• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.795-814
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• 2021

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: fⁿ + P(f) = R(z)e^α(z) and fⁿ + P_*(f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z) in the complex plane, where P(f) and P_*(f) are differential polynomials in f of degree n - 1 with coefficients being small functions and rational functions respectively, R is a non-vanishing small function of f, α is a nonconstant entire function, p₁, p₂ are non-vanishing rational functions, and α₁, α₂ are nonconstant polynomials. Particularly, we consider the solutions of the second equation when p₁, p₂ are nonzero constants, and deg α₁ = deg α₂ = 1. Our results are improvements and complements of Liao ([9]), and Rong-Xu ([11]), etc., which partially answer a question proposed by Li ([7]).

• Journal of the Mineralogical Society of Korea
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• v.16 no.4
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• pp.307-320
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• 2003

Garnet has been considered as a possible matrix for the immobilization of radioactive actinides. It is expected that Fe­based garnet be able to have the high substitution ability of actinide elements because ionic radius of Fe in tetrahedral site is larger than that of Si of Si­based garnet. Accordingly, we synthesized Fe­garnet with the batch composition of $Ca_{2,5}$C $e_{0.5}$Z $r_2$F $e_3$ $O_{12}$ and $Ca_2$CeZrFeF $e_3$ $O_{12}$ and studied their phase relations and properties. Mixed samples were fabricated in pellet forms under the pressure of 400 kg/$\textrm{cm}^2$ and were sintered in the temperature range of 1100∼140$0^{\circ}C$ in atmospheric conditions. Phase identification and chemical composition of synthesized samples were analyzed by XRD and SEM/EDS. In results, where the compounds were sintered at 130$0^{\circ}C$, we optimally obtained Fe­garnets as the main phase, even though some minor phases like perovskite were included. The compositions of Fe­garnets synthesized from the batch compositions of $Ca_{2,5}$C $e_{0.5}$Z $r_2$F $e_3$ $O_{12}$ and $Ca_2$CeZrFeF $e_3$ $O_{12}$, are $Ca_{2.5­3.2}$C $e_{0.3­0.7}$Z $r_{1.8­2.8}$F $e_{1.9­3.2}$ $O_{12}$ and $Ca_{2.2­2.5}$C $e_{0.8­1.0}$Z $r_{1.3­1.6}$ F $e_{0.4­.07}$ F $e_{3­3.2}$ $O_{12}$, respectively. Ca contents were exceeded and Ce contents were exceeded or depleted in 8­coodinated site, comparing to the initial batch composition. These results were caused by the compensation of the difference of ionic radius between Ca and Ce.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.819-829
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• 2016

In this paper, we study the order of growth of solutions to the non-homogeneous linear differential equation $$f^{(k)}+A_{k-1}e^{az}f^{(k-1)}+{\cdots}+A_1e^{az}f^{\prime}+A_0e^{az}f=F_1e^{az}+F_2e^{bz}$$, where $A_j(z)$ (${\not\equiv}0$) ($j=0,1,{\cdots},k-1$), $F_j(z)$ (${\not\equiv}0$) (j = 1, 2) are entire functions and a, b are complex numbers such that $ab(a-b){\neq}0$.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.357-366
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• 2007

Let G(V, E) be a simple graph, and let f be an integer function on V with $1{\leq}f(v){\leq}d(v)$ to each vertex $v{\in}V$. An f-edge cover-coloring of a graph G is a coloring of edge set E such that each color appears at each vertex $v{\in}V$ at least f(v) times. The f-edge cover chromatic index of G, denoted by ${\chi}'_{fc}(G)$, is the maximum number of colors such that an f-edge cover-coloring of G exists. Any simple graph G has an f-edge cover chromatic index equal to ${\delta}_f\;or\;{\delta}_f-1,\;where\;{\delta}_f{=}^{min}_{v{\in}V}\{\lfloor\frac{d(v)}{f(v)}\rfloor\}$. Let G be a connected and not complete graph with ${\chi}'_{fc}(G)={\delta}_f-1$, if for each $u,\;v{\in}V\;and\;e=uv{\nin}E$, we have ${\chi}'_{fc}(G+e)>{\chi}'_{fc}(G)$, then G is called an f-edge covered critical graph. In this paper, some properties on f-edge covered critical graph are discussed. It is proved that if G is an f-edge covered critical graph, then for each $u,\;v{\in}V\;and\;e=uv{\nin}E$ there exists $w{\in}\{u,v\}\;with\;d(w)\leq{\delta}_f(f(w)+1)-2$ such that w is adjacent to at least $d(w)-{\delta}_f+1$ vertices which are all ${\delta}_f-vertex$ in G.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1095-1113
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• 2020

In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^nf^{(k)}+Q_{d_*}(z,f)=R(z)e^{{\alpha}(z)}$ and fⁿf^(k) + Q_d(z, f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z), where $Q_{d_*}(z,f)$ and Q_d(z, f) are differential polynomials in f with small functions as coefficients, of degree d_* (≤ n - 1) and d (≤ n - 2) respectively, R, p₁, p₂ are non-vanishing small functions of f, and α, α₁, α₂ are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.293-299
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• 2005

Let ${\cal{L}}$ be a commutative subspace lattice on a Hilbert space ${\cal{H}}$ and X and Y be operators on ${\cal{H}}$. Let $${\cal{M}}_X=\{{\sum}{\limits_{i=1}^n}E_{i}Xf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}$$ and $${\cal{M}}_Y=\{{\sum}{\limits_{i=1}^n}E_{i}Yf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}.$$ Then the following are equivalent. (i) There is an operator A in $Alg{\cal{L}}$ such that AX = Y, Ag = 0 for all g in ${\overline{{\cal{M}}_X}}^{\bot},A^*A=AA^*$ and every E in ${\cal{L}}$ reduces A. (ii) ${\sup}\;\{K(E, f)\;:\;n\;{\in}\;{\mathbb{N}},f_i\;{\in}\;{\cal{H}}\;and\;E_i\;{\in}\;{\cal{L}}\}\;<\;\infty,\;{\overline{{\cal{M}}_Y}}\;{\subset}\;{\overline{{\cal{M}}_X}}$and there is an operator T acting on ${\cal{H}}$ such that ${\langle}EX\;f,Tg{\rangle}={\langle}EY\;f,Xg{\rangle}$ and ${\langle}ET\;f,Tg{\rangle}={\langle}EY\;f,Yg{\rangle}$ for all f, g in ${\cal{H}}$ and E in ${\cal{L}}$, where $K(E,\;f)\;=\;{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Y\;f_{i}{\parallel}/{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Xf_{i}{\parallel}$.

• Journal of the Institute of Electronics Engineers of Korea TC
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• v.45 no.6
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• pp.27-33
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• 2008

Given a polynomial ${\hbar}(x){\in}F_q[x]$ of degree h, it is an important problem to determine the size of multiplicative subgroup of $\(F_q[x]/({\hbar(x))\)*$ generated by $x-s_1,\;x-s_2,\;{\cdots},\;x-s_n$, where $\{s_1,\;s_2,\;{\cdots},\;s_n\}{\sebseteq}F_q$, and for all ${\hbar}(x){\neq}0$. So far the best known asymptotic lower bound is $(rh)^{O(1)}\(2er+O(\frac{1}{r})\)^h$, where $r=\frac{n}{h}$ and e(=2.718...) is the base of natural logarithm. In this paper, we exploit the coding theory connection of this problem and prove a better lower bound $(rh)^{O(1)}\(2er+{\frac{e}{2}}{\log}r-{\frac{e}{2}}{\log}{\frac{e}{2}}+O{(\frac{{\log}^2r}{r})}\)^h$, where log stands for natural logarithm We also discuss about the limitation of this approach.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.473-481
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• 2009

In this paper, we study the uniqueness of entire functions and prove the following theorem. Let n(${\geq}$ 5), k be positive integers, and let $S_1$ = {z : $z^n$ = 1}, $S_2$ = {$a_1$, $a_2$, ${\cdots}$, $a_m$}, where $a_1$, $a_2$, ${\cdots}$, $a_m$ are distinct nonzero constants. If two non-constant entire functions f and g satisfy $E_f(S_1,2)$ = $E_g(S_1,2)$ and $E_{f^{(k)}}(S_2,{\infty})$ = $E_{g^{(k)}}(S_2,{\infty})$, then one of the following cases must occur: (1) f = tg, {$a_1$, $a_2$, ${\cdots}$, $a_m$} = t{$a_1$, $a_2$, ${\cdots}$, $a_m$}, where t is a constant satisfying $t^n$ = 1; (2) f(z) = $de^{cz}$, g(z) = $\frac{t}{d}e^{-cz}$, {$a_1$, $a_2$, ${\cdots}$, $a_m$} = $(-1)^kc^{2k}t\{\frac{1}{a_1},{\cdots},\frac{1}{a_m}\}$, where t, c, d are nonzero constants and $t^n$ = 1. The results in this paper improve the result given by Fang (M.L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sc. Soc. 24(2001), 7-16).

• Analytical Science and Technology
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• v.26 no.1
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• pp.51-60
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• 2013

The geometrical parameters, vibrational frequencies, and adiabatic electron affinities (AEAs) for c-$C_nF_{2n}$ (n=8, 9) and $C_{10}F_{18}$ (perfluorodecalin) have been investigated using various quantum mechanical techniques. The possible structures for the neutrals and anions of c-PFA are fully optimized and electron affinities are predicted using energy difference between the neutral and anion. The harmonic vibrational frequencies are also determined and zero-point vibrational energies (ZPVEs) are considered for the better prediction of the electron affinities. The electron affinities are predicted to be 1.18 eV for c-$C_8F_{16}$ (ortho), 1.37 eV for c-$C_9F_{18}$, and 1.38 eV for $C_{10}F_{18}$ (perfluorodecalin) at the MP2 level of theory after ZPVE correction.

• The Journal of the Korean bone and joint tumor society
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• v.14 no.2
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• pp.106-118
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• 2008

Purpose: Disturbed cell cycle regulatory proteins are key events underlying the development and/or progression of human malignancies. The aim of this study is to evaluate the protein expression status involved in G1/S cell cycle in human soft tissue sarcoma. Materials and Methods: We simultaneously evaluated the expression of Cyclin D1, Cyclin E, CDK4, CDK2, p16, p27, Rb, E2F1, p53 and Ki-67 by immunohistochemistry in 43 cases of soft tissue sarcoma Results: The Cyclin D1, Cyclin E, CDK4, CDK2, E2F1, and p53 were expressed in 25 (58.1%), 18 (41.9%), 13 (30.2%), 33 (76.7%), 20 (46.5%), and 18 cases (41.9%). The p16, p27, and Rb expressions were decreased in 26 (60.5%), 22 (51.2%) and 19 cases (44.2%). All of the cases showed alterations of more than one out of the above proteins. The increased Cyclin E expression and Ki-67 labeling index (LI) were significantly associated with histologic grade. The Cyclin E and E2F-1 expressions were increased in relapsed cases and the CDK4 expression was increased in cases of metastasis. Conclusion: Alterations of G1/S cell cycle regulatory proteins may play an important role in the tumoriogensis of soft tissue sarcomas. Our results suggest that increased expressions of Cyclin E, E2F1, and CDK4 were associated with tumor relapse or metastasis and could be considered as parameters of prognosis of soft tissue sarcoma.