• Journal for History of Mathematics
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• v.23 no.3
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• pp.21-31
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• 2010

A latin square of order n is an $n{\times}n$ array with entries from a set of n numbers arrange in such a way that each number occurs exactly once in each row and exactly once in each column. Two latin squares of the same order are orthogonal latin square if the two latin squares are superimposed, then the $n^2$ cells contain each pair consisting of a number from the first square and a number from the second. In Europe, Orthogonal Latin squares are the mathematical concepts attributed to Euler. However, an Euler square of order nine was already in existence prior to Euler in Korea. It appeared in the monograph Koo-Soo-Ryak written by Choi Seok-Jeong(1646-1715). He construct a magic square by using two orthogonal latin squares for the first time in the world. In this paper, we explain Choi' s orthogonal latin squares and the history of the Orthogonal Latin squares.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1981-1988
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• 2013

We say a positive integer n satisfies the Lehmer property if ${\phi}(n)$ divides n - 1, where ${\phi}(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime p and a positive integer n, or of the form ${\alpha}2^{\beta}+1$ for ${\alpha}{\leq}{\beta}$ does not satisfy the Lehmer property.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.263-273
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• 2014

Ever since Euler solved the so-called Basler problem of ${\zeta}(2)=\sum_{n=1}^{\infty}1/n^2$, numerous evaluations of ${\zeta}(2n)$ ($n{\in}\mathbb{N}$) as well as ${\zeta}(2)$ have been presented. Very recently, Ritelli [61] used a double integral to evaluate ${\zeta}(2)$. Modifying mainly Ritelli's double integral, here, we aim at evaluating certain interesting alternating series.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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• pp.243-249
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• 2015

Let X be a closed surface for which the Euler characteristic $_{\mathcal{X}}(X)$ is negative, and let $f:X{\rightarrow}X$ be a self-map that is not surjective. In this short paper, we prove that we can compute the Nielsen number of f, N(f), under some algebraic conditions.

• East Asian mathematical journal
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• v.29 no.5
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• pp.545-550
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• 2013

A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.321-348
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• 2009

In this paper, we give two lower bounds on the diameter of the boundary slope set of a Montesinos knot. One is described in terms of the minimal crossing numbers of the knots, and the other is related to the Euler characteristics of essential surfaces with the maximal/minimal boundary slopes.

• Journal of applied mathematics & informatics
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• v.37 no.3_4
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• pp.317-328
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• 2019

This paper defines Carlitz's type (p, q)-Genocchi polynomials and Carlitz's type (h, p, q)-Genocchi polynomials, and explains fourteen properties which can be complemented by Carlitz's type (p, q)-Genocchi polynomials and Carlitz's type (h, p, q)-Genocchi polynomials, including distribution relation, symmetric property, and property of complement. Also, it explores alternating powers sums by proving symmetric property related to Carlitz's type (p, q)-Genocchi polynomials.

• Journal of Mechanical Science and Technology
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• v.17 no.4
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• pp.590-598
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• 2003

A computational aero-acoustic (CAA) method is used to predict the tonal noise generated from a cavity of automobile door seals or gaps at low flow Mach numbers (A$\_$$\infty$/=0.077 and 0.147) In the present method, the acoustically perturbed Euler equations are solved with the acoustic source term obtained from the unsteady incompressible Navier-Stokes calculations of the cavity flow in self-sustained oscillations. The aerodynamic and acoustic fields are computed for the Reynolds numbers based on the displacement thickness, Re$\_$$\delta$*/=850 and 1620 and their fundamental mode characteristics are investigated. The present method is also verified with the experimentally measured sound pressure level (SPL) spectra.

• Journal of KSNVE
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• v.6 no.5
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• pp.607-616
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• 1996

A combined computational fluid dynamics(CFD)-Kirchhoff method is presented for predicting high-speed impulsive noise generated by a hovering blade. Two types of Kirchhoff integral formula are used; one for the classical linear Kirchhoff formulation and the other for the nonlinear Kirchhoff formulation. An Euler finite difference solver is solved first to obtain the flow field close to the blade, and then this flow field is used as an input to a Kirchhoff formulation to predict the acoustic far-field. These formulas are used at Mach numbers of 0.90 and 0.95 to investigate the effectiveness of the linear and nonlinear Kirchhoff formulas for delocalized flow. During these calculiations, the retarded time equation is also carefully examined, in particular, for the cases of the control surface located outside of the sonic cylinder, where multiple roots are obtained. Predicted results of acoustic far-field pressure with the linear Kirchhoff formulation agree well with experimental data when the control surface is at the certain location(R=1.46), but the correlation is getting worse before or after this specific location of the control surface due to the delocalized nonlinear aerodynamic flow field. Calculations based on the nonlinear Kirchhoff equation using a linear sonic cylinder as a control surface show a reasonable agreement with experimental data in negative amplitudes for both tip Mach numbers of 0.90 and 0.95, except some computational integration problems over a shock. This concliudes that a nonlinear formulation is necessary if the control surface is close to the blade and the flow is delocalized.

• 한국전산유체공학회:학술대회논문집
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• 2008.03b
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• pp.354-357
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• 2008

Drag reduction is one of main concerns for commercial aircraft companies than ever because fuel price has been tripled in ten years. In this research, Natural Laminar Flow airfoil is designed and tested to reduce drag at cruise condition, $c_l$=0.3, Re=3.4${\times}$10$^6$ and M=0.6. NLF airfoil is characterized by delayed transition from laminar to turbulent flow, which comes from maintaining favorable pressure gradient to downstream. Transition is predicted by solving Boundary Layer equations in viscous boundary layer and by solving Euler Equation outside the boundary layer. Once boundary layer thickness and momentum thickness are obtained, $e^N$-method is used for transition point prediction. As results, KARI's NLF airfoil is designed and shows better characteristics than NLF-0115. The characteristics are tested and verified at low Reynolds numbers, but at high Reynolds numbers, laminar flow characteristics are not obtainable because of fully turbulent flow over airfoil surfaces. Precious experiences, however, relating NLF airfoil design, subsonic and transonic tests are acquired.