• Journal of the Korean Mathematical Society
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• v.38 no.2
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• pp.275-281
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• 2001

The trajectory of a classical dynamics is determined by the least action principle. As soon as we come to quantum dynamics, we have to consider all possible trajectories which are proposed to be a sum of the classical trajectory and Brownian fluctuation. Thus, the action involves the square of the derivative B(t) (white noise) of a Brownian motion B(t). The square is a typical example of a generalized white noise functional. The Feynman propagator should therefore be an average of a certain generalized white noise functional. This idea can be applied to a large class of dynamics with various kinds of Lagrangians.

• Korean Journal of Mathematics
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• v.8 no.2
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• pp.163-172
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• 2000

Let $\mathcal{S}^{\prime}(\mathbb{R})$ be the dual of the Schwartz spaces $\mathcal{S}(\mathbb{R})$), A be a self-adjoint operator in $L^2(\mathbb{R})$ and ${\Gamma}(A)^*$ be the adjoint operator of ${\Gamma}(A)$ which is the second quantization operator of A. It is proven that under a suitable condition on A there exists a nuclear subspace $\mathcal{S}$ of a fundamental space $\mathcal{S}_A$ of Hida's type on $\mathcal{S}^{\prime}(\mathbb{R})$) such that ${\Gamma}(A)\mathcal{S}{\subset}\mathcal{S}$ and $e^{-t{\Gamma}(A)}\mathcal{S}{\subset}\mathcal{S}$, which enables us to show that a stochastic differential equation: $$dX(t)=dW(t)-{\Gamma}(A)^*X(t)dt$$, arising from the central limit theorem for spatially extended neurons has an unique solution on the dual space $\mathcal{S}^{\prime}$ of $\mathcal{S}$.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.993-1008
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• 1996

The white noise analysis, initiated by Hida [3] in 1975, has been developed to an infinite dimensional distribution theory on Gaussian space $(E^*, \mu)$ as an infinite dimensional analogue of Schwartz distribution theory on Euclidean space with Legesgue measure. The mathematical framework of white noise analysis is the Gel'fand triple $(E) \subset (L^2) \subset (E)^*$ over $(E^*, \mu)$ where $\mu$ is the standard Gaussian measure associated with a Gel'fand triple $E \subset H \subset E^*$.

• Journal of The Korean Astronomical Society
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• v.23 no.1
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• pp.31-42
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• 1990

Fine structures of a quiescent prominence are studied by analyzing high resolution H alpha filtergrams and H alpha line spectra observed at the Hida Observatory of Kyoto University. We have found two kinds of downward motions in the prominence. One of them is a movement with a constant acceleration below the solar gravity(${\simeq}1/4g_s$) and the other with an uniform velocity(${\simeq}16Km/s$). The average life time and the size of prominence knots are estimated to be about 7 minutes and 4000Km, respectively. Spatial and brightness distribution of knots are also presented in this paper. With the analytical solutions derived from magnetostatic equilibrium in the prominence, we have examined the filamentary structure based on the Kippenhahn-Schluter model. Sag angles of the magnetic fields supporting the prominence matter are predicted from the observed density profile.