• Applied Microscopy
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• v.47 no.3
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• pp.110-120
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• 2017

Non-isothermal thermogravimetry (TG) measurements on melt-quenched $Bi_xSe_{100-x}$ specimens (x=0, 2.5, 7.5 at%) were made at a heating rate ${\beta}=10^{\circ}C/min$ in the range $T=35^{\circ}C{\sim}950^{\circ}C$. The as-measured TG curves confirm that $Bi_xSe_{100-x}$ samples were thermally stable with minor loss at $T{\leq}400^{\circ}C$ and mass loss starts to decrease up to $600^{\circ}C$, beyond which trivial mass loss was observed. These TG curves were used to estimate molar (Se/Bi)-ratios of $Bi_xSe_{100-x}$ samples, which were not in accordance with initial composition. Shaping features of conversion curves ${\alpha}(T)-T$ of $Bi_xSe_{100-x}$ samples combined with a reliable flow chart were used to reduce kinetic mechanisms that would have caused their thermal mass loss to few nth-order reaction models of the form $f[{\alpha}(T)]{\propto}[1-{\alpha}(T)]^n$ (n=1/2, 2/3, and 1). The constructed ${\alpha}(T)-T$ and $(d{\alpha}(T)/dT)-T$ curves were analyzed using Coats-Redfern (CR) and Achar-Brindley-Sharp (ABS) kinetic formulas on basis of these model functions, but the linearity of attained plots were good in a limited ${\alpha}(T)-region$. The applicability of CR and ABS methods, with model function of kinetic reaction mechanism R0 (n=0), was notable as they gave best linear fits over much broader ${\alpha}(T)-range$.

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.67-75
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• 2004

Paul Turan observed that the Legendre polynomials satisfy the inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)$ > 0, $-1{\leq}x{\leq}1$. And G. Gasper(ref. [6], ref. [7]) proved such an inequality for Jacobi polynomials and J. Bustoz and N. Savage (ref. [2]) proved $P^{\alpha}_n(x)P^{\beta}_{n+1}(x)-P^{\alpha}_{n+1}(x)P{\beta}_n(x)$ > 0, $\frac{1}{2}{\leq}{\alpha}$ < ${\beta}{\leq}{\alpha}+2.0$ < $x$ < 1, for the ultraspherical polynomials (respectively, Laguerre ploynomials). The Bustoz-Savage inequalities hold for Laguerre and ultraspherical polynomials which are symmetric. In this paper, we prove some similar inequalities for non-symmetric Jacobi polynomials.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.439-442
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• 1995

Let ${X_n : n = 1,2,\cdots}$ be a sequence of pairwise independent identically distributed random vectors taking values in a separable Hilbert space H such that $E \Vert X_1 \Vert = \infty$. Let $S_n = X_1 + X_2 + \cdots + X_n$ and for any real $\alpha$ with $0 < \alpha < 1$ define a sequence ${\gamma_n(\alpha)}$ as $\gamma_n(\alpha) = inf {r : P(\Vert S_n \Vert \leq r) \geq \alpha}$. Then $$ lim_{n \to \infty} sup \Vert S_n \Vert/\gamma_n(\alpha) = \infty $$ holds. This is a generalization of Vvedenskaya[2].

• Bulletin of the Korean Mathematical Society
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• v.33 no.2
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• pp.303-310
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• 1996

The Gottlieb group of a compact connected ANR X, G(X), consists of all $\alpha \in \prod_{1}(X)$ such that there is an associated map $A : S^1 \times X \to X$ and a homotopy commutative diagram $$ S^1 \times X \longrightarrow^A X $$ $$incl \uparrow \nearrow \alpha \vee id $$ $$ S^1 \vee X $$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.267-278
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• 2013

We consider zero range processes with density dependent jump rates g given by $g=g(n,k)=g_1(n)g_2(k/n)$ with $g_1(x)=x^{-\alpha}$ and $$g_2(x)=\{^{x^{-\alpha}\;if\;a&lt;x}_{Mx^{-\alpha}\;if\;x{\leq}a}$$. (0.1) In this case, with 1/2 < a < 1 and ${\alpha}$ > 0, we show that non-complete condensation occurs with maximum cluster size an. More precisely, for any ${\epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0 < M ${\leq}M^*$, the maximum cluster size is between (a - ${\epsilon}$)n and (a + ${\epsilon}$)n for large n. This provides a simple example of non-complete condensation under perturbation of rates which are deep in the range of perfect condensation (e.g. ${\alpha}$ >> 1) and supports the instability of the condensation transition.

• Journal of Korean Ophthalmic Optics Society
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• v.6 no.2
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• pp.65-70
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• 2001

We obtained the sum of two curvature ($C_x+C_y$) in toric lens which two toroidal surface is the right angle each other. $$C_x+C_y=\frac{x^2+y^2}{2r_1}+\frac{x^2}{2}(\frac{1}{r_2}-\frac{1}{r_1})$$ and the sum of two curvature ($C_a+C_b$) in toric lens about the cross angle. $$(C_a+C_b)=\frac{x^2cos^2{\alpha}_1}{2r_1}+\frac{x^2cos^2{\alpha}_2}{2r_2}+\frac{y^2sin^2{\alpha}_1}{2r_1}+\frac{y^2sin^2{\alpha}_2}{2r_2}$$ and claculated the parameter S, C, ${\theta}$ of a combination power in toric lens of the cross angle including surface curvature ($C_x$, $C_y$) values. $$S=(n-1)\[\frac{C_x}{x^2}+\frac{C_y}{y^2}\]-\frac{C}{2},\;C=-\frac{2(n-1)}{sin2{\theta}}\[\frac{C_x}{x^2}+\frac{C_y}{y^2}\]$$ $${\theta}=\frac{1}{2}tan^{-1}\[-\frac{{C_xy^2sin2{\theta}_1}+{C_yx^2sin2{\theta}_2}}{{C_xy^2cos2{\theta}_1}+{C_yx^2cos2{\theta}_2}}\]$$.

• Kyungpook Mathematical Journal
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• v.57 no.4
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• pp.623-630
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• 2017

For any nonzero elements x, y in a normed space X, the angular and skew-angular distance is respectively defined by ${\alpha}[x,y]={\parallel}{\frac{x}{{\parallel}x{\parallel}}}-{\frac{y}{{\parallel}y{\parallel}}}{\parallel}$ and ${\beta}[x,y]={\parallel}{\frac{x}{{\parallel}y{\parallel}}}-{\frac{y}{{\parallel}x{\parallel}}}{\parallel}$. Also inequality ${\alpha}{\leq}{\beta}$ characterizes inner product spaces. Operator version of ${\alpha}$ has been studied by $ Pe{\check{c}}ari{\acute{c}}$, $ Raji{\acute{c}}$, and Saito, Tominaga, and Zou et al. In this paper, we study the operator version of ${\beta}$ by using Douglas' lemma. We also prove that the operator version of inequality ${\alpha}{\leq}{\beta}$ holds for commutating normal operators. Some examples are presented to show essentiality of these conditions.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1105-1119
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• 2010

We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the pro cesses are interpreted as crashes and we assume that the jump intensity is a nondecreasing function of the current process say $\lambda$(X) (X = X(t) process). For the case of $\lambda$(X) = $X^{\alpha}$, $\alpha$ > 0, we show that the process X shold explode in finite time, say $t_e$, conditional on no crash For the case of $\lambda$(X) = (lnX)$^{\alpha}$, we show that $\alpha$ = 1 is the borderline of two different classes of processes. We generalize the model by adding a Brownian noise and examine the blow up properties of the sample paths.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.187-194
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• 2011

Examples of a quartically hyponormal weighted shift which is not 3-hyponormal are discussed in this note. In [7] Exner-Jung-Park proved that if ${\alpha}$(x) : $\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},{\cdots}$ with 0 < x ${\leq}\;\frac{53252}{100000}$, then $W_{\alpha(x)}$ is quartically hyponormal but not 4-hyponormal. And, Curto-Lee([5]) improved their result such as that if ${\alpha}(x)$ : $\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},{\cdots}$ with 0 < x ${\leq}\;\frac{667}{990}$, then $W_{\alpha(x)}$ is quartically hyponormal but not 3-hyponormal. In this note, we improve slightly Curto-Lee's extremal value by using an algorithm and computer software tool.

• The Korean Journal of Applied Statistics
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• v.5 no.2
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• pp.243-254
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• 1992

In the present study two sets of unbalanced two-way cross-classification data with and without empty cell(s) were used to evaluate empirically the various sums of squares in the analysis of variance table. Searle(1977) and Searle et.al.(1981) developed a method of computing R($\alpha$\mid$\mu, \beta$) and R($\beta$\mid$\mu, \alpha$) by the use of partitioned matrix of X'X for the model of no interaction, interchanging the columns of X in order of $\alpha, \mu, \beta$ and accordingly the elements in b. An alternative way of computing R($\alpha$\mid$\mu, \beta$), R($\beta$\mid$\mu, \alpha$) and R($\gamma$\mid$\mu, \alpha, \beta$) without interchanging the columns of X has been found by means of,$(X'X)^-$ derived, using $W_2 = Z_2Z_2-Z_2Z_1(Z_1Z_1)^-Z_1Z_2$. It is true that $R(\alpha$\mid$\mu,\beta,\gamma)\Sigma = SSA_W and R(\beta$\mid$\mu,\alpha,\gamma)\Sigma = SSB_W$ where $SSA_W$ and means analysis and $R(\gamma$\mid$\mu,\alpha,\beta) = R(\gamma$\mid$\mu,\alpha,\beta)\Sigma$ for the data without empty cell, but not for the data with empty cell(s). It is also noticed that for the datd with empty cells under W - restrictions $R(\alpha$\mid$\mu,\beta,\gamma)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W = R(\alpha$\mid$\mu) and R(\beta$\mid$\mu,\alpha,\gamma)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W = R(\beta$\mid$\mu) but R(\gamma$\mid$\mu,\alpha,\beta)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W \neq R(\gamma$\mid$\mu,\alpha,\beta)$. The hypotheses $H_o : K' b = 0$ commonly tested were examined in the relation with the corresponding sums of squares for $R(\alpha$\mid$\mu), R(\beta$\mid$\mu), R(\alpha$\mid$\mu,\beta), R(\beta$\mid$\mu,\alpha), R(\alpha$\mid$\mu,\beta,\gamma), R(\beta$\mid$\mu,\alpha,\gamma), and R(\gamma$\mid$\mu,\alpha,\beta)$ under the restrictions.