• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.493-511
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• 2009

Let $q\;{\geq}\;3$ be an odd integer and a be an integer coprime to q. Denote by N(a, q) the number of pairs of integers b, c with $bc\;{\equiv}\;a$ (mod q), $1\;{\leq}\;b$, $c\;{\leq}\;{\frac{q-1}{2}}$ and with b, c having different parity. The main purpose of this paper is to study the sum ${\sum}^{'q}_{a=1}\;\(N(a,\;q)\;-\;\frac{{\phi}(q)}{8}\)^2$ and obtain a sharp asymptotic formula.

• Journal of the Korean Mathematical Society
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• v.46 no.4
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• pp.733-746
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• 2009

For any integer k $\geq$ 2, let P(c, k + 1;q) be the number of all k+1-tuples with positive integer coordinates ($a_1,a_2,...,a_{k+1}$) such that $1{\leq}a_i{\leq}q$, ($a_i,q$) = 1, $a_1a_2...a_{k+1}{\equiv}$ c (mod q) and 2 $\nmid$ ($a_1+a_2+...+a_{k+1}$), and E(c, k+1; q) = P(c, k+1;q) - $\frac{{\phi}^k(q)}{2}$. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k+1,r;q) and E(c,k+1;q), and give an interesting mean value formula.

• Korean Journal of Mathematics
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• v.29 no.4
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• pp.665-677
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• 2021

The main object of this study is to introduce the spaces $c_0({\hat{F}^q)$ and $c({\hat{F}^q)$ derived by the matrix ${\hat{F}^q$ which is the multiplication of Riesz matrix and Fibonacci matrix. Moreover, we find the 𝛼-, 𝛽-, 𝛾- duals of these spaces and give the characterization of matrix classes (${\Lambda}({\hat{F}^q)$, Ω) and (Ω, ${\Lambda}({\hat{F}^q)$) for 𝚲 ∈ {c₀, c} and Ω ∈ {ℓ₁, c₀, c, ℓ_∞}.

• Korean Journal of Food Science and Technology
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• v.46 no.2
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• pp.180-188
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• 2014

This study focused on assessing the solubility and structural characteristics of two types of coenzyme $Q_{10}$ ($CoQ_{10}$) complexes: the $CoQ_{10}$-starch and the $CoQ_{10}$-cyclodextrin complexes. The solubility of $CoQ_{10}$-starch complex increased significantly as the temperature was increased. However, the solubility of $CoQ_{10}$-cyclodextrin complex reached a peak at $37^{\circ}C$, and strong aggregation occurred at $50^{\circ}C$. When the temperature was raised to $80^{\circ}C$, the $CoQ_{10}$-cyclodextrin complex dissociated owing to the weakening of bonds, resulting in $CoQ_{10}$ emerging at the surface of water. Therefore, $CoQ_{10}$-cyclodextrin complexes have lower solubility, due to their reduced heat-stability, than do the $CoQ_{10}$-starch complexes. Structural differences between the two $CoQ_{10}$ complexes were confirmed by Fourier transform infrared (FT-IR) spectroscopy, X-ray diffractometer (XRD), and differential scanning calorimeter (DSC). The $CoQ_{10}$-cyclodextrin complex included an isoprenoid chain of $CoQ_{10}$, while the $CoQ_{10}$-starch complex included both the benzoquinone ring and the isoprenoid chain of $CoQ_{10}$. These results suggest that $CoQ_{10}$-starch complexes possess higher heat-stability and solubility than do the $CoQ_{10}$-cyclodextrin complexes.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.4
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• pp.475-490
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• 2019

Let C^(q) be the q-commuting arithmetic table of (x + y)ⁿ with noncommuting variables. We study sequential properties of diagonal sums of C^(q) with various q > 0. One of our results shows that each diagonal sum plays like Fibonacci number with some minor conditions.

• Journal of Korean Society of Forest Science
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• v.78 no.2
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• pp.151-159
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• 1989

The present study has been designed to define the effects of photosynthetically active radiation, leaf temperature, and water stress on photosynthesis and respiration of leaves of four oak species (Quercus mongolica, Quercus aliens, Quercus variabilis, and Quercus serrate). The results obtained are as follows : 1. The estimated light compensation points at which Pn approached zero were 38, 24, 20, and $18{\mu}Em^{-2}s^{-1}$ for Q. aliens, Q. variabilis, Q, mongolica, and Q. serrate, respectively. The light saturation points occurred at $500{\mu}Em^{-2}s^{-1}$ in three oak species except Q, aliens. 2. The maximum rates of Pn were 19.7, 15.2, 11.2, and 11.0 mg $CO_2$ $dm^{-2}h^{-1}$ for Q. variabilis, Q. serrate, Q. monglica, and Q. aliens leaves, respectively. 3. The transpiration rates of Q. variabilis and Q. serrate leaves were slightly higher than those of Q. mongolica and Q. aliens leaves at various photosynthetically active radiations(PAR), but cuticular transpiration rates at dark were similar in four oak species. 4. The optimum photosynthesis occurred at $25^{\circ}C$ in Q. aliens, Q. variabilis, and Q. serrate leaves, but $20^{\circ}C$ in Q. mongolica leaves. In four oak species, the net photosynthesis approached zero at about $40^{\circ}C$. 5. The dark respiration rates of leaves exhibited the following ranking of species : Q, variabilis > Q. mongolica > Q. aliens > Q. serrate. 6. The maximum productive efficiency (Pg/Rd) of leaves occurred highest in Q, serrate at $20^{\circ}C$, then in Q. mongolica at $20^{\circ}C$, then in Q, aliens at $25^{\circ}C$, and finally in Q. variabilis at $15^{\circ}C$. 7. The decrease of net photosynthesis in Q. serrate began at about -1.2 MPa, and then approached zero at -2.9 MPa of leaf water potential. The decrease of net photosynthesis began at 3% of water loss, and then approached zero at 17.5% of water loss. 8. As indicated by tissue-water relations parameters, it may be suggested that Q. aliens and Q. variabilis are more tolerant and favored on xeric forest soils than Q. mongolica and Q. serrate.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.603-617
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• 1997

Consider a Sobolev inner product on the space of polynomials such as $$ \phi(p,q) = \lambda p(c)q(c) + <\tau,p'(x)q'(x)> $$ where $\tau$ is a moment functional and c and $\lambda$ are real constants. We investigate properties of orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and give necessary and sufficient conditions under which such Sobolev orthogonal polynomials satisfy a spectral type differential equation with polynomial coefficients.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.131-138
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• 2000

Let D be the open unit disk in the complex plane $\mathbb{C}$. For any q > 0, the operator $Q_q$ defined by $$Q_qf(z)=q\int_{D}\frac{f(\omega)}{(1-z{\bar{\omega}})^{1+q}}d{\omega},\;z{\in}D$$. maps $L^{\infty}(D)$ boundedly onto $B_q$ for each q > 0. In this paper, weighted Bloch spaces $\mathcal{B}_q$ (q > 0) are considered on the open unit ball in $\mathbb{C}^n$. In particular, we will investigate the possibility of extension of this operator to the Weighted Bloch spaces $\mathcal{B}_q$ in $\mathbb{C}^n$.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.277-283
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• 1992

Let GF(q) be a finite field with q elements where q=p$^{n}$ for a prime number p and a positive integer n. Consider an arbitrary function .phi. from GF(q) into GF(q). By using the Largrange's Interpolation formula for the given function .phi., .phi. can be represented by a polynomial which is congruent (mod x$^{q}$ -x) to a unique polynomial over GF(q) with the degree < q. In [3], Wells characterized all polynomial over a finite field which commute with translations. Mullen [2] generalized the characterization to linear polynomials over the finite fields, i.e., he characterized all polynomials f(x) over GF(q) for which deg(f) < q and f(bx+a)=b.f(x) + a for fixed elements a and b of GF(q) with a.neq.0. From those papers, a natural question (though difficult to answer to ask is: what are the explicit form of f(x) with zero terms\ulcorner In this paper we obtain the exact form (together with zero terms) of a polynomial f(x) over GF(q) for which satisfies deg(f) < p$^{2}$ and (1) f(x+a)=f(x)+c for the fixed nonzero elements a and c in GF(q).

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.543-557
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• 2017

In this paper, we investigate the notion of q-pseudoconvexity to discuss and describe some geometric characterizations of q-pseudoconvex domains ${\Omega}{\subset}{\mathbb{C}}^n$. In particular, we establish that ${\Omega}$ is q-pseudoconvex, if and only if, for every boundary point, the Levi form of the boundary is semipositive on the intersection of the holomorphic tangent space to the boundary with any (n-q+1)-dimensional subspace $E{\subset}{\mathbb{C}}^n$. Furthermore, we prove that the Kiselman's minimum principal holds true for all q-pseudoconvex domains in ${\mathbb{C}}^p{\times}{\mathbb{C}}^n$ such that each slice is a convex tube in ${\mathbb{C}}^n$.