• Kyungpook Mathematical Journal
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• 제50권4호
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• pp.473-482
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• 2010

In the present paper we define a q-extension of the Leibniz rule for q-derivatives via Weyl type q-derivative operator. Expansions and summation formulae for the generalized basic hypergeometric functions of one and more variables are deduced as the applications of the main result.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2003년도 ICCAS
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• pp.2707-2712
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• 2003

In this paper, with the utilization of a three-dimensional version of Leibniz’s rule, the procedure of deriving the time rate of change of an energy functional for axially moving continua is investigated. It will be shown that the method in [14], which describes the way of getting the time rate of change of an energy functional in Eulerian description, and subsequent results in [10, 11] are not complete. The key point is that the time derivatives at boundaries in the Eulerian description of axially moving continua should take into account the velocity of the moving material itself. A noble way of deriving the time rate of change of the energy functional is proposed. The correctness of the proposed method has been confirmed by other approaches. Two examples, one-dimensional axially moving string and beam equations, are provided for the purpose of demonstration. The results following the procedure proposed and the results in [14] are compared.

• East Asian mathematical journal
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• 제17권1호
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• pp.135-146
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• 2001

Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

• Communications for Statistical Applications and Methods
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• 제25권4호
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• pp.411-430
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• 2018

The Bayesian approach is a suitable alternative in constructing appropriate models for observed record values because the number of these values is small. This paper provides an objective Bayesian analysis method for upper record values arising from the Rayleigh distribution. For the objective Bayesian analysis, the Fisher information matrix for unknown parameters is derived in terms of the second derivative of the log-likelihood function by using Leibniz's rule; subsequently, objective priors are provided, resulting in proper posterior distributions. We examine if these priors are the PMPs. In a simulation study, inference results under the provided priors are compared through Monte Carlo simulations. Through real data analysis, we reveal a limitation of the appropriate confidence interval based on the maximum likelihood estimator for the scale parameter and evaluate the models under the provided priors.

• 한국경영과학회지
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• 제31권2호
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• pp.129-140
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• 2006

Quantity Flexibility contract coordinates individually motivated supplier and buyer to the systemwide optimal outcome by effectively allocating the costs of market demand uncertainty. The main feature of the contract is to couple the buyer's commitment to purchase no less than a certain percentage below the forecast with the supplier's guarantee to deliver up to a certain percentage above. In this paper we refine the previous models by adding some realistic features including the upper and lower limits of the purchase. We also incorporate purchase and canceling costs in a cost function to reflect the real world contracting process more accurately. To obtain the solution of the model, we derive a condition for extreme points using the Leibniz's rule and construct an algorithm for finding the optimal solution of the model. Several examples illustrating the algorithm show that the approach is valid and efficient.

• East Asian mathematical journal
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• 제18권1호
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• pp.111-125
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• 2002

Many different families of infinite series were recently observed to be summable in closed forms by means of certain operators of fractional calculus(that is, calculus of integrals and derivatives of any arbitrary real or complex order). In this sequel to some of these recent investigations, the authors present yet another instance of applications of certain fractional calculus operators. Alternative derivations without using these fractional calculus operators are shown to lead naturally a family of analogous infinite sums involving hypergeometric functions.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제8권2호
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• pp.127-135
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• 2001

The main object of this paper is to present a transformation formula for a finite series involving $_3F_2$ and some identities associated with the binomial coefficients by making use of the theory of Legendre polynomials $P_{n}$(x) and some summation theorems for hypergeometric functions $_pF_q$. Some integral formulas are also considered.

• 한국해양환경ㆍ에너지학회지
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• 제12권3호
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• pp.165-172
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• 2009

연직 2차원 평면을 대상으로 하는 연속방정식을 수심방향으로 이중적분 하여 수평1차원 파랑방정식을 구하였다. 새 방정식은 복소수 포텐셜 함수로 구성되어 있으며, 파랑의 진폭과 위상경사함수를 도입하여 한 세트의 실수방정식으로도 변형되었다. 파랑진폭과 위상경사함수를 포함한 한 세트의 식은 각각 1차, 2차 상미분방정식이며, 한쪽 경계에서 경계조건을 적절히 지정하여 전 영역에서의 해를 한 방향으로 진행하면서 구할 수 있다. 이때 경계조건으로는 파랑진폭 값, 파랑진폭의 경사, 위상 경사 값이다. 단순한 중앙차분식을 이용하여 식을 차분화 하였다. 새 방정식을 Booij의 경사판, Massel의 부드러운 저면, Bragg의 싸인 함수의 저면에 대하여 적용하여 보았다. 본 방정식은 Massel의 수정완경사방정식, Berkhoff의 완경사방정식, 완전 선형방정식과 비교하여 유사한 결과를 나타내었으며, 유용함을 보였다.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제31권2호
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• pp.139-152
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• 2017

본 논문에서 미적분을 다루는 중등교과서가 미적분 역사를 어떻게 소개하고 있는지를 알아보았다. 문제점을 파악하기 위하여 우리는 기원전 350~기원전 50년에 목성의 위치를 계산하기 위하여 이루어진 바빌로니아인의 사다리꼴을 사용한 구분구적법 그리고 1000년경 이집트에서 이루어진 이븐 알 하이탐(ibn al-Haytham)의 원판을 이용한 구분구적법 등을 고찰하였다. 이를 바탕으로 미적분 역사에 대한 건설적인 서술 방안을 제시하였다. 결론적으로 우리나라 중등수학 교과서는 뉴턴과 라이프니츠가 미적분을 창안한 것으로 설명하고 그 뿌리를 고대 그리스에 둔다. 미적분의 창안은 바빌로니아와 파티마 왕조(Faṭimah Dynasty: 909-1171)(이집트)에 있으며 인도에서 멱급수의 발전이 이루어진 후 미적분이 유럽에서 발전된 것으로 교과서에 아시아 아프리카의 가치가 소개되는 것이 바람직하다.

• Structural Engineering and Mechanics
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• 제83권3호
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• pp.293-304
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• 2022

In this work, a novel reliability approach for combined high and low cycle fatigue (CCF) estimation is developed by combining active learning strategy with least squares support vector machines (LS-SVM) (named as ALS-SVM) surrogate model to address the multi-resources uncertainties, including working loads, material properties and model itself. Initially, a new active learner function combining LS-SVM approach with Monte Carlo simulation (MCS) is presented to improve computational efficiency with fewer calls to the performance function. To consider the uncertainty of surrogate model at candidate sample points, the learning function employs k-fold cross validation method and introduces the predicted variance to sequentially select sampling. Following that, low cycle fatigue (LCF) loads and high cycle fatigue (HCF) loads are firstly estimated based on the training samples extracted from finite element (FE) simulations, and their simulated responses together with the sample points of model parameters in Coffin-Manson formula are selected as the MC samples to establish ALS-SVM model. In this analysis, the MC samples are substituted to predict the CCF reliability of turbine blades by using the built ALS-SVM model. Through the comparison of the two approaches, it is indicated that the reliability model by linear cumulative damage rule provides a non-conservative result compared with that by the proposed one. In addition, the results demonstrate that ALS-SVM is an effective analysis method holding high computational efficiency with small training samples to gain accurate fatigue reliability.