• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.983-1002
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• 2021

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]ⁿ + P²(z)f^m(z + 𝜂) = Q(z) and [f(z)f'(z)]ⁿ + P(z)[∆_𝜂f(z)]^m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P²(z) [f^(k)(z)]² + [αf(z + 𝜂) - 𝛽f(z)]² = e^r(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.

• Journal of the Korean Mathematical Society
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• v.40 no.5
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• pp.769-788
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• 2003

In this paper, we study the optimal control for the damped semilinear hyperbolic systems with unknown parameters (C(t)y')'＋ $A_2$(t, q)y'＋ $A_1$(t, q)y = f(t, q, y, u). We will prove the existence of weak solution of this system and is to find the optimal control pair (q, u) $\in$ $Q_{t}$ ${\times}$ $U_{ad}$ such that in $f_{u}$$\in$ $Q_{t}$/ J(q, u) = J(q, u).$_{t}$/ J(q, u) = J(q, u).

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.83-98
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• 2014

During the last decade, several papers have focused on linear q-difference equations of the form ${\sum}^n_{j=0}a_j(z)f(q^jz)=a_{n+1}(z)$ with entire or meromorphic coefficients. A tool for studying these equations is a q-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order ${\rho}_{log}$. It is shown, under certain assumptions, that ${\rho}_{log}(f)$ = max${{\rho}_{log}(a_j)}$ + 1. Moreover, it is illustrated that a q-Casorati determinant plays a similar role in the theory of linear q-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the q-gamma function and the q-exponential functions all have logarithmic order two.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.1
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• pp.9-16
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• 2014

Let $\mathbb{A}=\mathbb{F}_q[T]$ be a polynomial ring over $\mathbb{F}_q$. In this paper we determine an asymptotic mean value of quadratic Dirich-let L-functions L(s, ${\chi}_{{\gamma}D}$) at the central point s=$\frac{1}{2}$, where D runs over all monic square-free polynomials of even degree in $\mathbb{A}$ and ${\gamma}$ is a generator of $\mathbb{F}_q^*$.

• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.57-66
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• 2009

In this paper, we characterize the boundedness and compactness of the extended $Ces{\grave{a}}ro$ operators from general function spaces F(p, q, s) to Bloch-type spaces ${\mathcal{B}}_{\mu}$ where $\mu$ is normal function on [0,1).

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.557-564
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• 2012

This study concerns the existence of positive solution for the following nonlinear system $$\{-div(|x|^{-ap}|{\nabla}u|^{p-2}{\nabla}u)=|x|^{-(a+1)p+c_1}({\alpha}_1f(v)+{\beta}_1h(u)),x{\in}{\Omega},\\-div(|x|^{-bq}|{\nabla}v|q^{-2}{\nabla}v)=|x|^{-(b+1)q+c_2}({\alpha}_2g(u)+{\beta}_2k(v)),x{\in}{\Omega},\\u=v=0,x{\in}{\partial}{\Omega}$$, where ${\Omega}$ is a bounded smooth domain of $\mathbb{R}^N$ with $0{\in}{\Omega}$, 1 < $p,q$ < N, $0{{\leq}}a<\frac{N-p}{p}$, $0{{\leq}}b<\frac{N-q}{q}$ and $c_1$, $c_2$, ${\alpha}_1$, ${\alpha}_2$, ${\beta}_1$, ${\beta}_2$ are positive parameters. Here $f,g,h,k$ : $[0,{\infty}){\rightarrow}[0,{\infty})$ are nondecresing continuous functions and $$\lim_{s{\rightarrow}{\infty}}\frac{f(Ag(s)^{\frac{1}{q-1}})}{s^{p-1}}=0$$ for every A > 0. We discuss the existence of positive solution when $f,g,h$ and $k$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2165-2182
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• 2017

Let G be a group and $\mathbb{C}$ the field of complex numbers. Suppose ${\sigma}:G{\rightarrow}G$ is an endomorphism satisfying ${{\sigma}}({{\sigma}}(x))=x$ for all x in G. In this paper, we first determine the central solution, f : G or $G{\times}G{\rightarrow}\mathbb{C}$, of the functional equation $f(xy)+f({\sigma}(y)x)=2f(x)+2f(y)$ for all $x,y{\in}G$, which is a variant of the quadratic functional equation. Using the central solution of this functional equation, we determine the general solution of the functional equation f(pr, qs) + f(sp, rq) = 2f(p, q) + 2f(r, s) for all $p,q,r,s{\in}G$, which is a variant of the equation f(pr, qs) + f(ps, qr) = 2f(p, q) + 2f(r, s) studied by Chung, Kannappan, Ng and Sahoo in [3] (see also [16]). Finally, we determine the solutions of this equation on the free groups generated by one element, the cyclic groups of order m, the symmetric groups of order m, and the dihedral groups of order 2m for $m{\geq}2$.

• Journal of Korean Society of Forest Science
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• v.90 no.1
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• pp.19-27
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• 2001

From two to five-year-old hybrids of Q. serrate ${\times}$ Q. dentate; Q. dentata ${\times}$ Q. serrata, Q. dentate ${\times}$ Q. mongolica var. crispula, and Q. dentata ${\times}$ Q. aliena, intermediate-size of stellate hairs or both types of their parents have been observed. Especially the hybrids between Q. serrate and Q. aliena and Q. aliena and Q. serrate had both types of their parents. The trichomes of the hybrids between Q. fabri and Q. serrate was the characteristics of Q. serrate. It was possible to identify two to five-year-old of interspecific hybrids based on trichome characteristics.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.319-339
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• 2010

In this paper, we prove that if $f^nf'\;-\;P$ and $g^ng'\;-\;P$ share 0 CM, where f and g are two distinct transcendental meromorphic functions, $n\;{\geq}\;11$ is a positive integer, and P is a nonzero polynomial such that its degree ${\gamma}p\;{\leq}\;11$, then either $f\;=\;c_1e^{cQ}$ and $g\;=\;c_2e^{-cQ}$, where $c_1$, $c_2$ and c are three nonzero complex numbers satisfying $(c_1c_2)^{n+1}c^2\;=\;-1$, Q is a polynomial such that $Q\;=\;\int_o^z\;P(\eta)d{\eta}$, or f = tg for a complex number t such that $t^{n+1}\;=\;1$. The results in this paper improve those given by M. L. Fang and H. L. Qiu, C. C. Yang and X. H. Hua, and other authors.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.267-280
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• 2017

In this paper, we study analytic and geometric properties of the solution q(z) to the differential equation q(z) + zq'(z)/q(z) = h(z) with the initial condition q(0) = 1 for a given analytic function h(z) on the unit disk |z| < 1 in the complex plane with h(0) = 1. In particular, we investigate the possible largest constant c > 0 such that the condition |Im [zf"(z)/f'(z)]| < c on |z| < 1 implies starlikeness of an analytic function f(z) on |z| < 1 with f(0) = f'(0) - 1 = 0.