• Korean Journal of Mathematics
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• v.7 no.2
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• pp.183-198
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• 1999

Let $\mathcal{F}(B)$ be the Fresnel class on an abstract Wiener space (B, H, ${\omega}$) which consists of functionals F of the form : $$F(x)={\int}_H\;{\exp}\{i(h,x)^{\sim}\}df(h),\;x{\in}B$$ where $({\cdot}{\cdot})^{\sim}$ is a stochastic inner product between H and B, and $f$ is in $\mathcal{M}(H)$, the space of all complex-valued countably additive Borel measures on H. We introduce the concepts of an $L_p$ analytic Fourier-Feynman transform ($1{\leq}p{\leq}2$) and a convolution product on $\mathcal{F}(B)$ and verify the existence of the $L_p$ analytic Fourier-Feynman transforms for functionls in $\mathcal{F}(B)$. Moreover, we verify that the Fresnel class $\mathcal{F}(B)$ is closed under the $L_p$ analytic Fourier-Feynman transform and the convolution product, respectively. And we investigate some interesting properties for the $n$-repeated $L_p$ analytic Fourier-Feynman transform on $\mathcal{F}(B)$. Finally, we show that several results in [9] come from our results in Section 3.

• Korean Institute of Interior Design Journal
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• v.16 no.2 s.61
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• pp.87-96
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• 2007

This study analyzes the place of modem architecture based on the place theory of C. N, Schultz. For applying Schultz's theory to the modern architecture, It is required to examine the modern cityscape, features of inner space of architecture and features of program. By analyzing the avant-garde architecture of Rem Koolhaas on such basis, the potentiality of placeness of modern architecture could be verified and the alternatives would be searched. It is inferred that the placeness features of Rem Koolhaas' public architecture is under the influence of the interpretation of program based on the humane background rather than the physical aspects of surroundings. The inner space shows the non-linear features, the metaphor of city. The obscurity of physical boundary illustrates the flexible features with ambiguous boundary. Consequently, the inner space expresses the surreal atmosphere that doesn't match the purposes of usage of architecture, the traditional concept. The outer shape is recognized as the by-product from the interpretation of internal program rather than it considered the surrounding context. The outer shape has the relatively simple formative shape and contrasts against the complicated inner space by using the non-physical materials. It is found that Koolhaas' architecture doesn't pursue the features of placeness of traditional concept. However, It is inferred that his architecture has the possibility of placeness by attaching the meaning through the social roles of each architecture. It gives the substantial suggestion to the modern architecture that can't easily acquire the placeness of traditional concept due to the environment of modern city.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.151-158
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• 1989

Let (H, B, .nu.) be an abstract Wiener space where H is a separable Hilbert space with the inner product <.,.> and the norm vertical bar . vertical bar=.root.<.,.>, which is densely and continuously imbedded into a separable Banach space B with the norm ∥.∥ , and .nu. is a probability measure on the Borel .sigma.-algebra B(B) of B which satisfies (Fig.) where $B^{*}$ is the topological dual of B and (.,.) is the natural dual pairing between B and $B^{*}$. We will regard $B^{*}$.contnd.H.contnd.B in the natural way. Thus we have =(y, x) for all y in $B^{*}$ and x in H. Let $R^{n}$ and C denote the n-dimensional Euclidean space and the complex numbers respectively.ctively.

• East Asian mathematical journal
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• v.34 no.5
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• pp.643-649
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• 2018

Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigenvectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we proved that a connected 2-type hypersurface M in $E^{n+1}$ whose postion vector x satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle}$, ${\rangle}$ is the usual inner product in $E^{n+1}$, is of null 2-type and has constant mean curvature and scalar curvature.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.787-811
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• 1999

Let M be the maximal ideal space of the Banach algebra $H^{\infty}$ of bounded analytic functions on the open unit disc $\triangle$. For a positive singular measure ${\mu}\;on\;{\partial\triangle},\;let\;{L_{+}}^1(\mu)$ be the set of measures v with $0\;{\leq}\;{\nu}\;{\ll}\;{\mu}\;and\;{{\psi}_{\nu}}$ the associated singular inner functions. Let $R(\mu)\;and\;R_0(\mu)$ be the union sets of $\{$\mid$\psiv$\mid$\;<\;1\}\;and\;\{$\mid${\psi}_{\nu}$\mid$\;<\;0\}\;in\;M\;{\setminus}\;{\triangle},\;{\nu}\;\in\;{L_{+}}^1(\mu)$, respectively. It is proved that if $S(\mu)\;=\;{\partial\triangle}$, where $S(\mu)$ is the closed support set of $\mu$, then $R(\mu)\;=\;R0(\mu)\;=\;M{\setminus}({\triangle}\;{\cup}\;M(L^{\infty}(\partial\triangle)))$ is generated by $H^{\infty}\;and\;\overline{\psi_{\nu}},\;{\nu}\;{\in}\;{L_1}^{+}(\mu)$. It is proved that %d{\theta}(S(\mu))\;=\;0$ if and only if there exists as Blaschke product b with zeros $\{Zn\}_n$ such that $R(\mu)\;{\subset}\;{$\mid$b$\mid$\;<\;1}\;and\;S(\mu)$ coincides with the set of cluster points of $\{Zn\}_n$. While, we proved that $\mu$ is a sum of finitely many point measure such that $R(\mu)\;{\subset}\;\{$\mid${\psi}_{\lambda}$\mid$\;<\;1}\;and\;S(\lambda)\;=\;S(\mu)$. Also it is studied conditions on \mu for which $R(\mu)\;=\;R0(\mu)$.

• East Asian mathematical journal
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• v.33 no.5
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• pp.571-585
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• 2017

Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigen vectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we showed that a 2-type surface M in $E^3$ satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle},{\rangle}$ is the usual inner product in $E^3$, then M is an open part of a circular cylinder. Also we showed that if a quadric hypersurface M in a Euclidean space satisfies ${\langle}{\Delta}x,x{\rangle}=c$ for a constant c, then it is one of a minimal quadric hypersurface, a genaralized cone, a hypersphere, and a spherical cylinder.

• Journal of KIISE
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• v.42 no.2
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• pp.235-241
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• 2015

Locally linear embedding (LLE) [1] is a type of manifold algorithms, which preserves inner product value between high-dimensional data when embedding the high-dimensional data to low-dimensional space. LLE closely embeds data points on the same subspace in low-dimensional space, because the data points have significant inner product values. On the other hand, if the data points are located orthogonal to each other, these are separately embedded in low-dimensional space, even though they are in close proximity to each other in high-dimensional space. Meanwhile, it is well known that the facial images of the same person under varying illumination lie in a low-dimensional linear subspace [2]. In this study, we suggest an improved LLE method for face recognition problem. The method maximizes the characteristic of LLE, which embeds the data points totally separately when they are located orthogonal to each other. To accomplish this, all of the subspaces made by each class are forced to locate orthogonally. To make all of the subspaces orthogonal, the simultaneous Diagonalization (SD) technique was applied. From experimental results, the suggested method is shown to dramatically improve the embedding results and classification performance.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.37 no.2
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• pp.55-64
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• 2000

In this paper, we propose an efficient datapath placement algorithm to minimize track density. Here, we consider each datapath element as a cluster, and merge the most strongly connected two clusters to a new cluster until only one cluster remains. As nodes in the two clusters to be merged are already linearly ordered respectively, we can merge two clusters with connecting them. The proposed algorithm produces circular linear ordering by connecting starting point and end point of the final cluster, and n different linear ordering by cutting between two contiguous elements of the circular linear ordering. Among the n different linear ordering, the linear ordering to minimize track density is final solution. In this paper, we show and utilize that if two clusters are strongly connected in a graph, the inner product of the corresponding vectors mapped in d-dimensional space using spectral method is maximum. Compared with previous datapath placement algorithm GA/S $A^{［2］}$, the proposed algorithm gives similar results with much less computation time.

• Journal of Astronomy and Space Sciences
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• v.20 no.2
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• pp.109-122
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• 2003

To examine the causal relationship between geomagnetic storm and substorm, we investigate the correlation between dispersionless particle injection rate of proton flux observed from geosynchronous satellites, which is known to be a typical indicator of the substorm expansion activity, and Dst index during magnetic storms. We utilize geomagnetic storms occurred during the period of 1996 ~ 2000 and categorize them into three classes in terms of the minimum value of the Dst index ($Dst_{min}$); intense ($-200nT{$\leq$}Dst_{min}{$\leq$}-100nT$), moderate($-100nT{\leq}Dst_{min}{\leq}-50nT$), and small ($-50nT{\leq}Dst_{min}{\leq}-30nT$) -30nT)storms. We use the proton flux of the energy range from 50 keV to 670 keV, the major constituents of the ring current particles, observed from the LANL geosynchronous satellites located within the local time sector from 18:00 MLT to 04:00 MLT. We also examine the flux ratio ($f_{max}/f_{ave}$) to estimate particle energy injection rate into the inner magnetosphere, with $f_{ave}$ and $f_{max}$ being the flux levels during quiet and onset levels, respectively. The total energy injection rate into the inner magnetosphere can not be estimated from particle measurements by one or two satellites. However, the total energy injection rate should be at least proportional to the flux ratio and the injection frequency. Thus we propose a quantity, “total energy injection parameter (TEIP)”, defined by the product of the flux ratio and the injection frequency as an indicator of the injected energy into the inner magnetosphere. To investigate the phase dependence of the substorm contribution to the development of magnetic storm, we examine the correlations during the two intervals, main and recovery phase of storm separately. Several interesting tendencies are noted particularly during the main phase of storm. First, the average particle injection frequency tends to increase with the storm size with the correlation coefficient being 0.83. Second, the flux ratio ($f_{max}/f_{ave}$) tends to be higher during large storms. The correlation coefficient between $Dst_{min}$ and the flux ratio is generally high, for example, 0.74 for the 75~113 keV energy channel. Third, it is also worth mentioning that there is a high correlation between the TEIP and $Dst_{min}$ with the highest coefficient (0.80) being recorded for the energy channel of 75~113 keV, the typical particle energies of the ring current belt. Fourth, the particle injection during the recovery phase tends to make the storms longer. It is particularly the case for intense storms. These characteristics observed during the main phase of the magnetic storm indicate that substorm expansion activity is closely associated with the development of mangetic storm.