• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.28 no.2
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• pp.59-70
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• 2024

In this paper we present the approximation method of the circular helix by G² cubic polynomial curves. The approximants are G¹ Hermite interpolation of the circular helix and their approximation order is four. We obtain numerical examples to illustrate the geometric continuity and the approximation order of the approximants. The method presented in this paper can be extended to approximating the elliptical helix. Using the property of affine transformation invariance we show that the approximant has G² continuity and the approximation order four. The numerical examples are also presented to illustrate our assertions.

• The Transactions of the Korea Information Processing Society
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• v.13 no.2
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• pp.34-40
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• 2024

Secret sharing is a cryptographic technique that involves dividing a secret or a piece of sensitive information into multiple shares or parts, which can significantly increase the confidentiality of a secret. There has been a lot of research on secret sharing for different contexts or situations. Tassa's conjunctive secret sharing method employs polynomial derivatives to facilitate hierarchical secret sharing. However, the use of derivatives introduces several limitations in hierarchical secret sharing. Firstly, only a single group of participants can be created at each level due to the shares being generated from a sole derivative. Secondly, the method can only reconstruct a secret through conjunction, thereby restricting the specification of arbitrary secret reconstruction conditions. Thirdly, Birkhoff interpolation is required, adding complexity compared to the more accessible Lagrange interpolation used in polynomial-based secret sharing. This paper introduces the multi-compartment secret sharing method as a generalization of the conjunctive hierarchical secret sharing. Our proposed method first encrypts a secret using external groups' shares and then generates internal shares for each group by embedding the encrypted secret value in a polynomial. While the polynomial can be reconstructed with the internal shares, the polynomial just provides the encrypted secret, requiring external shares for decryption. This approach enables the creation of multiple participant groups at a single level. It supports the implementation of arbitrary secret reconstruction conditions, as well as conjunction. Furthermore, the use of polynomials allows the application of Lagrange interpolation.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.27 no.7
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• pp.859-866
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• 2003

The Lagrangian dispserion of fluid particles in inhomogeneous turbulence is investigated by a direct numerical simulation of turbulent channel flow. Fluid particle velocity and acceleration along a particle trajectory are computed by employing several interpolation schemes such as linear interpolation, high-order Lagrange polynomial interpolation and the Hermite interpolation schemes. The performances of the schemes are evaluated through comparison of errors in computed particle positions, velocities and accelerations against spectral interpolation. Adopting the four-point Hermite interpolation in the homogeneous directions and Chebyshev polynomials in the wall-normal direction appears to produce most reliable Lagrangian statistics including acceleration correlations with a reasonable amount of computational overhead.

• Journal of the Korean Association of Geographic Information Studies
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• v.13 no.3
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• pp.29-41
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• 2010

In this study, the prediction errors of various spatial interpolation methods used to model values at unmeasured locations was compared and the accuracy of these predictions was evaluated. The root mean square (RMS) was calculated by processing different parameters associated with spatial interpolation by using techniques such as inverse distance weighting, kriging, local polynomial interpolation and radial basis function to known elevation data of the east coastal area under the same condition. As a result, a circular model of simple kriging reached the smallest RMS value. Prediction map using the multiquadric method of a radial basis function was coincident with the spatial distribution obtained by constructing a triangulated irregular network of the study area through the raster mathematics. In addition, better interpolation results can be obtained by setting the optimal power value provided under the selected condition.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.605-612
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• 2004

The Gibbs’ phenomenon for the classical Fourier series is known. This occurs for almost all series expansions. This phenomenon has been observed even in sampling series. In this paper, we show the existence of Gibbs phenomenon for trigonometric interpolating polynomial by a simple and different manner from the wok[4].

• Structural Engineering and Mechanics
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• v.11 no.2
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• pp.221-236
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• 2001

A local point interpolation method (LPIM) is presented for the stress analysis of two-dimensional solids. A local weak form is developed using the weighted residual method locally in two-dimensional solids. The polynomial interpolation, which is based only on a group of arbitrarily distributed nodes, is used to obtain shape functions. The LPIM equations are derived, based on the local weak form and point interpolation. Since the shape functions possess the Kronecker delta function property, the essential boundary condition can be implemented with ease as in the conventional finite element method (FEM). The presented LPIM method is a truly meshless method, as it does not need any element or mesh for both field interpolation and background integration. The implementation procedure is as simple as strong form formulation methods. The LPIM has been coded in FORTRAN. The validity and efficiency of the present LPIM formulation are demonstrated through example problems. It is found that the present LPIM is very easy to implement, and very robust for obtaining displacements and stresses of desired accuracy in solids.

• Structural Engineering and Mechanics
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• v.59 no.6
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• pp.1071-1094
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• 2016

We employ the so-called problem-dependent linked interpolation concept to develop two cubic 4-node quadrilateral plate finite elements with 12 external degrees of freedom that pass the constant bending patch test for arbitrary node positions of which the second element has five additional internal degrees of freedom to get polynomial completeness of the cubic form. The new elements are compared to the existing linked-interpolation quadratic and nine-node cubic elements presented by the author earlier and to the other elements from literature that use the cubic linked interpolation by testing them on several benchmark examples.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.18 no.7
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• pp.907-912
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• 1993

In this Paper we have proposed a new method to implement the interpolation of the functions, using a neural network. The architecture of neural network is a three-layer perceptron and the training algorithm is a modified error back propagation algorithm adding neurons to hidden layer. The interpolated functions are sin(7 X), 3rd order polynomial 0.5$\times$3_2$\times$2+X+2.5 and rectangular pulse 0.99 U (X-0.2) -0.99 U(X-0.8) +0.01, where U(X) is the unit step. The root mean squred errors of the interpolated functions are 0.00258, 0.00164 and 0.00116 respectively.

• Proceedings of the IEEK Conference
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• 2008.06a
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• pp.83-84
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• 2008

It is efficient to use the time-domain beamforming to operate the various pulse with the different pulse length, frequency, bandwidth in active sonar system. In this paper, we propose a time-domain beamformer with the cumulative processing in the decomposed channel using the polynomial interpolation to solve the problem of the computational cost, high transmission data rate, and the lack of internal memory.

• Proceedings of the Korean Society of Broadcast Engineers Conference
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• 2002.11a
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• pp.81-84
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• 2002

본 논문에서는 간략화 된 형상학적 다항식 변환(Morphological Polynomial Transform)과 형상학적 보간법(Morphological Interpolation)을 이용하는 비선형 예측 방법을 제안한다. 형상학적 다항식 변환은 형상학적 연산을 통해 데이터를 구조함수들의 계수들로 표현하는 변환이며, 형상학적 보간법은 형상학적 다항식 변환에 의한 계수들을 이용하여 보간하는 방법이다. 형상학적 다항식 변환을 간략화 하여 정수 연산만으로 적용할 수 있도록 개선하였으며, 보다 영상에 적합한 형상학적 보간법에 기반 한 예측 방법을 사용한다. 제안하는 예측 방법과 허프만 부호화를 사용하여 적은 비트로 영상을 손실 없이 저장할 수 있음을 실험으로 검증한다.