• Journal for History of Mathematics
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• v.27 no.2
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• pp.127-138
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• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• Journal of the Korean Society for Precision Engineering
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• v.26 no.8
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• pp.62-71
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• 2009

In mobile robotics, ultrasonic sensors became one of the most popular devices for collision avoidance and navigation primarily due to data robustness, the easy availability of low-cost systems, their compact size, simple circuits, and their ease in interfacing with computers. However, ultrasonic sonic sensors are subject to noise which results in inaccuracy of mapping and localization of the robot. This paper introduces a new approach to enhance environmental maps based on ultrasonic range data using linear interpolation and Newton interpolation. The simulation and experimental results show that the proposed method improves of the accuracy of the map through better distance estimation between the mobile robot and obstacles.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.537-547
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• 2012

In this paper, we define a $p$, $q$-difference operator and obtain an explicit formula which is used to express the $p$, $q$-analogue of the unified generalization of Stirling numbers and its exponential generating function in terms of the $p$, $q$-difference operator. Explicit formulas for the non-central $q$-Stirling numbers of the second kind and non-central $q$-Lah numbers are derived using the new $q$-analogue of Newton's interpolation formula. Moreover, a $p$, $q$-analogue of Newton's interpolation formula is established.

• The KIPS Transactions:PartA
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• v.11A no.5
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• pp.401-406
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• 2004

Among many algorithms for the integration problems in which one wants to compute the approximation to the definite integral in the average case setting, we study the average case errors of numerical integration rules using interpolation. In particular, we choose the composite Newton-Cotes quadratures and the function values at equally spaced sample points on the given interval as information. We compute the average case error of composite Newton-Cotes quadratures and show that it is minimal(modulo a multiplicative constant).

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.53 no.12
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• pp.820-826
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• 2004

In this paper, studied error of various signal interpolation methods in vibration signal analysis with digital order tracking. Because, interpolation errors are related with sampling rate and amount of calculation. Appled Signal interpolation methods are Lagrange, Newton and Cubic-spline. This paper proposed more proper interpolation method. Also, we suggest guideline for adaptive application of signal interpolation methods with Calculated results.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2017.10a
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• pp.681-683
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• 2017

In the digital information age, image processing is essential for various digital devices such as smart phones, cameras, and TVs. However, degradation occurs in analyzing, recognizing, and processing image data, and salt & pepper noise occurs. Therefore, in this paper, we applied linear interpolation method, newton interpolation method, lagrange interpolation method, and spline interpolation method to the image damaged by salt & pepper noise in order to find more effective interpolation method in salt & pepper noise environment, The methods were compared using the PSNR (peak signal to noise ratio).

• The Journal of the Convergence on Culture Technology
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• v.6 no.1
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• pp.449-454
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• 2020

We propose an improved de-interlacing method that converts the interlaced images into the progressive images by one field. In the first, Inter-pixel values are calculated by applying Newton's forward difference, backward difference interpolation from upper and lower 5 pixel values. Using inter-pixel values obtained from upper and lower 5 pixel values, it makes more accurate a direction estimate by applying the correlation between upper and lower pixel. If an edge direction is determined from the correlation, a missing pixel value is calculated into the average of upper and lower pixel obtained from predicted direction of edge. From simulation results, it is shown that the proposed method improves subjective image quality at edge region and objective image quality at 0.2~0.3dB as quantitative calculation result of PSNR, compared to previous various de-interlacing methods.

• Kyungpook Mathematical Journal
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• v.34 no.2
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• pp.207-216
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• 1994

• The Journal of Korean Institute of Communications and Information Sciences
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• v.19 no.6
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• pp.1016-1026
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• 1994

In this paper, we have studied load balancing problems in distributed systems. The nodes of distributed systems exchange periodically system state information each other. The information is stored in history. Based on the information, we compute an expected load index(ELI) using a five-degree interpolation polynomial in Newton`s backward difference interpolation formula. A new location policy of dynamic load balancing systems makes use of the ELI. We show that its performance is better than that of the existing load balancing algorithm through a simulation study.

• Proceedings of the Korean Information Science Society Conference
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• 2012.06c
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• pp.236-238
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• 2012

Ad-hoc 네트워크에서 각 노드는 분산 및 협동을 통해 자체적으로 이웃노드와 무선네트워크를 구축하고 주고받게 된다. 그러나 Ad-hoc 네트워크에서 하위노드의 제한적인 저장/통신/계산 능력, 상호인증의 어려움 등으로 기존의 보안대책을 그대로 적용할 수 없어 Ad-hoc 네트워크 특성에 맞는 새로운 보안대책이 필요하다. 이를 위해 비밀분산기법 중의 일종인 (t,n) 임계치 기법을 통해 노드를 인증하는 방식이 제안되기도 하였으나, 이는 고정된 t개 노드의 분산정보가 모여야만 원래의 비밀을 복원할 수 있는 것으로 주로 적대적 환경에 배치되어야 하는 전술 Ad-hoc 네트워크의 요구사항과는 부합하지 않는다. 따라서, 본 논문에서는 기존의 (t,n) 임계치 기법에 Newton의 보간 다항식을 최초로 적용하여 임계값 t를 동적으로 변경할 수 있는 공개키 인증방식을 제안하고, 그 유효성을 증명하고자 한다.