• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.709-723
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• 2016

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.703-715
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• 2006

Recently, we proved the existence theorem of measure-valued Feynman-Kac formula and showed that it satisfied Volterra's integral equation. In this paper, we establish the operational calculus for a measure-valued Dyson series and give some examples related to the measure-valued Dyson series.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.477-486
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• 2007

This paper is the first in a series in which we consider bilinear integration with respect to measure-valued measure. We use the integration techniques to establish generalized Egorov theorem and Vitali theorem.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.317-334
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• 1997

The existence of an operator-valued function space integral as an operator on $L_p(R) (1 \leq p \leq 2)$ was established for certain functionals which involved the Labesgue measure [1,2,6,7]. Johnson and Lapidus showed the existence of the integral as an operator on $L_2(R)$ for certain functionals which involved any Borel measures [5]. J. S. Chang and Johnson proved the existence of the integral as an operator from L_1(R)$ to $C_0(R)$ for certain functionals involving some Borel measures [3]. K. S. Chang and K. S. Ryu showed the existence of the integral as an operator from $L_p(R) to L_p'(R)$ for certain functionals involving some Borel measures [4].

• Journal of the Korean Chemical Society
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• v.21 no.1
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• pp.23-31
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• 1977

For the metal complex of $d^1$ configuration with the octahedrally coordinated ligands, the crystal field parameter, 10Dq, is calculated from first principles within the framework of the crystal field theory. With the point charge model, the configuration interaction is introduced by use of the Shull-L$\"{o}$wdin functions. Through the Integral Hellmann-Feynman Theorem, the higher order effect is visualized. It is found that the higher order effect on 10Dq is about $50{\%}$ of the first order effect. Since 3d function is angularly undistorted and radially equally distorted in $E_g\;and\;T_{2g}$ states, due to the octahedral potential, the calculated 10Dq is still the unique parameter for the splitting.

• Journal of the Korean Society of Earth Science Education
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• v.16 no.1
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• pp.72-86
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• 2023

The purpose of this study is to examine what epistemological beliefs pre-service teachers have about science depending on the situation, and to explore in-depth changes in epistemological beliefs through disciplinary reading. For this purpose, 77 essays written by pre-service elementary school teachers after reading Feynman's 'the meaning of it all' were analyzed using an inductive analysis method. As a result of the study, the epistemological beliefs of pre-service teachers were divided into two situations: 'science in subject learning' and 'science in daily life', and the epistemological beliefs formed in the 'science handled by scientists' situation were analyzed after reading the book. Each situation was divided into sub-categories of 'Impression of Knowledge', 'Source of Knowledge', 'Justification of Knowledge', 'Variability of Knowledge', 'Structure of Knowledge', and 'Value of Knowledge Acquisition' to reveal differences in sophisticated beliefs and naive belief levels. As a result, it was derived that Feynman's science lecture influenced pre-service teachers in terms of establishing new perspectives and recontextualizing existing epistemological beliefs. This study is meaningful in that pre-service teachers' scientific epistemological beliefs may vary depending on the situation, and that the scope and depth of epistemological beliefs may be expanded to include scientists' beliefs in science through disciplinary reading.

• Journal of the Korean Mathematical Society
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• v.38 no.2
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• pp.337-348
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• 2001

The unitary Lie-Trotter-Kato product formula gives in a simplest way a meaning to the Feynman path integral for the Schroding-er equation. In this note we want to survey some of recent results on the norm convergence of the selfadjoint Lie-Trotter Kato product formula for the Schrodinger operator -1/2Δ + V(x) and for the sum of two selfadjoint operators A and B. As one of the applications, a remark is mentioned about an approximation therewith to the fundamental solution for the imaginary-time Schrodinger equation.

• Journal of the Korean Mathematical Society
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• v.33 no.2
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• pp.269-282
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• 1996

It has long been known that Wiener measure and Wiener measurbility behave badly under the change of scale transformation [3] and under translation [2]. However, Cameron and Storvick [4] obtained the fact that the analytic Feynman integral was expressed as a limit of Wiener integrals for a rather larger class of functionals on a classical Wienrer space.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.289-295
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• 2012

In this note we prove the space (A, ${\parallel}.{\parallel}$) is a Banach space and ${\parallel}ab{\parallel}{\leq}{\parallel}a{\parallel}{\parallel}b{\parallel}$ for $a,b{\in}A$ where $A:=\{a:=(a_t)_{t{\in}G}:{\sum}_{t{\in}G}{\parallel}a_t{\parallel}_t<{\infty}\}$, $G=\mathbb{N}^*$. Also we show some property in (A, ${\parallel}.{\parallel}$).

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.521-537
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• 1994

We consider the Schrodinger equation of quantum mechanics $$ i\hbar\frac{\partial t}{\partial}\Gamma(t, \vec{\eta}) = -\frac{2m}{\hbar}\Delta(t, \vec{\eta}) + V(\vec{\eta}\Gamma(t, \vec{\eta}) (1.1) $$ $$ \Gamma(0, \vec{\eta}) = \psi(\vec{\eta}), \vec{\eta} \in R^n $$ where $\Delta$ is the Laplacian on $R^n$, $\hbar$ is Plank's constant and V is a suitable potential.