• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.537-547
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• 2012

In this paper, we define a $p$, $q$-difference operator and obtain an explicit formula which is used to express the $p$, $q$-analogue of the unified generalization of Stirling numbers and its exponential generating function in terms of the $p$, $q$-difference operator. Explicit formulas for the non-central $q$-Stirling numbers of the second kind and non-central $q$-Lah numbers are derived using the new $q$-analogue of Newton's interpolation formula. Moreover, a $p$, $q$-analogue of Newton's interpolation formula is established.

• Honam Mathematical Journal
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• v.26 no.3
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• pp.281-286
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• 2004

We calculate the harmonic series $E_{2p}:=^P\;_{^{\infty}_{n=1}}{\frac{1}{n^{2p}}$ via Waring's formula.

• Tribology and Lubricants
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• v.12 no.3
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• pp.115-122
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• 1996

In this paper, a finite difference method and the Newton-Raphson method are used to evaluate the Hamrock and Dowson's EHL film thickness formulas in elliptical contact problems. The minimum and central film thicknesses are compared with the Hamrock and Dowson's numerical results for various dimensionless parameters and with their film thickness formulas. The results of present analysis are more accurate and physically reasonable. The minimum film thickness formula is similar with the Hamrock and Dowson's results, however, the central film thickness formula shows large differences. Therefore, the Hamrock and Dowson's central film thickness formula should be replaced by following equation. $H_{c} = 4.88U^{0.68}G^{0.44}W^{0.096}(1-0.58e^{-0.60k})$ More accurate film thickness formula for general elliptical contact problems can be expected using present numerical methods and further research should be required.

• Korean Journal of Mathematics
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• v.32 no.2
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• pp.349-364
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• 2024

We propose a new method of investigation of an integral equality associated with tempered fractional integrals. In addition to this, several Newton-type inequalities are considered for differentiable convex functions by taking the modulus of the newly established identity. Moreover, we establish some Newton-type inequalities with the help of Hölder and power-mean inequality. Furthermore, several new results are presented by using special choices of obtained inequalities.

• Journal of the Society of Naval Architects of Korea
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• v.45 no.4
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• pp.389-395
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• 2008

It is very well known that the natural frequency of an oscillating body on the free surface is determinable only after the added mass is given. However, it is hard to find analytical investigations in which actually the natural frequency is obtained. Difficulties arise from the fact that in order to determine the natural frequency we need to compute the added mass at least for a range of frequencies, and to solve an equation where the frequency is a variable. In this study, first, a formula is obtained for the added mass, and then an equation for finding the natural frequency is defined and solved by Newton's iteration. It is confirmed that the formula shows a good agreement with the results given by Ursell(1949), and the value of natural frequency is reduced by 21.5% compared to the pre-natural frequency, which is obtained without considering the effect of added mass.

• Kyungpook Mathematical Journal
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• v.52 no.3
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• pp.347-357
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• 2012

When Newton's method, or Halley's method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

• Honam Mathematical Journal
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• v.45 no.1
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• pp.160-183
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• 2023

In the present paper, we prove that our main inequality reduces to some trapezoid and Newton type inequalities for differentiable s-convex functions. These inequalities are established by using the well-known Riemann-Liouville fractional integrals. With the help of special cases of our main results, we also present some new and previously obtained trapezoid and Newton type inequalities.

• Kyungpook Mathematical Journal
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• v.34 no.2
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• pp.207-216
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• 1994

• Journal for History of Mathematics
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• v.27 no.2
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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.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.229-243
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• 2019

We use the so-called pseudoinversion of degenerate metrics technique on foliated compact null hypersurface, $M^{n+1}$, in Lorentzian manifold ${\overline{M}}^{n+2}$, to derive an integral formula involving the r-th order mean curvatures of its foliations, ${\mathcal{F}}^n$. We apply our formula to minimal foliations, showing that, under certain geometric conditions, they are isomorphic to n-dimensional spheres. We also use the formula to deduce expressions for total mean curvatures of such foliations.