• 대한수학회보
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• 제54권1호
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• pp.343-358
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• 2017

Let {$X_i$} be a sequence of stationary ${\alpha}-mixing$ random variables with probability density function f(x). The recursive kernel estimators of f(x) are defined by $$\hat{f}_n(x)={\frac{1}{n\sqrt{b_n}}{\sum_{j=1}^{n}}b_j{^{-\frac{1}{2}}K(\frac{x-X_j}{b_j})\;and\;{\tilde{f}}_n(x)={\frac{1}{n}}{\sum_{j=1}^{n}}{\frac{1}{b_j}}K(\frac{x-X_j}{b_j})$$, where 0 < $b_n{\rightarrow}0$ is bandwith and K is some kernel function. Under appropriate conditions, we establish the Berry-Esseen bounds for these estimators of f(x), which show the convergence rates of asymptotic normality of the estimators.

• 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]).

• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.377-389
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• 2019

We evaluate the convolution sum $W_{a,b}(n):=\sum_{al+bm=n}{\sigma}(l){\sigma}(m)$ for (a, b) = (1, 28),(4, 7),(2, 7) for all positive integers n. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form $x^2_1+x^2_2+x^2_3+x^2_4+7(x^2_5+x^2_6+x^2_7+x^2_8)$. Finally we express the modular forms ${\Delta}_{4,7}(z)$, ${\Delta}_{4,14,1}(z)$ and ${\Delta}_{4,14,2}(z)$ (given in [10, 14]) as linear combinations of eta quotients.

• 한국분말재료학회지
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• 제5권2호
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• pp.98-110
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• 1998

$Ti_{52}Al_{48}$ and $(Ti_{52}Al_{48})_{100-x}B_x(x=0.5, 2, 5)$ alloys have been Produced by mechanical milling in an attritor mill using prealloyed powders. Microstructure of binary $Ti_{52}Al_{48}$ powders consists of grains of hexagonal phase whose structure is very close to $Ti_2Al$. $(Ti_{52}Al_{48})_{95}B_5$ powders contains TiB2 in addition to matrix grains of hexagonal phase. The grain sizes in the as-milled powders of both alloys are nanocrystalline. The mechanically alloyed powders were consolidated by vacuum hot pressing (VHP) at 100$0^{\circ}C$ for 2 hours, resulting in a material which is fully dense. Microstructure of consolidated binary alloy consists of $\gamma$-TiAl phase with dispersions of $Ti_2AlN$ and $A1_2O_3$ phases located along the grain boundaries. Binary alloy shows a significant coarsening in grain and dispersoid sizes. On the other hand, microstructure of B containing alloy consists of $\gamma$-TiAl grains with fine dispersions of $TiB_2$ within the grains and shows the minimal coarsening during annealing. The vacuum hot pressed billets were subjected to various heat treatments, and the mechanical properties were measured by compression testing at room temperature. Mechanically alloyed materials show much better combinations of strength and fracture strain compared with the ingot-cast TiAl, indicating the effectiveness of mechanical alloying in improving the mechanical properties.

• 한국자기학회:학술대회 개요집
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• 한국자기학회 1994년도 추계연구발표회 논문개요집
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• pp.42-43
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• 1994

• 한국자기학회지
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• 제11권2호
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• pp.58-62
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• 2001

L $i_{0.5}$ F $e_{2.5-x}$A $l_{x}$ $O_4$ 페라이트계에 대한 XRD와 Mossbauer 스펙트럼을 측정 분석하였다. $Al^{3+}$ 씨온의 치환량 x가 증가할수록 격자상수는 작아지고, 값이 0, 0.3, 0.6인 시료에서는 두 site의 F $e^{3+}$ 이온에 의한 두 개의 6선 흡수선이 나타났고, x가 0.9, 1.2인 시료에서는 6선 흡수선외에 열에 의한 전자적 완화현상에 기인한 2선 흡수선이 공존하였으며, x가 1.5인 시료는 2선 흡수선만 나타났다. 공명흡수면적으로 계산된 L $i_{0.5}$ F $e_{2.5-x}$A $l_{x}$ $O_4$ 페라이트계의 금속 양이온 분포식은(L $i_{1-a}$$^{+}$F $e_{a}$ $^{3+}$ ) z으로 나타낼 수 있었고, 시료내의 $Al^{3+}$ 의 증가는 B-site의 F $e^{3+}$ - $O^{2-}$ 결합거리를 증가시켜 공유 결합성을 약화시키는 것으로 확인되었으며, $Al^{3+}$ 씨 증가에 따라 B-site의 F $e^{3+}$ 이온의 수가 감소하여 A-B 초 교환 상호 작용이 약화되는 것을 알 수 있었다.다.

• 대한수학회보
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• 제47권3호
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• pp.645-651
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• 2010

In this note we study positive solutions of the mth order rational difference equation $x_n=(a_0+\sum{{m\atop{i=1}}a_ix_{n-i}/(b_0+\sum{{m\atop{i=1}}b_ix_{n-i}$, where n = m,m+1,m+2, $\ldots$ and $x_0,\ldots,x_{m-1}$ > 0. We describe a sufficient condition on nonnegative real numbers $a_0,a_1,\ldots,a_m,b_0,b_1,\ldots,b_m$ under which every solution $x_n$ of the above equation tends to the limit $(A-b_0+\sqrt{(A-b_0)^2+4_{a_0}B}$/2B as $n{\rightarrow}{\infty}$, where $A=\sum{{m\atop{i=1}}\;a_i$ and $B=\sum{{m\atop{i=1}}\;b_i$.

• 한국자기학회:학술대회 개요집
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• 한국자기학회 2004년도 동계학술연구발표회 논문개요집
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• pp.241-242
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• 2004

• 한국자기학회:학술대회 개요집
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• 한국자기학회 1994년도 추계연구발표회 논문개요집
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• pp.54-55
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• 1994

• 한국자기학회:학술대회 개요집
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• 한국자기학회 1997년도 추계연구발표회 논문개요집
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• pp.60-61
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• 1997