• 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 Magnetics
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• v.1 no.1
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• pp.31-36
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• 1996

An externally applied magnetic field during heat treating the $Nd_2Fe_{14}B/Fe_3B$ based spring magnet was found to enhance the exchange coupling between the hard and soft magnetic grains. More than 30% increase in $M_r/M_s$ values for melt-spun $Nd_2Fe_{73.5}Co_3$$(Hf_{1-x}Ga_x)B_{18.5}$ (x=0, 0.5, 1) alloys was resulted from a uniform distribution of $Fe_3B, \alpha-Fe$ and $Nd_2Fe_{14}B$ phases, and also from a reduced grain size of those phases by 20%. The externally applied magnetic field induced a uniform distribution of fine grains. A study of Mossbauer effect also report that the enhancement of total magnetization of nanocomposite $Nd_2Fe_{14}B/Fe_3B$ alloys is attributed to an increased formation of $Fe_3$B after magnetic annealing.

• Electrical & Electronic Materials
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• v.10 no.6
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• pp.533-540
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• 1997

The microwave dielectric properties of ((P $b_{1-x}$ C $a_{x}$)Zr $O_3$ and (P $b_{0.63}$,C $a_{0.37-x}$ $M_{x}$)Zr $O_3$(M=Mg,Sr) ceramics were investigated. In (P $b_{1-x}$ C $a_{x}$)Zr $O_3$ (X=0.33~0.40) ceramics, high quality factor and small temperature coefficient of resonant frequency were obtain in (P $b_{0.63}$C $a_{0.37}$)Zr $O_3$with perovskite structure. In the case of (P $b_{0.63}$C $a_{0.37-x}$M $g_{x}$)Zr $O_3$ dielectric constant temperature coefficient of resonant frequency increased and quality factor decreased due to increase of polarization of A-O bonding. When replacing Ca ion with Sr ion with large ion radius, polarization decreased with increased of bonding length and thus dielectric constant and temperature coefficient of resonant frequency decreased.decreased.creased.

• 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.

• Electrical & Electronic Materials
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• v.10 no.5
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• pp.434-439
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• 1997

It this paper, the piezoelectric and electro-induced strain properties of (P $b_{1-}$2x/3/B $i_{x}$ )[N $i_{1}$3/N $b_{2}$3/)$_{0.4}$( $Ti_{0.6}$Z $r_{0.4}$)$_{0.6}$] $O_3$ceramics (x=0, 0.005, 0.02) were investigated with the substitution of B $i^{3+}$, and the feasibility of the application for bimorph actuator was evaluated by measuring the dynamic properties of the piezoelectric bimorph fabricated with above ceramics. Dielectric constant was enhanced with the increase of B $i^{3+}$ substitution, and appeared the maximum value of 5032 at x=0.01 composition. Increasing the substitution of B $i^{3+}$, the electromechanical coefficient( $k_{p}$ , $k_{31}$ ) was increased up to the substitution of 0.5 mol% B $i^{3+}$, showed the value of 0.656, 0.439, respectively. The piezoelectric constant( $d_{33}$ $d_{31}$ ) had the highest value of 344, 825 with the substitution of 0.5 mol% B $i^{3+}$. The strain, generated by 60 Hz AC electric field, had the largest value of 1200($\times$10$^{-6}$ $\Delta$1/1) in the composition with the substitution of 0.5 mol% B $i^{3+}$. The dynamic properties of the bimorph actuator, fabricated with the composition substitution of 0.5 mol% B $i^{3+}$, showed the largest value of 325 $\mu$m at $\pm$150 V square pulse. square pulse.are pulse..

• Journal of the Korean Magnetics Society
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• v.13 no.1
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• pp.36-40
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• 2003

We have grown D $y_{x}$B $i_{3-x}$F $e_{5}$ $O_{12}$ (x = 0.5,1.0, 1.5,2.0) magnetic garnet thin films upon $Al_2$O3i and GGG substrate using Pechini process. The annealing temperature to get single phase D $y_{x}$B $i_{3-x}$F $e_{5}$ $O_{12}$ garnet is dependent on substrate, i.e. the annealing temperature for GGG substrate il 5$0^{\circ}C$ lower than that for $Al_2$ $O_3$ substrate. The grains of garnet thin film grown on GGG (111) plane align along [111] direction, and in this case the hysteresis curve does not saturate up to H : 5000 Oe. We attribute this phenomenon to rotation magnetization process. The maximum amount of Bi substitution in polycrystalline D $y_{x}$B $i_{3-x}$F $e_{5}$ $O_{12}$ thin film prepared by Pechini process is restricted to 2.0 Bi atom/unit cell, and this value is less than that in single garnet crystall grown by LPE method.own by LPE method.ethod.

• 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.2 no.3
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• pp.211-215
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• 1992

The temperature-dependent saturation magnetization curves of amorphous $Fe_{80-x}Co_xB_{20}$ (8 ${\le}$ x ${\le}$ 40, the step of x is 8) alloys were measured using a vibrating sample magnetometer from 77 K up to 1000 K. Curie temperature and the Bloch coefficient were estimated from the saturation magnetization curves. The low temperature dependence of magnetization is in good agreement with Bloch relation, $M_s(T)$=$M_s(O)$ (1-$BT^{3/2}$ - ${\cdots}.$). The spin wave stiffness constant, tne range of the exchange interaction, and the probable atomic spin were calculated from the saturation magnetization values.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
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• 1997.11a
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• pp.68-72
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• 1997

In this paper, to obtain the YbB $a_2$C $u_3$ $O_{x}$ superconductor, the mixed Powders of Y $b_2$ $O_3$, BaC $O_3$, CuO and Y $b_2$BaCu $O_{5}$, BaCu $O_2$ were used and the various calcining conditions were applied for the 123 phase of YbB $a_2$C $u_3$ $O_{x}$. Samples were prepared by the mixed oxide method and calcined with various temperatures of 88$0^{\circ}C$ ~91$0^{\circ}C$ . It was observed that the distribution of YbB $a_2$C $u_3$ $O_{x}$ phase which was calcined at 90$0^{\circ}C$ for 12 hours and 99 hours. But the result of long time calcination(99 hrs), the 123 phase of YbB $a_2$C $u_3$ $O_{x}$ was existed between 89$0^{\circ}C$ and 91$0^{\circ}C$ . And the best case could be obtained at the calcination temp. of 90$0^{\circ}C$ from the mixed Powder of YbB $a_2$C $u_3$ $O_{5}$ and Bacu $O_2$ which were prepared individually.idually.

• Journal of the Korean Ceramic Society
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• v.40 no.10
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• pp.998-1004
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• 2003

With starting materials of two different powder mixtures, $Al_2$(S $O_4$)$_3$+xNa$_2$B$_4$ $O_{7}$$.$10$H_2O$(㏖ ratio; x=0.1, 0.7) and ${\gamma}$-Al$_2$ $O_3$+xNa$_2$B$_4$ $O_{7}$$.$10$H_2O$(㏖ ratio; x=0.1, 0.7), whisker-type $Al_{18}$B$_4$ $O_{33}$ particles were synthesized by using conventional and microwave heat-treatment. The effects of microwave, amount of flux and temperature on the growth of whisker-type $Al_{18}$B$_4$ $O_{33}$ particles were investigated by using X-Ray Diffractometry (XRD) and Scanning Electron Microscopy (SEM). With increase of heat-treatment temperature and amount of flux, the size of whisker-type $Al_{18}$B$_4$ $O_{33}$ particles increased in both conventional and microwave heat-treated samples. However, compared to the conventional heat-treated samples, whisker-type $Al_{18}$B$_4$ $O_{33}$ particles were well grown for the microwave heat-treated samples.ted samples.