• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.279-287
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• 1995

Let $F$ be a set of distributions on R with the topology of weak convergence, and let $A$ be the $\sigma$-field generated by the open sets. We denote by $F_1^\infty$ the space consisting of all infinite sequence $(F_1, F_2, \cdots), F_n \in F and R_1^\infty$ the space consisting of all infinite sequences $(x_1, x_2, \cdots)$ of real numbers. Take the $\sigma$-field $F_1^\infty$ to be the smallest $\sigma$-field of subsets of $F_1^\infty$ containing all finite-dimensional rectangles and take $B_1^\infty$ to be the Borel $\sigma$-field $R_1^\infty$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.807-816
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• 2007

For the stochastic differential inclusion on infinite dimensional space of the form $dX_t{\in}\sigma(X_t)dW_t+b(X_t)dt$, where ${\sigma}$, b are set-valued maps, W is an infinite dimensional Hilbert space valued Q-Wiener process, we prove the boundedness and continuity of solutions under the assumption that ${\sigma}$ and b are closed convex set-valued satisfying the Lipschitz property using approximation.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.229-241
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• 1999

The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $\sigma$-multifractal from $\sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $\sigma$-multifractal. We show that the non-differentiable function on the $\sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $\sigma$-multifractal spaces.

• Journal of the Korean Chemical Society
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• v.26 no.4
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• pp.195-204
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• 1982

Semi-empirical MO calculations, EHT, CNDO/2, MINDO/3, and MNDO met hods, were performed on various geometries of n-butane, n-alkyl radical and tetramethylene diracal (triplet) in order to compare eigenvalue and eigenvector properties with those obtained by STO-3G method. All methods predicted the same relative order of stabilities of various geometries for n-butane; geometrical preferences were found to be dominated by one-electron factor, ${\pi}$-orbital energy changes being more impotant in the semi-empirical methods. The hyperconjugative energy changes accompanying structural changes from $(n-{\sigma}{\ast})_{trans}$ to (n-{\sigma}{\ast})cis were underestimated in the EHT, CNDO/2 and MINDO/3, whereas those were overestimated in the MNDO. The net destabilizing effect of $(n-{\sigma}{\ast})_{trans}$ structure was mainly due to the large internuclear energy involved in the structure. Through-space interaction between $n_1$ and $n_2$ orbitals of diradical caused energy gap narrowing of ${\Delta}E_{sp}$ and ${\Delta}{\varepsilon}={\varepsilon}_0$-${\varepsilon}_{av}$; through-space interaction had opposing effect to that of through-bond interaction. Due to the less severe neglect of differential overlaps in the MNDO, this energy gap narrowing effect appeared amplified in the MNDO. In general orbital properties were found to be reproduced satisfactorily, but eigenvalue properties were not, in all the semi-empirical methods especially when ${\sigma}-{\sigma}{\ast}$ and n-$n-{\sigma}{\ast}$interactions were involved.

• Journal of the Korean Mathematical Society
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• v.34 no.3
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• pp.633-651
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• 1997

For a non-degenerate symmetric bilinear form $\sigma$ on a finite dimensional vector space E, the Jordan algebra of $\sigma$-symmetric operators has a symmetric cone $\Omega_\sigma$ of positive definite operators with respect to $\sigma$. The cone $C_\sigma$ of elements (x,y) \in E \times E with \sigma(x,y) \geq 0$ gives the compression semigroup. In this work, we show that in the sutomorphism group of the tube domain over $\Omega_\sigma$, this semigroup has a UDL and Ol'shanskii decompositions and is exactly the compression semigroup of $\Omega_sigma$.

• The Transactions of the Korean Institute of Power Electronics
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• v.14 no.1
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• pp.54-61
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• 2009

This paper proposes the 2nd-order SDSDM (Space Dithered Sigma-Delta Modulation) for performance improvement of a buck converter. The PWM (Pulse Width Modulation) has a drawback in that power spectrum tends to be concentrated around the switching frequency. The resulting harmonic spikes cause a EMI(Electromagnetic Interference) and switching loss in semiconductor, etc. The 1st-order SDSDM scheme is a kind of DSDM for reducing these harmonic spikes. In this scheme, a switching frequency is spread through random dither generator placed on input part. In experimental result, the proposed 2nd-order SDSDM is confirmed by applying to a buck converter.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.677-685
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• 2003

Let M be the vector space of all real S-measurable functions defined on a measure space (X, S, $\mu$). In this paper, we investigate some topological structure of T on M. Indeed, (M, T) becomes a topological vector space. Moreover, if $\mu$, is ${\sigma}-finite$, we can define a complete invariant metric on M which is compatible with the topology T on M, and hence (M, T) becomes a F-space.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.71-76
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• 1988

For each continuous convex function ${\phi}$ defined on an open convex subset $A_{\phi}$ of a Banach space X, if we define a positively homogeneous sublinear functional ${\sigma}_x$ on X by ${\sigma}_x(y)=\sup{\lbrace}f(y)\;:\;f{\in}{\partial}{\phi}(x){\rbrace}$, where ${\partial}{\phi}(x)$ is a subdifferential of ${\phi}$ at x, then we get the following characterization theorem of Gateaux differentiability (weak Asplund) sapce. THEOREM. For every ${\phi}$ above, $D_{\phi}={\lbrace}x{\in}A\;:\;\sup_{||u||=1}\;{\sigma}_x(u)+{\sigma}_x(-u)=0{\rbrace}$ contains dense (dense $G_{\delta}$) subset of $A_{\phi}$ if and only if X is a Gateaux differentiability (weak Asplund) space.

• Korean Journal of Mathematics
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• v.13 no.1
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• pp.43-50
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• 2005

We introduce the Joint spectrum on the complex Banach space and on the complex Hilbert space and the tensor product spectrums on the tensor product spaces. And we will show ${\sigma}[P(T_1,T_2,{\ldots},T_n)]={\sigma}(T_1{\otimes}T_2{\otimes}{\cdots}{\otimes}T_n)$ on $X_1{\overline{\otimes}}X_2{\overline{\otimes}}{\cdots}{\overline{\otimes}}X_n$ for a polynomial P.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.709-721
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• 2004

A punctured torus ${\Sigma}$(1, 1) is a building block of oriented surfaces. In this paper we formulate the matrix presentations of elements of the Teichmuller space of a punctured torus using the matrix presentations of a pair of pants ${\Sigma}$(0, 3) and the gluing method.