• Journal of the Korean Statistical Society
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• 제36권4호
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• pp.479-488
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• 2007

The elliptic random field is an extension to the Gaussian random field. We proved a theorem which characterizes the elliptic random field. We proposed a heuristic approach to derive an approximation to the distribution of the size of one connected component of its excursion set above a high threshold. We used this approximation to approximate the distribution of the largest cluster size. We used simulation to compare the approximation with the exact distribution.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 2003년도 The Fifth Asian Computational Fluid Dynamics Conference
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• pp.156-157
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• 2003

A simple method is proposed to split the flux vector of the Euler equations by introducing two artificial wave speeds. The direction of wave propagation can be adjusted by these two wave speeds. This idea greatly simplifies the upwinding, and leads to a new family of upwind schemes. Numerical flux function for multi-dimensional Euler equations is formulated for any grid system, structured or unstructured. A remarkable simplicity of the scheme is that it successfully achieves one-sided approximation for all waves without recourse to any matrix operation. Moreover, its accuracy is comparable with the exact Riemann solver. For 1-D Euler equations, the scheme actually surpasses the exact solver in avoiding expansion shocks without any additional entropy fix. The scheme can exactly resolve stationary contact discontinuities, and it is also freed of the carbuncle problem in multi­dimensional computations.

• Journal of applied mathematics & informatics
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• 제34권5_6호
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• pp.421-434
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• 2016

The main aim of this paper is to discuss the difference between the Euler-Maruyama's approximate solutions and the accurate solution to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and weakened linear growth condition. Furthermore, we give the pth moment continuous of the approximate solution for the delay equation.

• Coupled systems mechanics
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• 제11권2호
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• pp.167-198
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• 2022

In this paper we deal with classical instability problems of heterogeneous Euler beam under conservative loading. It is chosen as the model problem to systematically present several possible solution methods from simplest deterministic to more complex stochastic approach, both of which that can handle more complex engineering problems. We first present classical analytic solution along with rigorous definition of the classical Euler buckling problem starting from homogeneous beam with either simplified linearized theory or the most general geometrically exact beam theory. We then present the numerical solution to this problem by using reduced model constructed by discrete approximation based upon the weak form of the instability problem featuring von Karman (virtual) strain combined with the finite element method. We explain how such numerical approach can easily be adapted to solving instability problems much more complex than classical Euler's beam and in particular for heterogeneous beam, where analytic solution is not readily available. We finally present the stochastic approach making use of the Duffing oscillator, as the corresponding reduced model for heterogeneous Euler's beam within the dynamics framework. We show that such an approach allows computing probability density function quantifying all possible solutions to this instability problem. We conclude that increased computational cost of the stochastic framework is more than compensated by its ability to take into account beam material heterogeneities described in terms of fast oscillating stochastic process, which is typical of time evolution of internal variables describing plasticity and damage.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.113-124
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• 2006

In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point at a = ao, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point at ah = ao + O(h).

• 호남수학학술지
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• 제35권1호
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• pp.101-107
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• 2013

The Gamma function ${\Gamma}$ which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers $\mathbb{R}$, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권3호
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• pp.138-155
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• 2022

To solve the initial value problem we present a new single-step implicit method based on the Euler method. We prove that the proposed method has convergence order 2. In practice, numerical results of the proposed method for some selected examples show an error tendency similar to the second-order Taylor method. It can also be found that this method is useful for stiff initial value problems, even when a small number of nodes are used. In addition, we extend the proposed method by using weighted averages with a parameter and show that its convergence order becomes 2 for the parameter near $\frac{1}{2}$. Moreover, it can be seen that the extended method with properly selected values of the parameter improves the approximation error more significantly.

• Communications for Statistical Applications and Methods
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• 제17권3호
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• pp.367-376
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• 2010

최근 확산모형의 추정을 위한 매우 다양한 방법론들이 제시되고 연구 되어 왔다. 본 연구에서는 제안된 확산모형의 추정 방법 중에서, 안장점근사법을 이용한 확산모형의 모수 추정방법에 대하여 살펴보게 되고, 가장 단순한 형태의 안장점근사법인 단일항 안장점근사법의 사용을 제안하게 된다. 단일항 안장점근사법은 오일러근사법과 마찬가지로 계산속도가 빠르고, 다양한 모형에 적용이 가능하면서도 최대우도추정량과 마찬가지로 성능이 우수한 특성을 갖고 있음을 살펴보게 된다. OU 확산모형을 대상으로 한 시뮬레이션 연구를 통하여 단일항 안장점근사를 이용한 추정량과 다른 추정량들과의 성질을 비교한다.

• 한국전산구조공학회논문집
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• 제20권2호
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• pp.127-136
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• 2007

본 논문에서는 비동질 반무한 평면에 대한 비례경계유한요소법의 식을 유도하고 수치예제를 해석하였다. 비례경계유한 요소법은 편미분 방정식을 경계방향으로는 유한요소와 같은 근사를 통해서 약화시키고 방사방향으로는 정확해를 사용하는 반 해석적인 방법으로, 방사방향으로 멱함수를 따라 탄성계수가 변화되는 반무한 평면에 대해서 관계식을 가상일의 원리에 근거하여 새로이 유도하였다. 이 과정에서 반무한평면의 거동이 Euler-Cauchy방정식을 따름을 보이고, 기존의 동질 반무한평면의 해석시 도입되던 로그모드가 비동질 반무한 평면의 해석에는 유효하지 않음을 보였다. 수치예제를 통하여 유도된 식이 타당한 거동을 보임을 증명하고 이 접근법이 실제 공학적 문제의 해결에 있어서 유용함을 보였다.

• 응용통계연구
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• 제24권6호
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• pp.1007-1020
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• 2011

현대금융공학에 있어서 확산모형은 중요한 역할을 담당하고 있다. 다양한 형태의 확산모형이 제안되어왔고 현실에 응용되어 왔다. 확산모형을 이용하여 금융자료를 분석하기 위하여는 확산모형의 모수를 추정하는 것이 필수불가결한 단계이다. 이들 모수에 대한 다양한 추정방법들이 제안되어 왔고, 많은 연구에서 이러한 추정방법들이 갖는 성질에 대하여 연구되어져왔다. 이 연구에서는 그 적용방법이 단순하여 가장 자주 사용되는 것으로 알려진, 오일러 근사법과 신국소근사법(NLL) 그리고 일반화 적률법(GMM)과 같은 세 가지 추정방법들에 대한 통계적 성질을 검토하게 될 것이다. 모의실험연구를 통하여 오일러근사법이나 NLL방법이 GMM 방법에 비하여 훨씬 좋은 성질을 가지고 있음을 보이게 된다. 특히 GMM은 적용방법이 단순할 뿐만 아니라 강건성(robustness)이라는 좋은 성질을 가지고 있는 것으로 알려져 있어서 많은 연구에서 매우 자주 사용되는 추정방법이다. 그러나 본 연구에서 확인해 본 바와 같이 GMM은 그 사용법이 오히려 더욱 단순한 NLL이나 오일러방법에 비하여 열등한 통계적 성질을 보여주고 있었다. 특히나 확산계수에 추정모수가 포함된 경우에 GMM은 매우 좋지 못한 성질을 보이게 된다.