• Bulletin of the Korean Chemical Society
• /
• v.32 no.6
• /
• pp.2045-2050
• /
• 2011

Using first principles density functional theory the formation energies of various binary compounds of lithium graphite and its homologues were calculated. Lithium and graphite react to form $Li_1C_6$ (+141 mV) but not form $LiC_4$ (-143 mV), $LiC_3$ (-247 mV) and $LiC_2$ (-529 mV) because they are less stable than lithium metal itself. Properties of structure and reaction potentials of $C_5B$, $C_5N$ and $B_3N_3$ materials as iso-structural graphite were studied. Boron and nitrogen substituted graphite and boron-nitrogen material as a iso-electronic structured graphitic material have longer graphene layer spacing than that of graphite. The layer spacing of $Li_xC_6$, $Li_xC_5B$, $Li_xC_5N$ materials increased until to x=1, and then decreased until to x=2 and 3. Nevertheless $Li_xB_3N_3$ has opposite tendency of layer spacing variation. Among various lithium compositions of $Li_xC_5B$, $Li_xC_5N$ and $Li_xB_3N_3$, reaction potentials of $Li_xC_5B$ (x=1-3) and $Li_xC_5$ (x=1) from total energy analyses have positive values against lithium deposition.

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

• Honam Mathematical Journal
• /
• v.30 no.3
• /
• pp.567-579
• /
• 2008

We obtain generalized super stability of Cauchy's gamma-beta functional equation B(x, y) f(x + y) = f(x)f(y), where B(x, y) is the beta function and also generalize the stability in the sense of R. Ger of this equation in the following setting: ${\mid}{\frac{B(x,y)f(x+y)}{f(x)f(y)}}-1{\mid}$ < H(x,y), where H(x,y) is a homogeneous function of dgree p(0 ${\leq}$ p < 1).

• Kyungpook Mathematical Journal
• /
• v.55 no.1
• /
• pp.1-12
• /
• 2015

Let R be a ring and $D:R^n{\longrightarrow}R$ be n-additive mapping. A map $d:R{\longrightarrow}R$ is said to be the trace of D if $d(x)=D(x,x,{\ldots}x)$ for all $x{\in}R$. Suppose that ${\alpha},{\beta}$ are endomorphisms of R. For any $a,b{\in}R$, let < a, b > $_{({\alpha},{\beta})}=a{\alpha}(b)+{\beta}(b)a$. In the present paper under certain suitable torsion restrictions it is shown that D = 0 if R satisfies either < d(x), $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$ or ${\ll}$ d(x), x > $_{({\alpha},{\beta})}$, $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$. Further, if < d(x), x > ${\in}Z(R)$, the center of R, for all $x{\in}R$ or < d(x)x - xd(x), x >= 0, for all $x{\in}R$, then it is proved that d is commuting on R. Some more related results are also obtained for additive mapping on R.

• Journal of Magnetics
• /
• v.14 no.4
• /
• pp.150-154
• /
• 2009

The effect of Ga and, Nb addition on the kinetics and mechanism of the disproportionation of a Nd-Fe-B alloy were investigated by isothermal thermopiezic analysis (TPA) using $Nd_{12.5}Fe_{(81.1-(x+y))}B_{6.4}Ga_xNb_y$ (x=0 and 0.3, y= 0 and 0.2) alloys. The addition of Ga and Nb retarded the disproportionation kinetics of the Nd-Fe-B alloy significantly, and increased the activation energy of the disproportionation reaction. The disproportionation kinetics of the $Nd_{12.5}Fe_{(81.1-(x+y))}B_{6.4}Ga_xNb_y$ alloys measured under an initial hydrogen pressure of 0.02 MPa were fitted to a parabolic rate law. This suggested that during the disproportionation of $Nd_{12.5}Fe_{(81.1-(x+y))}B_{6.4}Ga_xNb_y$ alloys with an initial hydrogen pressure of 0.02 MPa, a continuous disproportionation product is formed and the overall reaction rate is limited by the diffusion of hydrogen atoms (or ions).

• Journal of applied mathematics & informatics
• /
• v.15 no.1_2
• /
• pp.457-466
• /
• 2004

Let{$X_{ni}$\mid$\;1\;{\leq}\;i\;{\leq}\;k_n,\;n\;{\geq}\;1$} be an array of rowwise negatively associated random variables such that $P$\mid$X_{ni}$\mid$\;>\;x)\;=\;O(1)P($\mid$X$\mid$\;>\;x)$ for all $x\;{\geq}\;0,\;and\; \{k_n\}\;and\;\{r_n\}$ be two sequences such that $r_n\;{\geq}\;b_1n^r,\;k_n\;{\leq}\;b_2n^k$ for some $b_1,\;b_2,\;r,\;k\;>\;0$. Then it is shown that $\frac{1}{r_n}\;max_1$\mid${\Sigma_{i=1}}^j\;X_{ni}$\mid$\;{\rightarrow}\;0$ completely convergence and the strong convergence for weighted sums of N A arrays is also considered.

• Bulletin of the Korean Mathematical Society
• /
• v.22 no.2
• /
• pp.121-126
• /
• 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 the Korean Crystal Growth and Crystal Technology
• /
• v.9 no.3
• /
• pp.286-288
• /
• 1999

When concentration of vacancies in a CZ silicon crystal is defined by molar fraction $X_{B}$, the degree for supersaturation $\sigma$ is given by $[X_{B}-X_{BS}]/X_{BS}=X_{B}/X_{BS}-1=ln(X_{B}/X_{BS})$ because $X_{B}/X_{BS}$ is nearly equal to unity. Here, $X_{BS}$ is the saturated concentration of vacancies in a silicon crystal and $X_{B}$ is a little larger than $X_{BS}$. According to Bragg-Williams approximation, the chemical potential of the vacancies in the crystal is given by ${\mu}_{B}={\mu}^{0}+RT$ ln $X_{B}+RT$ ln ${\gamma}$, where R is the gas constant, T is temperature, ${\mu}^{0}$ is an ideal chemical potential of the vacancies and ${\gamma}$ is and adjustable parameter similar to the activity of solute in a solute in a solution. Thus, ${\sigma}(T)$ is equal to $({\mu}_{B}-{\mu}_{BS})/RT$. Driving force of nucleation for the vacancy agglomeration will be proportional to the chemical potentialdifference $({\mu}_{B}-{\mu}_{BS})/RT$ or ${\sigma}(T)$, while growth of the vacancy agglomeration is proportaional to diffusion of the vacancies and grad ${\mu}_{B}$.

• Kyungpook Mathematical Journal
• /
• v.13 no.2
• /
• pp.235-240
• /
• 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]).

• Korean Journal of Soil Science and Fertilizer
• /
• v.34 no.5
• /
• pp.311-315
• /
• 2001

In order to improve a soil test-based fertilization model for nitrogen, the Mitscherlich-Baule-Spillman equation modified by Bray and his coworkers was introduced and tested for its possibility of application. The coefficients $C_1$ and $C_2$ in the equation A were calculated by applying 10 pot experimental results done by the National Institute of Agricultural Sciences and Technology. Then, a regression(B) between C_1 and x, the fertilizer level was estimated. log(Ymax-Y) = logYmax-$C_1x-C_2b$ (A) where Ymax : maximum yield. Y : yield without or with a limited amount of fertilizer applied, x : application level of fertilizer, b : content of a nutrient in soil, $C_1$ and $C_2$ : coefficients of x and b, respectively. log(0.05)=$-C_1x-C_2b$ (B) Fertilizer requirement calculated by the equation B was significantly correlated with the real level of fertilizer applied for 95 percentage of the maximum yield. A more extensive and precise study on $C_1$ and $C_2$ in the equation A in expected to improve significantly the fitness of soil analysis to the fertilizer recommendation.