• Communications of the Korean Mathematical Society
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• v.16 no.1
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• pp.153-161
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• 2001

In the course of working on the preconditioning $C^1$ polynomial spline collocation method, one has to deal with the exponential decay of $C^1$ Lagrange polynomial splines. In this paper we show the exponential decay of $C^1$ Lagrange polynomial splines using the Chebyshev-Gauss points as the local data points.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.13 no.7
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• pp.3690-3713
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• 2019

According to the main group key management schemes logical key hierarchy (LKH), exclusion basis systems (EBS) and other group key schemes are limited in network structure, collusion attack, high energy consumption, and the single point of failure, this paper presents a group key management scheme for wireless sensor networks based on Lagrange interpolation polynomial characteristic (AGKMS). That Chinese remainder theorem is turned into a Lagrange interpolation polynomial based on the function property of Chinese remainder theorem firstly. And then the base station (BS) generates a Lagrange interpolation polynomial function f(x) and turns it to be a mix-function f(x)' based on the key information m(i) of node i. In the end, node i can obtain the group key K by receiving the message f(m(i))' from the cluster head node j. The analysis results of safety performance show that AGKMS has good network security, key independence, anti-capture, low storage cost, low computation cost, and good scalability.

• Journal for History of Mathematics
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• v.27 no.3
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• pp.165-182
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• 2014

After algebraic expressions for the roots of 3rd and 4th degree polynomial equations were given in the mid 16th century, seeking such a formula for the 5th and greater degree equations had been one main problem for algebraists for almost 200 years. Lagrange made careful and thorough investigation of various solving methods for equations with the purpose of finding a principle which could be applicable to general equations. In the process of doing this, he found a relation between the roots of the original equation and its auxiliary equation using permutations of the roots. Lagrange's ingenious idea of using permutations of roots of the original equation is regarded as the key factor of the Abel's proof of unsolvability by radicals of general 5th degree equations and of Galois' theory as well. This paper intends to examine Lagrange's contribution in the theory of polynomial equations, providing a detailed analysis of various solving methods of Lagrange and others before him.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.51 no.6
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• pp.235-240
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• 2002

This paper presents a new method for estimating the block pulse series coefficients by using the Lagrange's second order interpolation polynomial. Block pulse functions have been used in a variety of fields such as the analysis and controller design of the systems. When the block pulse functions are used, it is necessary to find the more exact value of the block pulse series coefficients. But these coefficients have been estimated by the mean of the adjacent discrete values, and the result is not sufficient when the values are changing extremely. In this paper, the method for improving the accuracy of the block pulse series coefficients by using the Lagrange's second order interpolation polynomial is presented.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.51 no.7
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• pp.286-293
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• 2002

This paper presents a new method for finding the Block Pulse series coefficients and deriving the Block Pulse integration operational matrices which are necessary for the control fields using the Block Pulse functions. In order to apply the Block Pulse function technique to the problems of continuous-time dynamic systems more efficiently, it is necessary to find the more exact value of the Block Pulse series coefficients and drives the related integration operational matrices by using the Lagrange second order interpolation polynomial.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.52 no.6
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• pp.351-358
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• 2003

This paper presents a new method for finding the Block Pulse series coefficients, deriving the Block Pulse integration operational matrices and generalizing the integration operational matrices which are necessary for the control fields using the Block Pulse functions. In order to apply the Block Pulse function technique to the problems of state estimation or parameter identification more efficiently, it is necessary to find the more exact value of the Block Pulse series coefficients and integral operational matrices. This paper presents the method for improving the accuracy of the Block Pulse series coefficients and derives the related integration operational matrices and generalized integration operational matrix by using the Lagrange second order interpolation polynomial.

• Journal of the Korean Institute of Telematics and Electronics
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• v.15 no.5
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• pp.29-33
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• 1978

In this paper, the properties of Galois fields defined over any finite field are analysed to derive Galois switching functions and the arithmetic operation methods over any finite field are showed. The polynomial expansions over finite fields by Lagrange's interpolation method are derived and proved. The results are applied to multivalued single variable logic networks.

• Proceedings of the KIEE Conference
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• 2005.05a
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• pp.161-163
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• 2005

In this paper, we propose an efficient motion vector recovery algorithm for the new coding standard H.264, which makes use of the Lagrange interpolation formula. In H.264/AVC, a 16$\times$16 macroblock can be divided into different block shapes for motion estimation, and each block has its own motion vector. In the natural video the motion vector is likely to move in the same direction, hence the neighboring motion vectors are correlative. Because the motion vector in H.264 covers smaller area than previous coding standards, the correlation between neighboring motion vectors increases. We can use the Lagrange interpolation formula to constitute a polynomial that describes the motion tendency of motion vectors, and use this polynomial to recover the lost motion vector. The simulation result shows that our algorithm can efficiently improve the visual quality of the corrupted video.

• The KIPS Transactions:PartA
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• v.11A no.7 s.91
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• pp.571-576
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• 2004

In this paper, by using Lagrange polynomial interpolation with a trick such that for $f(x)^{3}$ we shall use $f(x_i)^{3}I_i(x)^{3}$ instead of $I(x)^{3}$ where $I{x}{\;}={\;}\sum_{i}^{f}(x_i)I_i(x)$. We show the convergence and stability and calculate errors. These errors are approximately less than $C(\frac{1}{N})^{N-1} hN(N-1)(\frac{N}{2})^{N-1} /(\frac{N}{2})!$ where N is a polynomial degree.

• ETRI Journal
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• v.6 no.2
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• pp.18-26
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• 1984

Among the various forms of interpolating polynomial for approximation, this paper is a study about the characteristics of piecewise Lagrange interpolating polynomials. And throughout the study, an attempt is made to construct the two-dimensional ap proximating function over Rectangular Grid and Triangular Grid by using the one-dim ensional interpolating polynomials.