• 한국수학사학회지
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• 제31권1호
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• pp.19-35
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• 2018

Taylor's Theorem is an important theorem which is applied to several disciplines. It is usually taught in a college-level calculus course for the first time. Many students have a hard time to understand or to make applications. In this paper, we look into the history of the development of Taylor's theorem and consider a teaching strategy of the theorem.

• 한국수학사학회지
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• 제27권2호
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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.

• 대한기계학회:학술대회논문집
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• 대한기계학회 2004년도 춘계학술대회
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• pp.180-185
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• 2004

This paper investigates asimply supported orthotropic rectangular laminate with viscous interfaces subjected to bending. Additional mathematical difficulty is involved due to the presence of viscous interfaces because the behavior of the laminate depends on time. A step-by-step state-space approach is suggested, which is directly based on the threedimensional theory of elasticity. In particular, Taylor's expansion theorem is employed to model the variations of field variables with time. The proposed method is suitable for analyzing laminated plate of arbitrary thickness. Numerical calculations are performed and it is shown that the viscous interfaces have a significant fluence on the response.

• 대한조선학회논문집
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• 제30권2호
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• pp.43-53
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• 1993

부유체 강제동요문제를 수치해석하는 데에는 두가지 어려움이 따른다. 첫째는 교차점주위의 급격한 유동이고 둘째는 무한원방처리이다. 본 논문에서는 무한원방처리에 주안점을 두어, 자유표면의 Taylor 전개 및 F.F.T. 적용으로 계산시간을 단축시켜 수치해석하였다. 즉 Green 정리를 이용하여 해를 표현한 후, 축차방법에 의하여 해를 구한다. 축차단계에서는 계산식을 자유표면 기울기에 대하여 Taylor 전개하여 Convolution 형태로 변형한 후 F.F.T. 를 적용함으로써 계산시간을 O(Nlog N)으로 유지할 수 있었다. 수치검증을 위하여 부유체 선형문제와 압력장의 비선형 문제를 수치해석하여 비교하였고, 이를 확장하여 부유체의 강제동요문제를 수치해석하였는데, 계산시간을 O(Nlog N)으로 유지하면서 수치해석할 수 있었다.

• 한국수자원학회:학술대회논문집
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• 한국수자원학회 1988년도 제30회 수공학연구발표회논문초록집
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• pp.29-36
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• 1988

A general scheme is developed which determines the Lagrangian motions of water particles by the Eulerian velocity at their mean positions by use of Taylor's theorem. Utilizing the Stokes finite-amplitude wave theory, the mass transport velocity which includes the effects of higher-order wave components is determined. The fifth-order theory predicts the mass transport velocity less than that given by the existing second-order theory over the whole depth. Limited experimental data for changes in wave celerity in closed wave flumes are compared with the theoretical predictions.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1101-1121
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• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• 한국해안해양공학회지
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• 제4권4호
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• pp.187-200
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• 1992

Taylor 정리를 이용하여 평균위치에서의 Eulerian유속으로부터 수립자의 시간에 따른 Lagrangian 운동궤적을 결정하는 방법이 제안되었다. 이 방법을 Stokes 유한진폭파이론에 적용하여 고차 파성분을 포함하는 수립자의 운동궤적과 질양이동속도를 결정하였다. Stokes 5차파이론의 적용 결과, 평균위치에서의 Eulerian 유속으로부터 결정한 수립자의 운동궤적은 자유수면 부근을 제외하고는 5차파이론에 의한 순간속도에 의해 계산된 값과 매우 좋은 일치를 보여주었다. Stokes 5차파 이론에 의한 질양이동속도는 종전의 2차파이론에 의한 질양이동속도보다 전수심에서 작은 값을 보여준다.

• 제어로봇시스템학회논문지
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• 제5권7호
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• pp.794-799
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• 1999

This study deals with robustness bounds estimation for uncertain linear systems with structured perturbations where the eigenvalues of the perturbed systems are guaranteed to stay in a prescribed region. Based upon the Lyapunov approach, new theorems to estimate allowable perturbation parameter bounds are derived. The theorems are referred to as the zero-order or first-order asymmetric robustness measure depending on the order of the P matrix in the sense of Taylor series expansion of perturbed Lyapunov equation. It is proven that Gao's theorem for the estimation of stability robustness bounds is a special case of proposed zero-order asymmetric robustness measure for eigenvalue assignment. Robustness bounds of perturbed parameters measured by the proposed techniques are asymmetric around the origin and less conservative than those of conventional methods. Numerical examples are given to illustrate proposed methods.