• Kyungpook Mathematical Journal
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• 제13권2호
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• pp.235-240
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• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• 대한수학회보
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• 제56권6호
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• pp.1525-1537
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• 2019

Let ${\tilde{M}}$ be a Riemannian manifold equipped with a concircular vector field ${\tilde{X}}$ and M a submanifold (with its induced metric) of ${\tilde{M}}$. Denote by X the restriction of ${\tilde{X}}$ on M and by $X^T$ the tangential component of X, called the canonical field of M. In this article we study submanifolds of ${\tilde{M}}$ whose canonical field $X^T$ is also concircular. Several characterizations and classification results in this respect are obtained.

• 제어로봇시스템학회논문지
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• 제9권10호
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• pp.774-781
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• 2003

The important factors that cause position error in X-Y robot are inertial force, frictions and spring distortion in screw or coupling. We have to estimate these factors precisely to correct position errors, Which is very difficult. In this paper, we makes systems to inspect metal stencil which is used to print solder paste on pads of SMD of PCB with low precision X-Y robot and vision system. To correct position error that is caused by low precision X-Y robot, we defines position error vector that is formed with position of objects that exist in reference and camera image. We apply MHT(Modified Hough Transformation) for the aim of determining the dominant position error vector. We modify reference image using extracted dominant position error vector and obtain reference image that is the same with camera image. Effectiveness and performance of this method are verified by simulation and experiment.

• 품질경영학회지
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• 제22권2호
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• pp.89-97
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• 1994

We consider the linear calibration model: $y_1={\alpha}+{\beta}x_i+{\sigma}{\varepsilon}_i$, i = 1, ${\cdots}$, n, $y={\alpha}+{\beta}x+{\sigma}{\varepsilon}$ where ($y_1$, ${\cdots}$, $y_n$, y) stands for an observation vector, {$x_i$} fixed design vector, (${\alpha}$, ${\beta}$) vector of regression parameters, x unknown true value of interest and {${\varepsilon}_i$}, ${\varepsilon}$ are mutually uncorrelated measurement errors with zero mean and unit variance but otherwise unknown distributions. On the basis of simple small-sample low-noise approximation, we introduce a new method of comparing the mean squared errors of the various competing estimators of the true value x for finite sample size n. Then we show that a class of estimators including the classical and the inverse estimators are consistent and first-order efficient within the class of all regular consistent estimators irrespective of type of measurement errors.

• Communications for Statistical Applications and Methods
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• 제16권1호
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• pp.185-193
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• 2009

Machine learning methods such as support vector machines and random forests yield nonparametric prediction functions of the form y = $f(x_1,{\ldots},x_p)$. As a sequel to the previous article (Huh and Lee, 2008) for visualizing nonparametric functions, I propose more sensible graphs for visualizing y = $f(x_1,{\ldots},x_p)$ herein which has two clear advantages over the previous simple graphs. New graphs will show a small number of prototype curves of $f(x_1,{\ldots},x_{j-1},x_j,x_{j+1}{\ldots},x_p)$, revealing statistically plausible portion over the interval of $x_j$ which changes with ($x_1,{\ldots},x_{j-1},x_{j+1},{\ldots},x_p$). To complement the visual display, matching importance measures for each of p predictor variables are produced. The proposed graphs and importance measures are validated in simulated settings and demonstrated for an environmental study.

• 대한수학회지
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• 제45권6호
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• pp.1561-1576
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• 2008

In this paper we shall prove some weighted norm inequalities of the form $${\int}_{R^n}\;|Tf(x)|^pu(x)dx\;{\leq}\;C_p\;{\int}_{R^n}\;|f(x)|^pNu(x)dx$$ for certain rough singular integral T and maximal singular integral $T^*$. Here u is a nonnegative measurable function on $R^n$ and N denotes some maximal operator. As a consequence, some vector valued inequalities for both T and $T^*$ are obtained. We shall also get a boundedness result of T on the Triebel-Lizorkin spaces.

• Journal of applied mathematics & informatics
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• 제15권1_2호
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• pp.415-423
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• 2004

In allelic model X = ($x_1,\;x_2,...x_{d}$), $M_f(t)$= f(p(t)) - ${{\int}^{t}}_0$Lf(p(t))ds is a P-martingale for diffusion operator L under the certain conditions. In this note, we can show uniqueness of martingale problem associated with mean vector and obtain a complete description of ergodic property by using of the semigroup method.

• Communications for Statistical Applications and Methods
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• 제10권2호
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• pp.399-422
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• 2003

In actual manufacturing industries, process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. Since these characteristics are related, it is a risky undertaking to represent variation of even a univariate characteristic by a single index. Therefore, the desirability of using vector-valued process capability index(PCI) arises quite naturally. In this paper, some vector-valued ${PCI}_p$ ${C}_p$=(${C}_{px}$, ${C}_{py}$),${C}_{pk}$=(${C}_{pkx}$, ${C}_{pky}$) and ${C}_{pm}$=(${C}_{pmx}$, ${C}_{pmy}$) considering univariate PCIs ${C}_p$,${C}_{pk}$ and ${C}_{pm}$ are studied. First, we propose some asymptotic confidence regions of our vector-valued PCIs with bootstrap. And we examine the performance of asymptotic confidence regions of our vector-valued PCIs ${C}_p$ and ${C}_{pk}$ under the assumption of bivariate normal distribution BN($\mu_{x}$, $\mu_{y}$, $\sigma_{x}^{2}$, $\sigma_{y}^{2}$, $\rho$) and bivariate chi-square distribution Bivariate $x^2$(5,5,$\rho$).

• Korean Journal of Mathematics
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• 제30권2호
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• pp.239-247
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• 2022

Let C₀[0, T] denote the Wiener space, the space of continuous functions x(t) on [0, T] such that x(0) = 0. Define a random vector $Z_{\vec{e},k}:C_0[0,\;T] {\rightarrow}{\mathbb{R}}^k$ by $$Z_{\vec{e},k}(x)=({\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^T}\;e_1(t)dx(t),\;{\ldots},\;{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^T}\;ek(t)dx(t))$$ where e_j ∈ L₂[0, T] with e_j ≠ 0 a.e., j = 1, …, k. In this paper we study the conditional Fourier-Feynman transform and a conditional convolution product for a cylinder type functionals defined on C0[0, T] with a general vector valued conditioning functions $Z_{\vec{e},k}$ above which need not depend upon the values of x at only finitely many points in (0, T] rather than a conditioning function X(x) = (x(t₁), …, x(t_n)) where 0 < t₁ < … < t_n = T. In particular we show that the conditional Fourier-Feynman transform of the conditional convolution product is the product of conditional Fourier-Feynman transforms.

• 통합자연과학논문집
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• 제10권1호
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• pp.33-39
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• 2017

Consider the problem of estimating a $p{\times}1$ mean vector ${\theta}(p-q{\geq}3)$, $q=rank(P_V)$ with a projection matrix $P_v$ under the quadratic loss, based on a sample $X_1$, $X_2$, ${\cdots}$, $X_n$. We find a James-Stein type decision rule which shrinks towards projection vector when the underlying distribution is that of a variance mixture of normals and when the norm ${\parallel}{\theta}-P_V{\theta}{\parallel}$ is restricted to a known interval, where $P_V$ is an idempotent and projection matrix and rank $(P_V)=q$. In this case, we characterize a minimal complete class within the class of James-Stein type decision rules. We also characterize the subclass of James-Stein type decision rules that dominate the sample mean.