• 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.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.915-923
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• 2022

The purpose of the present paper is to obtain commutativity of prime Γ-near-ring N with generalized derivations F and G with associated derivations d and h respectively satisfying one of the following conditions:(i) G([x, y]α = ±f(y)α(xoy)βγg(y), (ii) F(x)βG(y) = G(y)βF(x), for all x, y ∈ N, β ∈ Γ (iii) F(u)βG(v) = G(v)βF(u), for all u ∈ U, v ∈ V, β ∈ Γ,(iv) if 0 ≠ F(a) ∈ Z(N) for some a ∈ V such that F(x)αG(y) = G(y)αF(x) for all x ∈ V and y ∈ U, α ∈ Γ.

• Proceedings of the Korean Society of Potoscience Conference
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• spring
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• pp.20-20
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• 1996

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1185-1196
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• 2016

We show a class of homogeneous polynomials of Fermat-Waring type such that for a polynomial P of this class, if $P(f_1,{\ldots},f_{N+1})=P(g_1,{\ldots},g_{N+1})$, where $f_1,{\ldots},f_{N+1}$; $g_1,{\ldots},g_{N+1}$ are two families of linearly independent entire functions, then $f_i=cg_i$, $i=1,2,{\ldots},N+1$, where c is a root of unity. As a consequence, we prove that if X is a hypersurface defined by a homogeneous polynomial in this class, then X is a unique range set for linearly non-degenerate non-Archimedean holomorphic curves.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1483-1503
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• 2008

Let f : M ${\rightarrow}$ M be a self-map on the Klein bottle M. We compute the Lefschetz number and the Nielsen number of f by using the infra-nilmanifold structure of the Klein bottle and the averaging formulas for the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer n, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $NP_n(f)$ and $N{\Phi}_{n}(f)\;of\;f^{n}$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1529-1561
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• 2018

Let f and g be nonconstant meromorphic (entire, respectively) functions in the complex plane such that f and g are of finite order, let a and b be nonzero complex numbers and let n be a positive integer satisfying $n{\geq}21$ ($n{\geq}12$, respectively). We show that if the difference polynomials $f^n(z)+af(z+{\eta})$ and $g^n(z)+ag(z+{\eta})$ share b CM, and if f and g share 0 and ${\infty}$ CM, where ${\eta}{\neq}0$ is a complex number, then f and g are either equal or at least closely related. The results in this paper are difference analogues of the corresponding results from.

• East Asian mathematical journal
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• v.31 no.1
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• pp.103-107
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• 2015

We use some known contiguous function relations for $_2F_1$ to show how simply the following three recurrence relations for Jacobi polynomials $P_n^{({\alpha},{\beta)}(x)$: (i) $({\alpha}+{\beta}+n)P_n^{({\alpha},{\beta})}(x)=({\beta}+n)P_n^{({\alpha},{\beta}-1)}(x)+({\alpha}+n)P_n^{({\alpha}-1,{\beta})}(x);$ (ii) $2P_n^{({\alpha},{\beta})}(x)=(1+x)P_n^{({\alpha},{\beta}+1)}(x)+(1-x)P_n^{({\alpha}+1,{\beta})}(x);$ (iii) $P_{n-1}^{({\alpha},{\beta})}(x)=P_n^{({\alpha},{\beta}-1)}(x)+P_n^{({\alpha}-1,{\beta})}(x)$ can be established.

• Food Science and Preservation
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• v.12 no.3
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• pp.275-281
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• 2005

This study was carried out to investigate the cathepsin B inhibition effect by Achyranthes japonica N. root extract in vitro. The methanol/$H_{2}O$(4:1, v/v) extract was fractionated by ethyl acetate(F1), chloroform(F2), chloroform/methanol(3:1, v/v)(F3) and methanol(F4). The yield of F4 in Achyranthes japonica N. root was $8.27\%$. As an index material of Achyranthes japonica N. root, 20-hydroxy ecdysone was detected by TLC, and HPLC and it's content was $0.33\%$. Three isolates(F1, F3, F4) showed the cathepsin B inhibition activity, and F4 showed the highest inhibition activity among them. In the inhibition activity on cathepsin B, leupeptin, 20-hydroxy ecdysone and F4(at the same concentration of 20-hydroxy ecdysone.) were 92, 88 and $97\%$ on BANA($N{\alpha}$-benzoyl-DL-arginine ${\beta}$-naphthylamide) substrate, and 62, 36 and $67\%$ on CLN($N{\alpha}$-CBZ(carbobenzlyoxy)-L-lysine p-nitrophenyl ester HCI) substrate, respectively.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.495-505
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• 2018

The uniqueness problems on entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study an entire function f(z) that shares a nonzero polynomial a(z) with $f^{(1)}(z)$, together with its linear differential polynomials of the form: $L=L(f)=a_1(z)f^{(1)}(z)+a_2(z)f^{(2)}(z)+{\cdots}+a_n(z)f^{(n)}(z)$, where the coefficients $a_k(z)(k=1,2,{\ldots},n)$ are rational functions and $a_n(z){\not{\equiv}}0$.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.645-648
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• 2005

Let K be a Galois extension of a field F with G = Gal(K/F). Let L be an extension of F such that $K\;{\otimes}_F\;L\;=\; N_1\;{\oplus}N_2\;{\oplus}{\cdots}{\oplus}N_k$ with corresponding primitive idempotents $e_1,\;e_2,{\cdots},e_k$, where Ni's are fields. Then G acts on $\{e_1,\;e_2,{\cdots},e_k\}$ transitively and $Gal(N_1/K)\;{\cong}\;\{\sigma\;{\in}\;G\;/\;{\sigma}(e_1)\;=\;e_1\}$. And, let R be a commutative F-algebra, and let P be a prime ideal of R. Let T = $K\;{\otimes}_F\;R$, and suppose there are only finitely many prime ideals $Q_1,\;Q_2,{\cdots},Q_k$ of T with $Q_i\;{\cap}\;R\;=\;P$. Then G acts transitively on $\{Q_1,\;Q_2,{\cdots},Q_k\},\;and\;Gal(qf(T/Q_1)/qf(R/P))\;{\cong}\;\{\sigma{\in}\;G/\;{\sigma}-(Q_1)\;=\;Q_1\}$ where qf($T/Q_1$) is the quotient field of $T/Q_1$.