• Journal of applied mathematics & informatics
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• 제8권1호
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• pp.231-242
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• 2001

We showed in [2] that if r≤2, zero mean Gaussian of average error of the Trapezoidal rule is proportional to h/sub i//sup 2r+3/ on the interval [0,1]. In this paper, if r≥3, we show that zero mean Gaussian of average error of the Trapezoidal rule is bounded by Ch⁴/sub i/h⁴/sub j/.

• 대한수학회지
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• 제33권2호
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• pp.235-247
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• 1996

Many numerical computations in science and engineering can only be solved approximately since the available infomation is partial. For instance, for problems defined ona space of functions, information about f is typically provided by few function values, $N(f) = [f(x_1), f(x_2), \ldots, f(x_n)]$. Knwing N(f), the solution is approximated by a numerical method. The error between the true and the approximate solutions can be reduced by acquiring more information. However, this increases the cost. Hence there is a trade-off between the error and the cost.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.365-372
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• 2003

We show that if ${\gamma}$ $\leq$ 2, the average error of the composite Trapezoidal rule on two consecutive intervals is proportional to h$\^$2h+3/ where h is the length of each subinterval of the interval [0, 1]. As a result, we show that the Trapezoidal rule with equally spaced points is optimal in the average case setting when ${\gamma}$ $\leq$ 2.

• Journal of applied mathematics & informatics
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• 제12권1_2호
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• pp.107-117
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• 2003

We study the integration problem in which one wants to compute the approximation to the definite integral in the average case setting. We choose the composite Newton- Cotes quadratures as our algorithm and the function values at equally spaced sample points on the given interval［0, 1］as information. We compute the average case error of composite Newton-Cotes quadratures and show that it is minimal (modulo a multi-plicative constant).

• Communications for Statistical Applications and Methods
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• 제11권3호
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• pp.495-501
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• 2004

HJM representation of the term structure of interest rates sometimes produces the negative interest rates with positive probability. This paper shows that the condition of positive interest rates can be derived from the jump diffusion process, if a proper positive martingale process with the compensated jump process is chosen. As in Flesaker and Hughston, the condition is incorporated into the bond price process.

• 대한수학회논문집
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• 제16권4호
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• pp.691-701
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• 2001

We computed zero mean Gaussian of average error bounds pf Simpsons quadrature with convariances in [2]. In this paper, we compute zero mean Gaussian of average error bounds between Simpsons quadrature and composite Simpsons quadra-ture on four consecutive subintervals. The reason why we compute these on subintervals is because these results enable us to compute a posteriori error bounds on the whole interval in the later paper.

• 충청수학회지
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• 제26권2호
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• pp.323-342
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• 2013

Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $Xn:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}:C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\cdots},x(t_n),x(t_{n+1}))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions which have the form $${\int}_{L_2[0,t]}{{\exp}\{i(v,x)\}d{\sigma}(v)}{{\int}_{\mathbb{R}^r}}\;{\exp}\{i{\sum_{j=1}^{r}z_j(v_j,x)\}dp(z_1,{\cdots},z_r)$$ for $x{\in}C[0,t]$, where $\{v_1,{\cdots},v_r\}$ is an orthonormal subset of $L_2[0,t]$ and ${\sigma}$ and ${\rho}$ are the complex Borel measures of bounded variations on $L_2[0,t]$ and $\mathbb{R}^r$, respectively. We then investigate the inverse transforms of the function with their relationships and finally prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transforms of each function.

• 대한수학회논문집
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• 제35권3호
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• pp.809-823
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• 2020

Let C[0, T] denote the space of continuous, real-valued functions on the interval [0, T] and let C₀[0, T] be the space of functions x in C[0, T] with x(0) = 0. In this paper, we introduce a Banach algebra ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ on C[0, T] and its equivalent space ${\bar{\mathcal{F}}}({\mathcal{H}}) $, a space of transforms of equivalence classes of measures, which generalizes Fresnel class 𝓕(𝓗), where 𝓗 is an appropriate real separable Hilbert space of functions on [0, T]. We also investigate their properties and derive an isomorphism between ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ and ${\bar{\mathcal{F}}}({\mathcal{H}}) $. When C[0, T] is replaced by C₀[0, T], ${\bar{\mathcal{F}}}({\mathcal{H}}) $ and ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ reduce to 𝓕(𝓗) and Cameron-Storvick's Banach algebra 𝓢, respectively, which is the space of generalized Fourier-Stieltjes transforms of the complex-valued, finite Borel measures on L²[0, T].

• 전기학회논문지
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• 제62권2호
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• pp.239-245
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• 2013

In this paper, we propose a time-frequency domain reflectometry (TFDR) based measurement method for localizing concentric neutrals corrosion on live underground power cable. It consists of two inductive couplers which can transmit the reference signal into live underground power cable and measure the reflected signals from the impedance discontinuities of concentric neutrals corrosion. In order to compensate the dispersion of the measured reflected signal via inductive coupler, an equalizer based on Wiener filtering is designed. To improve the localizing performance of concentric neutrals corrosion in the vicinity of the measurement point, the reference signal is removed from the measured reflected signals. The localization performance of the proposed method is verified by the concentric neutrals corrosion localization experiment.

• 대한수학회논문집
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• 제12권2호
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• pp.293-303
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• 1997

The existence theorem for the operator valued function space integral has been studied, when the wave function was in $L_1(R)$ class and the potential energy function was represented as a double integra [4]. Johnson and Lapidus established the existence theorem for the operator valued function space integral, when the wave function was in $L_2(R)$ class and the potential energy function was represented as an integral involving a Borel measure [9]. In this paper, we establish the existence theorem for the operator valued function we establish the existence theorem for the operator valued function space integral as an operator from $L_1(R)$ to $L_\infty(R)$ for certain potential energy functions which involve double integrals with some Borel measures.