• The Pure and Applied Mathematics
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• v.18 no.4
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• pp.345-351
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• 2011

Taylor's formula is a powerful tool in analysis. In this study, we assume that an operator is m-times Fr$\acute{e}$chet-differentiable and satisfies a Lipschitz condition. We then obtain some Taylor formulas using only the Lipschitz constants. Applications are also provided.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.2
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• pp.43-52
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• 2002

By the use of Taylor's expansion formula with the integral remainder, an accurate approximation for the fiber refractive index profile is given.

• East Asian mathematical journal
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• v.17 no.2
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• pp.197-215
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• 2001

A generalised Taylor series with integral remainder involving a convex combination of the end points of the interval under consideration is investigated. Perturbed generalised Taylor series are bounded in terms of Lebesgue p-norms on $[a,b]^2$ for $f_{\Delta}:[a,b]^2{\rightarrow}\mathbb{R}$ with $f_{\Delta}(t,s)=f(t)-f(s)$.

• Journal of Applied and Pure Mathematics
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• v.5 no.5_6
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• pp.407-414
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• 2023

In this article we continue the study of smooth Picard singular integral operators that started in [2], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor's formula. We establish the convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive. Our results are pointwise and uniform.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.20 no.3
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• pp.277-290
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• 2008

In order to apply UTM coordinates conversion in zones larger than $14^{\circ}$ wide, a new conversion formula, based on the 12th expansion of Taylor series, is derived which is shown to be an extension of Thomas' formula(1952). Some examples of coordinate conversion between WGS84 and UTM are presented and convergences of computational results are also tested according to the order of formula. The present conversion formula can be used to make rectangular coordinate grid systems for numerical models to compute long wave propagation such as tide or tsunami around Korea.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.2
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• pp.1-15
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• 2005

In this article, a generalisation of Sard's inequality for Appell polynomials is obtained. Estimates for the remainder are also provided.

• Communications for Statistical Applications and Methods
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• v.21 no.3
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• pp.253-260
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• 2014

The intraclass correlation model has a long history of applications in several fields of research. Case deletion diagnostic methods for the intraclass correlation model are proposed. Based on the likelihood equations, we derive a formula for a case deletion diagnostic method which enables us to investigate the influence of observations on the maximum likelihood estimates of the model parameters. Using the Taylor series expansion we develop an approximation to the likelihood distance. Numerical examples are provided for illustration.

• Communications of the Korean Mathematical Society
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• v.29 no.2
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• pp.269-283
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• 2014

The aim of the paper is to present several new relationships involving the hyperharmonic function introduced by Mez$\ddot{o}$ in (I. Mez$\ddot{o}$, Analytic extension of hyperharmonic numbers, Online J. Anal. Comb. 4, 2009) which is an analytic extension of the hyperharmonic numbers. These relations are obtained by using some fractional calculus theorems as Leibniz rules and Taylor like series expansions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.3
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• pp.180-193
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• 2023

Approximating the implied volatilities and estimating the model parameters are important topics in quantitative finance. This study proposes an approximation formula for short-maturity near-the-money implied volatilities in stochastic volatility models. A general second-order nonlinear PDE for implied volatility is derived in terms of time-to-maturity and log-moneyness from the Feyman-Kac formula. Using regularity conditions and the Taylor expansion, an approximation formula for implied volatility is obtained for short-maturity nearthe-money call options in two stochastic volatility models: Heston model and SABR model. In addition, we proposed a novel numerical method to estimate model parameters. This method reduces the number of model parameters that should be estimated. Generating sample data on log-moneyness, time-to-maturity, and implied volatility, we estimate the model parameters fitting the sample data in the above two models. Our method provides parameter estimates that are close to true values.

• Transactions of the Korean Society of Mechanical Engineers
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• v.11 no.1
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• pp.154-159
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• 1987

In the previously reported Part I, the behavior of the flank wear for carbide tool was studied as a preceeding step to present a simple method for Quick Tool-Life Testing, and the following general equation was obtained $W_{f}$ =(a+bt) $V^{m}$ . In this study the following step-cutting formula for the constants a, b and m in the above general model is derived by using step-cutting data [a numerical formula] To check the validity of the above formula, the comparison is made between the tool-life equation inferred in this method and that inferred in the conventional tool-life testing method, when the wear criterion is 0.3mm. The equation obtained in the present method is V(T')$^{0.57}$=1763 whereas the equation obtained in the conventional tool-life testing method is V(T)$^{0.56}$=1605 The results of the above two formula are satisfactory and also verify the validity of the present research.earch.