• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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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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• v.13 no.2
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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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• v.59 no.3
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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.

• Journal of Powder Materials
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• v.5 no.2
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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.

• Proceedings of the Korean Magnestics Society Conference
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• 1994.10a
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• pp.42-43
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• 1994

• Journal of the Korean Magnetics Society
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• v.11 no.2
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• pp.58-62
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• 2001

The L $i_{0.5}$F $e_{2.5-x}$A $l_{x}$ $O_4$ systems (x=0, 0.3, 0.6, 0.9, 1.2, 1.5) were investigated by X-ray diffraction and Mossbauer spectroscopy. The structure of all the samples is cubic spinel type and lattice constant decrease with increasing Al content x. The Moissbauer spectra reveal two sextet for 0$\leq$x$\leq$0.6, two sextet and a doublet for 0.9$\leq$x$\leq$1.2, and a doublet for x=1.5. The cation distribution of the samples is (L $i_{1-a}$$^{+}$F $e_{a}$ $^{3+}$)$^{A}$[L $i_{a-0.5}$$^{+}$A $l_{2.5-a-x}$$^{+}$F $e_{2.5-a-x}$$^{3+}$]$^{B}$ $O_4$$^{2-}$ and substituted $Al^{3+}$ ions decrease the covalency of F $e^{3+}$- $O^{2-}$ bond in B-sites and A-B super-exchange interactions.tions.s.tions.ons.s.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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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 $B=\sum{{m\atop{i=1}}\;b_i$.

• Proceedings of the Korean Magnestics Society Conference
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• 2004.12a
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• pp.241-242
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• 2004

• Proceedings of the Korean Magnestics Society Conference
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• 1994.10a
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• pp.54-55
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• 1994

• Proceedings of the Korean Magnestics Society Conference
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• 1997.11a
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• pp.60-61
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• 1997