this paper we consider the tangent as the optimal linear approximation to a curve at a given point and make three discussions on pedagogical aspects of it. First, it provides a method of finding roots of real numbers which can be used as an application of tangent. This may help students improve their affective variables such as interest, attitude, motivation about the learning of tangent. Second, this concept reflects the modern point of view of tangent, the linear approximation of nonlinear problems. Third, it gives precise meaning of two tangent lines appearing two sides of a cusp point of a curve.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.689-706
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• 2008

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

• Journal of the Korean Mathematical Society
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• v.35 no.3
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• pp.539-557
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• 1998

This paper is concerned with the performance evaluation of loss networks. We shall review the Erlang Fixed Point (EFP) method for estimating the blocking probabilities, which is based on an assumption that links are blocked independently. For networks with linear structure, the behaviour of adjacent links can be highly correlated. We shall give particular attention to recently-developed fixed-point methods which specifically account for the dependencies between neighbouring links. For the network considered here, namely a ring network with two types of traffic, these methods produce relative errors typically $10^{-5}$ of that found using the basic EFP approximation.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.20 no.12
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• pp.3804-3821
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• 1996

The configuration optimization is a structural optimization method which includes the coordinates of a structure as well as the sectional properties in the design variable set. Effective reduction of the weight of discrete structures can be obrained by changing the geometry while satisfying stress, Ei;er bickling, displacement, and frequency constraints, etc. However, the nonlinearity due to the configuration variables may cause the difficulties of the convergence and expensive computational cost. An efficient approximation method for the configuration optimization has been developed to overcome the difficulties. The method approximates the constraint functions based onthe second-order Taylor series expansion with reciprocal design variables. The Hessian matrix is approzimated from the information on previous design points. The developed algotithms are coded and the examples are solved.

• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.433-446
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• 1997

In this paper we study an existence and the approxi-mation of the solution of the solution of the elliptic variational inequality from an abstract axiomatic point of view. We discuss convergence results and give an error estimate for the difference of the two solutions in an appropriate norm Also we present some computational results by using fixed point method.

• Journal of Mechanical Science and Technology
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• v.15 no.7
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• pp.876-888
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• 2001

For a large scaled optimization based on response surface methods, an efficient quadratic approximation method is presented in the context of the trust region model management strategy. If the number of design variables is η, the proposed method requires only 2η+1 design points for one approximation, which are a center point and tow additional axial points within a systematically adjusted trust region. These design points are used to uniquely determine the main effect terms such as the linear and quadratic regression coefficients. A quasi-Newton formula then uses these linear and quadratic coefficients to progressively update the two-factor interaction effect terms as the sequential approximate optimization progresses. In order to show the numerical performance of the proposed method, a typical unconstrained optimization problem and two dynamic response optimization problems with multiple objective are solved. Finally, their optimization results compared with those of the central composite designs (CCD) or the over-determined D-optimality criterion show that the proposed method gives more efficient results than others.

• Journal of Electrical Engineering and Technology
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• v.11 no.1
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• pp.1-8
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• 2016

It is well known that the behavior of PV curves is similar to a quadratic function. This is used in some papers to approximate PV curves and calculate the maximum-loading point by minimum number of power flow runs. This paper also based on quadratic approximation of the PV curves is aimed at completing previous works so that the computational efforts are reduced and the accuracy is maintained. To do this, an iterative method based on a quadratic function with two constant coefficients, instead of the three ones, is used. This simplifies the calculation of the quadratic function. In each iteration, to prevent the calculations from diverging, the equations are solved on the assumption that voltage magnitude at a selected load bus is known and the loading factor is unknown instead. The voltage magnitude except in the first iteration is selected equal to the one at the nose point of the latest approximated PV curve. A method is presented to put the mentioned voltage in the first iteration as close as possible to the collapse point voltage. This reduces the number of iterations needed to determine the maximum-loading point. This method is tested on four IEEE test systems.

• Bulletin of the Korean Chemical Society
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• v.30 no.3
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• pp.567-572
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• 2009

In this study, the effects of counter ion valency of the electrolyte on the colloidal repulsion between two parallel cylindrical particles were investigated. Electrostatic interactions of the cylindrical particles were calculated with the variation of counter ion valency. To calculate the electrical repulsive energy working between these two cylindrical particles, Derjaguin approximation was applied. The electrostatic potential profiles were obtained numerically by solving nonlinear Poission-Boltzmann (P-B) equation and calculating middle point potential and repulsive energy working between interacting surfaces. The electrical potential and repulsive energy were influenced by counter ion valency, Debye length, and surface potential. The potential profile and middle point potential decayed with the counter ion valency due to the promoted shielding of electrical charge. On the while, the repulsive energy increased with the counter ion valency at a short separation distance. These behaviors of electrostatic interaction agreed with previous results on planar or spherical surfaces.

• Journal of the Korean Data and Information Science Society
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• v.24 no.6
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• pp.1309-1317
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• 2013

There are various parameter estimation methods for the generalized exponential distribution under progressive type I interval censoring. Chen and Lio (2010) studied the parameter estimation method by the maximum likelihood estimation method, mid-point approximation method, expectation maximization algorithm and methods of moments. Among those, mid-point approximation method has the smallest mean square error in the generalized exponential distribution under progressive type I interval censoring. However, this method is difficult to derive closed form of solution for the parameter estimation using by maximum likelihood estimation method. In this paper, we propose two type of approximate maximum likelihood estimate to solve that problem. The simulation results show the obtained estimators have good performance in the sense of the mean square error. And proposed method derive closed form of solution for the parameter estimation from the generalized exponential distribution under progressive type I interval censoring.

• Journal of the Korean Statistical Society
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• v.30 no.4
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• pp.679-691
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• 2001

Two truncated sequential tests are developed for the two-sample scale problem based on the usual Wilcoxon rank-sum statistic for two different dispersion indices - absolute median deviations, when the medians of the two populations X and Y are equal or known and sums of squared mean deviations, when the medians are either unknown or unequal. The first test is briefly called SWAMD test and the second SWSMD test. For the SWAMD test, the percentile points for both the one-sided and two-sided alternatives, (equation omitted) have been found by Wiener approximation and their values computed for a range of values of a and N; analytical expression for the power function has been derived through Wiener process and its performance studied for various sequential designs for exponential distribution. This test has been illustrated by a numerical example. All the results of the SWAMD test, being directly applicable to the SWSMD test, are not dealt with separately Both the tests are compared and their suitable applications indicated.