• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.383-388
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• 2007

A (g, f)-factor F of a graph G is Called a Hamiltonian (g, f)-factor if F contains a Hamiltonian cycle. The binding number of G is defined by $bind(G)\;=\;{min}\;\{\;{\frac{{\mid}N_GX{\mid}}{{\mid}X{\mid}}}\;{\mid}\;{\emptyset}\;{\neq}\;X\;{\subset}\;V(G)},\;{N_G(X)\;{\neq}\;V(G)}\;\}$. Let G be a connected graph, and let a and b be integers such that $4\;{\leq}\;a\;<\;b$. Let g, f be positive integer-valued functions defined on V(G) such that $a\;{\leq}\;g(x)\;<\;f(x)\;{\leq}\;b$ for every $x\;{\in}\;V(G)$. In this paper, it is proved that if $bind(G)\;{\geq}\;{\frac{(a+b-5)(n-1)}{(a-2)n-3(a+b-5)},}\;{\nu}(G)\;{\geq}\;{\frac{(a+b-5)^2}{a-2}}$ and for any nonempty independent subset X of V(G), ${\mid}\;N_{G}(X)\;{\mid}\;{\geq}\;{\frac{(b-3)n+(2a+2b-9){\mid}X{\mid}}{a+b-5}}$, then G has a Hamiltonian (g, f)-factor.

• Journal of the Korean Magnetic Resonance Society
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• v.9 no.1
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• pp.61-66
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• 2005

$^{11}B MAS NMR$ spectra of binary glass system $xV_2O5-B_2O_3$ and ternary glass system $xV_2O5-B_2O_3-yNa_2O$ (x = $V_2O_5 mol%/$B_2O_3$ mol%, y = $Na_2O$ mol$/$B_2O_3$ mol%) were acquired. $BO_3$ units are dominant components in the spectra of $xV_2O_5-B_2O_3$glass systems while both $BO_3$ and $BO_4$ unit appear in comparable amounts in the spectra of $xV_2O_5-B_2O_3-yNa_2O$ glass systems. More $BO_3$ units were monitored for higher $V_2O_5$ contents while more $BO_4$ unit for higher $Na_2O$ contents. Quadrupole parameters such as $e^2qQ$ and $\eta$ obtained form spectral simulation indicate that $e^2qQ$ has a maximum value at x = y 1 and $\eta$ decreases and increases as x or y grows, respectively. Our results suggest that $V_2O_5$ and $Na_2O$ play opposite roles in the ternary glasses.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1149-1159
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• 2018

A quadratic polynomial ${\Phi}_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation ${\Phi}_{a,b,c}(x,y,z)=n$ has an integer solution x, y, z for any nonnegative integer n. In this article, we show that if (a, b, c) = (2, 2, 6), (2, 3, 5) or (2, 3, 7), then ${\Phi}_{a,b,c}(x,y,z)$ is universal. These were conjectured by Sun in [8].

• Journal of applied mathematics & informatics
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• v.35 no.1_2
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• pp.121-130
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• 2017

This study is concerned with the existence of positive solution for the following nonlinear elliptic system $$\{-M_1(\int_{\Omega}{\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^pdx)div({\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)\\{\hfill{120}}={\mid}x{\mid}^{-(a+1)p+c_1}\({\alpha}_1A_1(x)f(v)+{\beta}_1B_1(x)h(u)\),\;x{\in}{\Omega},\\-M_2(\int_{\Omega}{\mid}x{\mid}^{-bq}{\mid}{\nabla}v{\mid}^qdx)div({\mid}x{\mid}^{-bq}{\mid}{\nabla}v{\mid}^{q-2}{\nabla}v)\\{\hfill{120}}={\mid}x{\mid}^{-(b+1)q+c_2}\({\alpha}_2A_2(x)g(u)+{\beta}_2B_2(x)k(v)\),\;x{\in}{\Omega},\\{u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}$ is a bounded smooth domain of ${\mathbb{R}}^N$ with $0{\in}{\Omega}$, 1 < p, q < N, $0{\leq}a$ < $\frac{N-p}{p}$, $0{\leq}b$ < $\frac{N-q}{q}$ and ${\alpha}_i,{\beta}_i,c_i$ are positive parameters. Here $M_i,A_i,B_i,f,g,h,k$ are continuous functions and we discuss the existence of positive solution when they satisfy certain additional conditions. Our approach is based on the sub and super solutions method.

• Korean Journal of Materials Research
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• v.11 no.1
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• pp.3-7
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• 2001

$M\"{o}ssbauer$ spectra of the $Ni_{1+x}Ti_xFe_{2-2x}O_4$ systems ($0{\leqq}x{\leqq}0.7$), which appear as single phase spinel structure, were examined at RT. The $M\"{o}ssbauer$ spectra reveal two sextet for $0{\leqq}x{\leqq}0.3$, two sextet and a doublet for $0.4{\leqq}x{\leqq}0.6$, and a doublet for x=0.7 As x increases, the area ratio of B-site and A-site($A_B/A_A$) of the sextet decreases, and the area ratio of the doublet and the total areas($A_{doublet}/A_{tot.}$) increases. The isomer shift(I.S.) of A-site slightly increases and magnetic hyperfine fields($H_{hf}$) of two sites decrease as the increasing x. From these results, we have obtained the cation distributions of the samples and concluded that the increasing x leads to the decrease of covalency of $Fe^{3+}-O^{2-}$ bond in A-sites and A-B superexchange interactions.eractions.

• Journal of the Korean Crystal Growth and Crystal Technology
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• v.11 no.1
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• pp.20-26
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• 2001

Titanium boride ($\textrn{TiB}_{x}$) films were deposited on (100) silicon substrates at the substrate temperature of $500^{\circ}C$ by means of the co-evaporation of titanium and boron evaporants during deposition. The co-evaporation method makes it possible to deposit the non-stoichiometric films with different boron-to-titanium ratio($0{\le}B/Ti \le 2.5$). The resistivity increases linearly as the boron-to-titanium ratio in the as-deposited films is increased. The surface roughness of $\textrn{TiB}_{x}$ films is changed as a function of the boron-to-titanium ratio. The XRD spectrum for pure titanium film shows a highly (002) preferred orientation. For B/Ti=0.59 ratio only a single TiB phase that shows a (111) preferred orientation is observed. However, the $\textrn{TiB}_{x}$ phase with the hexagonal structure of the $AlB_2$(C32) type appears as the boron concentration increase, and only a single $\textrn{TiB}_{x}$ phase is observed for $B/Ti \ge 2.0$ ratio. The $\textrn{TiB}_{x}$/Si samples reveal a tensile stress (3~$20{\times}^9$dyn/$\textrm{cm}^2$) in the overall composition of the films, although the magnitude of the residual stresses is depended on the nominal B/Ti ratio.

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.121-126
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• 1985

D.K. Harrison [5] has shown that if R and S are fields of characteristic different from 2, then two Witt rings W(R) and W(S) are isomorphic if and only if W(R)/I(R)$^{3}$ and W(S)/I(S)$^{3}$ are isomorphic where I(R) and I(S) denote the fundamental ideals of W(R) and W(S) respectively. In [1], J.K. Arason and A. Pfister proved a corresponding result when the characteristics of R and S are 2, and, in [9], K.I. Mandelberg proved the result when R and S are commutative semi-local rings having 2 a unit. In this paper, we prove the result when R and S are 2-fold full rings. Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is nondegenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$ (V, R) induced by B is an isomorphism), and with a quadratic mapping .phi.:V.rarw.R such that B(x,y)=(.phi.(x+y)-.phi.(x)-.phi.(y))/2 and .phi.(rx)= $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U(R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$, .., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2, we reserve the notation [ $a_{11}$, $a_{22}$] for the space.the space.e.e.e.

• Journal of Magnetics
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• v.7 no.4
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• pp.143-146
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• 2002

Mechanical milling technique is considered to be a useful way of processing the fine Nd-Fe-B-type powder with high coercivity. In the present study, phase evolution of the $Nd_{15}(Fe_{1-x}Co_{x})_{77}B_{8}$ (x=0-0.6) alloys during the high energy mechanical milling and annealing was investigated. The effect of Co-substitution on the crystallization of the mechanically milled $Nd_{15}(Fe_{1-x}Co_{x})_{77}B_{8}$ amorphous material was examined. The Nd-Fe-B-type alloys can be amorphized completely by a high-energy mechanical milling. On annealing of the amorphous material, fine $\alpha$-Fe crystallites form first from the amorphous. These fine $\alpha$-Fe crystallites reacts with the remaining amorphous afterwards, leading to crystallization to $Nd_2Fe_{14}$B phase. The Co-substitution for Fe in $Nd_{15}(Fe_{1-x}Co_{x})_{77}B_{8}$ ($\mu$x=0∼0.6) alloys lower significantly the crystallization temperature of the amorphous phase to the $Nd_2Fe_{14}$B phase. The mechanically milled and annealed $Nd_{15}Fe_{77}B_8$ alloy without Co-substitution exhibits consistently better magnetic properties with respect to the alloys with Co-substitution.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.573-586
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• 2018

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

• Bulletin of the Korean Mathematical Society
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• v.22 no.1
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• pp.11-15
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• 1985

Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) = $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$,.., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2 we reserve the notation [a $a_{11}$, $a_{22}$] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple $a_{1}$, $a_{2}$, $a_{3}$ of elements in R with ( $a_{1}$, $a_{2}$, $a_{3}$)=R, there is an element w in R such that $a_{1}$+ $a_{2}$w+ $a_{3}$ $w^{2}$=unit.TEX>=unit.t.t.t.