• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.3
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• pp.194-206
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• 2023

We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype function with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the numerical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.243-251
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• 2003

This note Presents sufficient conditions for Lyapunov's stability and instability of a class of nonlinear nonautonomous second-order ordinary differential equations. Such a class includes as particular cases a remarkably large number of differential equations with specific physical applications. Two successive nonlinear transformations are applied to the original differential equation in order to convert it into a more convenient form for stability analysis purposes. The obtained stability / instability conditions depend closely on the parameterization of the original differential equation.

• Journal of Ocean Engineering and Technology
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• v.21 no.2 s.75
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• pp.12-21
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• 2007

As the economy grows and the population increases, we need to develop our coastal area and make use of it for various purposes. Specifically, investigation of the wave interactions on and around the vertical cylinders is very important in the design of the offshore or coastal structures. The nonlinear potential analysis developed so far, although very useful, has been found to be limited in application, as strong nonlinear waves generated by the interference between multilayered cylinders and wave impact forces by breaking waves can hardly be estimated. In this study, using a 3-Dimensional volume tracking method VOF(Volume of Fluid), based on Namer-Stokes equations, was developed to simulate highly nonlinear effects, such as breaking waves at the interface or complicated interference waves among structures. A numerical method for nonlinear interaction wave and vertical cylinders is newly proposed. The wave forces and wave transformations computed by the newly proposed numerical simulation method were compared to the other researcher's experimental results, and the results agree well. Based on the validation of this study, this numerical method is applied to the two vertical cylinders to discuss their nonlinear wave forces and wave transformations, according to the variations of separate distance of vertical cylinders.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.20 no.1
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• pp.1-13
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• 2008

In the present work, wave transformation by vertical columns and its wave forces acting on them are discussed using a direct 3-D numerical model based on the VOF (Volume Of Fluid) method. The numerical results for wave transformations and wave forces are critically compared to an advanced experimental data, and provide the verification of the numerical model used in the present study. Overall model-data comparisons are good. After verification of the numerical model, it is used to simulate wave fields around dual vertical columns with arbitrary cross section, and the characteristics of nonlinear wave forces and wave transformations according to the variations of different cross section types of vertical columns, an interval of vertical columns and incident wave angle are discussed.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.133-146
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• 2011

We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations compared with those of existing non-linear transformation methods.

• 제어로봇시스템학회:학술대회논문집
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• 1997.10a
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• pp.159-162
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• 1997

This paper describes robust stabilization of nonlinear single-input uncertain systems without matching conditions. We consider nonlinear systems with a vector of unknown constant parameters perturbed about a known value. The approach utilizes the generalized controller canonical form to lump the unmatched uncertainties recursively into the matched ones. This can be achieved via nonlinear coordinate transformations which depend not only on the states of the nonlinear system but also on the control input. Then the dynamic robust control law is derived and the stability result is also presented.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.329-336
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• 2006

In the category of the group theoretic methods using invertible discrete group transformation, we give a useful relation between Emden-Fowler equations and nonlinear heat equation. In this paper, by means of appropriate transformations of discrete group analysis, the nonlinear hate equation transformed into the class of the Emden-Fowler equations. This approach shows that, under the group action, the solution of reference equation can be transformed into the solution of the transformed equation.

• Journal of applied mathematics & informatics
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• v.42 no.2
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• pp.321-331
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• 2024

We consider the numerical treatment of blow-up problems having non-monotonic singular solutions that tend to infinity at some point in the domain. The use of standard numerical methods for solving problems with blow-up solutions can lead to significant errors. The reason being that solutions of such problems have singularities whose positions are unknown in advance. To be able to integrate such non-monotonic blow-up problems, we describe and use a method of non-local transformations. To show the efficiency of the method, we present a comparison of exact and numerical solutions in addition to some comparison with the work of other authors.

• International Journal of Contents
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• v.10 no.1
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• pp.18-22
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• 2014

Hidden nodes have a key role in the information processing of feed-forward neural networks in which inputs are processed through a series of weighted sums and nonlinear activation functions. In order to understand the role of hidden nodes, we must analyze the effect of the nonlinear activation functions on the weighted sums to hidden nodes. In this paper, we focus on the effect of nonlinear functions in a viewpoint of information theory. Under the assumption that the nonlinear activation function can be approximated piece-wise linearly, we prove that the entropy of weighted sums to hidden nodes decreases after piece-wise linear functions. Therefore, we argue that the nonlinear activation function decreases the uncertainty among hidden nodes. Furthermore, the more the hidden nodes are saturated, the more the entropy of hidden nodes decreases. Based on this result, we can say that, after successful training of feed-forward neural networks, hidden nodes tend not to be in linear regions but to be in saturated regions of activation function with the effect of uncertainty reduction.

• Journal of the Korean Society for Precision Engineering
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• v.13 no.2
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• pp.94-101
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• 1996

The equations of motion for a basic industrial robotic system which has a rigid or a flexible arm are derived by Lagrange's equation, respectively. Especially, for the deflection of the flexible arm, the assumed mode method is employed. These equations are highly nonlinear equations with nonlinear coupling between the variables of motion. In order to design the control law for the rigid-arm robot, Hunt-Su's nonlinear transformation method and Marino's feedback equivalence condition are used with linear quadratic regulator(LQR) theory. The control law for the rigid-arm robot is employed to input the desired path and to provide the required nonlinear transformations for the flexible-arm robot to follow. By using the implicit Euler method to solve the nonlinear equations, the comparison of the motions between the flexible and the rigid robots and the effect of flexibility are examined.