• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1909-1920
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• 2018

In the paper, the authors find a common solution to three series of differential equations related to the generating function of the Bernoulli numbers of the second kind and present a recurrence relation, an explicit formula in terms of the Stirling numbers of the first kind, and a determinantal expression for the Bernoulli numbers of the second kind.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.549-555
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• 2021

We show the solvability of the ordinary differential equations describing curves on sphere or hyperbolic space. Making use of the geometric properties of the equations, we derive explicit solution formulae.

• Bulletin of the Society of Naval Architects of Korea
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• v.23 no.3
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• pp.10-16
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• 1986

This paper deals with a simple, approximate formula to predict the natural frequencies of the sagged cable structures. Assuming that the propagation velocity of the lateral wave is dependent only on the local mass per unit length and local tension, the explicit simple formula to predict the fundamental period is newly derived. The modified form of these formula is also presented for the prediction of the fundamental period of general shaped cable structures. The results of comparison shows fairly good agreements with experimental results and with theoretical ones. This formula is also used to predict the natural frequencies of a long vertical cable and the derived approximate formula in that case, becomes identical to the exact solution.

• Industrial Engineering and Management Systems
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• v.4 no.2
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• pp.123-128
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• 2005

The author derives a general explicit formula and presents an heuristic algorithm for solving Baker’s model. The examples show that this new approximate solution procedure for determining near optimum inspection intervals is more accurate than the ones suggested by Chung (1993) and Vaurio (1994), and is more efficient computationally than the one suggested by Hariga (1996). The construction and solution of the simplest profit model for an exponential failure distribution were presented in Baker (1990), and approximate analytical results were obtained by Chung (1993) and Vaurio (1994). The author will therefore mainly devote the following discussion to the problem of further approximating optimum inspection intervals.

• Journal of the Korean Statistical Society
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• v.11 no.1
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• pp.59-68
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• 1982

We propose that, in order to control the inflation and general instability associated with the least squares estimates, we can use the ridge estimator $$ \hat{B}^* = (X'X+kI)^{-1}X'Y : k \leq 0$$ for the regression coefficients B in multivariate regression. Our hope is that by accepting some bias, we can achieve a larger reduction in variance. We show that such a k always exists and we derive the formula obtaining k in multivariate ridge regression.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.507-516
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• 2016

In this paper, we introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation $$x_{n+1}={\frac{ax_{n-3}}{b-cx_{n-1}x_{n-3}}}$$, $n=0,1,{\ldots}$ where a, b, c are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.471-479
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• 2008

Let $R\;{\in}\;C^{m{\times}m}$ and $S\;{\in}\;C^{n{\times}n}$ be nontrivial unitary involutions, i.e., $R^*\;=\;R\;=\;R^{-1}\;{\neq}\;I_m$ and $S^*\;=\;S\;=\;S^{-1}\;{\neq}\;I_m$. We say that $G\;{\in}\;C^{m{\times}n}$ is a generalized reflexive matrix if RGS = G. The set of all m ${\times}$ n generalized reflexive matrices is denoted by $GRC^{m{\times}n}$. In this paper, an efficient method for the least squares solution $X\;{\in}\;GRC^{m{\times}n}$ of the matrix equation AXB = D with arbitrary coefficient matrices $A\;{\in}\;C^{p{\times}m}$, $B\;{\in}\;C^{n{\times}q}$and the right-hand side $D\;{\in}\;C^{p{\times}q}$ is developed based on the canonical correlation decomposition(CCD) and, an explicit formula for the general solution is presented.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1497-1530
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• 2016

External barrier options are two-asset options with stochastic variables where the payoff depends on one underlying asset and the barrier depends on another state variable. The barrier state variable determines whether the option is knocked in or out when the value of the variable is above or below some prescribed barrier level. This paper derives the explicit analytic solution of the chained option with an external single or double barrier by utilizing the probabilistic methods - the reflection principle and the change of measure. Before we do this, we examine the closed-form solution of the external barrier option with a single or double-curved barrier using the methods of image and double Mellin transforms. The exact solution of the external barrier option price enables us to obtain the pricing formula of the chained option with the external barrier more easily.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.4C
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• pp.377-385
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• 2010

In this paper, a general and explicit technique is presented for determining the filter coefficients of maximally flat (MAXFLAT) FIR filter with the magnitude response exactly passing through a prescribed cutoff point. This technique is based on a general formula (i.e. impulse response) with an arbitrary cutoff point and permits direct computation of the coefficients of this filter type with a specified cutoff point. The technique provides an explicit method for choosing the order of flatness of the filter with the specified cutoff point. Also, in the paper, it is shown to give a computationally efficient and accurate solution to the design of the filters with the desired cutoff point.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.25 no.4A
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• pp.489-495
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• 2000

In this paper we propose an approximate method to estimate the cell loss probability(CLP) due to buffer overflow in ATM networks. The main idea is to relate the buffer capacity with the CLP target in explicit formula by using the approximate upper bound for the tail distribution of a queue. The significance of the proposition lies in the fact that we can obtain the expected CLP by using only the source traffic data represented by mean rate and its variance. To that purpose we consider the problem of estimating the cell loss measures form the statistical viewpoint such that the probability of cell loss due to buffer overflow does not exceed a target value. In obtaining the exact solution we use a typical matrix analytic method for GI/D/1B queue where B is the queue size. Finally, in order to investigate the accuracy of the result, we present both the approximate and exact results of the numerical computation and give some discussion.