• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1143-1157
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• 2019

Let $T_aD_b(n)$ and $T_aD^{\prime}_b(n)$ denote respectively the number of representations of a positive integer n by $a(x^2-x)/2+b(4y^2-3y)$ and $a(x^2-x)/2+b(4y^2-y)$. Similarly, let $S_aD_b(n)$ and $S_aD^{\prime}_b(n)$ denote respectively the number of representations of n by $ax^2+b(4y^2-3y)$ and $ax^2+b(4y^2-y)$. In this paper, we prove 162 formulas for these functions.

• Journal of Powder Materials
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• v.22 no.5
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• pp.344-349
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• 2015

$MgB_2$ bulk superconductors are synthesized by the solid state reaction of ($MgB_4$+xMg) precursors with excessive Mg compositions (x=1.0, 1.4, 2.0 and 2.4). The $MgB_4$ precursors are synthesized using (Mg+B) powders. The secondary phases ($MgB_4$ and MgO) present in the synthesized $MgB_4$ are removed by $HNO_3$ leaching. It is found that the formation reaction of $MgB_2$ is accelerated when Mg excessive compositions are used. The magnetization curves of $Mg_1+_xB_2$ samples show that the transition from the normal state to the superconducting state of the Mg excessive samples with x=0.5 and x=0.7 are sharper than that of $MgB_2$. The highest $J_c-B$ curve at 5 K and 20 K is achieved for x=0.5. Further addition of Mg decreases the $J_c$ owing to the formation of more pores in the $MgB_2$ matrix and smaller volume fraction of $MgB_2$.

• The Pure and Applied Mathematics
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• v.16 no.4
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• pp.327-344
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• 2009

In this paper, we deal with the following system of nonlinear singular boundary value problems(BVPs) on time scale $\mathbb{T}$ $$\{{{{{{x^{\bigtriangleup\bigtriangleup}(t)+f(t,\;y(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}\atop{y^{\bigtriangleup\bigtriangleup}(t)+g(t,\;x(t))=0,\;t{\in}(a,\;b)_{\mathbb{T}},}}\atop{\alpha_1x(a)-\beta_1x^{\bigtriangleup}(a)=\gamma_1x(\sigma(b))+\delta_1x^{\bigtriangleup}(\sigma(b))=0,}}\atop{\alpha_2y(a)-\beta_2y^{\bigtriangleup}(a)=\gamma_2y(\sigma(b))+\delta_2y^{\bigtriangleup}(\sigma(b))=0,}}$$ where $\alpha_i$, $\beta_i$, $\gamma_i\;{\geq}\;0$ and $\rho_i=\alpha_i\gamma_i(\sigma(b)-a)+\alpha_i\delta_i+\gamma_i\beta_i$ > 0(i = 1, 2), f(t, y) may be singular at t = a, y = 0, and g(t, x) may be singular at t = a. The arguments are based upon a fixed-point theorem for mappings that are decreasing with respect to a cone. We also obtain the analogous existence results for the related nonlinear systems $x^{\bigtriangledown\bigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangledown\bigtriangledown}(t)$ + g(t, x(t)) = 0, $x^{\bigtriangleup\bigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangleup\bigtriangledown}(t)$ + g(t, x(t)) = 0, and $x^{\bigtriangledown\bigtriangleup}(t)$ + f(t, y(t)) = 0, $y^{\bigtriangledown\bigtriangleup}(t)$ + g(t, x(t)) = 0 satisfying similar boundary conditions.

• The Korean Journal of Ceramics
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• v.2 no.2
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• pp.76-82
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• 1996

PLZT ceramics having two nominal compositions, $Pb_{1-3x/2}La_xV_{x/2}(Zr_{03}Ti_{03})O_3$ and $Pb_{1-x}La_x(Zr_{0.2}Ti_{0.5})_{1-x/4}V_{x/4}O_3$ (V: vacancy) with x=0.00~0.30, were prepared. The physical, structural, and dielectric properties were investigated by X-ray diffraction, scanning electron microscopy, Raman spectroscopy, and measurements of bulk density and dielectric constant. The two series with A-stie and B-site vacancies showed different physical, structural, dielectric properties, and, specially, Curie temperature. In comparison to PLZT with B-site vacancies, PLZT with A-site vacancies showed high Curie temperatures and low maxima of dielectric constant. Consequently, it is evident that the properties of PLZT ceramics depend on the vacancy formula adopted as a batch composition in preparation.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.263-275
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• 2010

Let {$X_n;n\;\geq\;1$} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n\;=\;{\sum}^n_{k=1}X_k$, $M_n\;=\;max_{k{\leq}n}|S_k|$, $n\;{\geq}\;1$. Suppose $\sigma^2\;=\;EX^2_1+2{\sum}^\infty_{k=2}EX_1X_k$ (0 < $\sigma$ < $\infty$). We prove that for any b > -1/2, if $E|X|^{2+\delta}$(0<$\delta$$\leq$1), then $$lim\limits_{\varepsilon\searrow0}\varepsilon^{2b+1}\sum^{\infty}_{n=1}\frac{(loglogn)^{b-1/2}}{n^{3/2}logn}E\{M_n-\sigma\varepsilon\sqrt{2nloglogn}\}_+=\frac{2^{-1/2-b}{\sigma}E|N|^{2(b+1)}}{(b+1)(2b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2(b+1)}}$$ and for any b > -1/2, $$lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum^{\infty}_{n=1}\frac{(loglogn)^b}{n^{3/2}logn}E\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8loglogn}}-M_n\}_+=\frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2b+2'}}$$, where $\Gamma(\cdot)$ is the Gamma function and N stands for the standard normal random variable.

• Journal of the Korean Magnetics Society
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• v.3 no.2
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• pp.135-138
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• 1993

Saturation rnagnetostriction and rnagnetoelastic properties of amorphous $Fe_{82}B_{18-x}Al_{x}$ and $Fe_{80}B_{20-x}Al_{x}$ alloys have been studied. Saturation magnetostriction of the alloys has increased according to the increase of AI content. The amorphous $Fe_{82}B_{14}Al_{4}$ alloy shows the highest saturation magnetostriction of 45 ppm among the alloy systems. The ratio of maximum magnetic induction change to tensile stress of this alloy is about $0.026\;T.mm^{2}/N$, and it is considered to be applicable for a high resolution mechanical sensor.

• Journal of the Korean Statistical Society
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• v.12 no.2
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• pp.100-109
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• 1983

Let ${X_i}$ be a process with stationary and independent increments whose log characteristic function is expressed as $ibut-2^{-1}\sigma^2u^2t+t\int_{{0 }^c}{(exp(iux)-1-iux(i+x^2)^{-1})dv(x)}$. Our main result is taht $x^2(\int_{\y\>x}{dv(y)})/(\int_{$\mid$y$\mid$\leqx}{y^2dv(y)+\sigma^2}) \to 1$ as $x \to 0 (resp. x \to \infty)$ is necessary, and sufficient for ${X-i}$ to have ${A_t}$ and ${B_t}$ such that $(X_t-A_t)/B_t \to^D n(0,1)$ as $t \to 0 (resp. t \to \infty)$.

• Journal of the Korean Magnetics Society
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• v.9 no.3
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• pp.154-158
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• 1999

Magnetic properties and microstructure of nanocrystalline $\alpha$-(Fe, Co)-based Nd-(Fe, Co)-B-Nb-Cu alloys have been investigated. $Nd_x(Fe_{0.9}Co_{0.1})_{90-x}B_6Nb_3Cu_1$(x=2, 3, 4, 5, 6) alloys prepared by rapid solidification process show amorphous phase except the one with x=2. By a proper annealing, the amorphous in the alloy is changed to a nanocrystalline phase. It is confirmed that the nanocrystalline alloys are composed of $\alpha$-(Fe, Co) and $Nd_2(Fe, Co)_{14}B_1$ phase. The optimally annealed $Nd_3(Fe_{0.9}Co_{0.1})_87B_6Nb_3Cu_1$ alloy shows the highest remanence of 1.55 T. The coercivity increases with the increase of Nd content The maximum coercivity of 4.6 kOe is obtained from an optimally annealed $Nd_6(Fe_{0.9}Co_{0.1})_84B_6Nb_3Cu_1$ alloy, resulting in the maximum energy product of 10.6 MGOe.

• Journal of Powder Materials
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• v.19 no.2
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• pp.146-150
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• 2012

The effects of nano $Fe_xC$ addition to superconducting properties of $in$ $situ$ processed $MgB_2$ superconductors was examined. 0.1 wt.% and 1 wt.% nano $Fe_xC$ powders were mixed with boron and magnesium powders by ball milling. The powder mixtures were made into pellets by uniaxial pressing. The pellets were heat-treated at $700^{\circ}C-900^{\circ}C$ in argon atmosphere for $MgB_2$ formation. It was found by powder X-ray diffraction that the raw powders were completely converted into $MgB_2$ after the heat treatment. The superconducting transition temperature ($T_c$) and critical current density ($J_c$), estimated from susceptibility-temperature and $M-H$ curves, were decreased by nano $Fe_xC$ addition. The $T_c$ and $J_c$ decrease by nano $Fe_xC$ addition are attributed to the incorporation of iron and carbon with $MgB_2$ lattices (Fe substitution for Mg and C substitution for B) due to the high reactivity of the nano $Fe_xC$ powder.

• Journal of the Korean Magnetics Society
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• v.7 no.1
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• pp.49-54
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• 1997

The effect of Co and annealing temperature on the magnetic properties, phase change and microstructure of melt-spun $(Nd_{14.73}Fe_{78.67}B_{6.60})_{100-x}Co_x$ (X=0, 1, 2, 3) ribbons has been studied. The Co containing ribbons were found to have higher coercivity ($_iH_c$) than the ribbons without Co. Intrinsic coercivity of 20.3 kOe has been obtained by addition of 2 at%Co. This effect by Co addition is also represented in the case of hot pressed and die-upseted magnets. The maximum intrinsic coercivities of hot press and die-upset $(Nd_{14.73}Fe_{78.67}B_{6.60})_{100- x}Co_x$ (X=0, 1, 2, 3) magnets are 16.9 kOe and 15.2 kOe when X=2.