• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.35-43
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• 2015

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with $1{\leq}a{\leq}q-1$, it is clear that the exists one and only one b with $0{\leq}b{\leq}q-1$ such that $ab{\equiv}c$ (mod q). Let N(c, q) denote the number of all solutions of the congruence equation $ab{\equiv}c$ (mod q) for $1{\leq}a$, $b{\leq}q-1$ in which a and $\bar{b}$ are of opposite parity, where $\bar{b}$ is defined by the congruence equation $b\bar{b}{\equiv}1$ (modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving $(N(c,q)-\frac{1}{2}{\phi}(q))$ and Kloosterman sums, and give a sharper asymptotic formula for it.

• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.367-379
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• 2009

For a prime number p, let $\mathbb{Q}_p$ denote the p-adic field and let $\mathbb{Q}_p^d$ denote a vector space over $\mathbb{Q}_p$ which consists of all d-tuples of $\mathbb{Q}_p$. For a function f ${\in}L_{loc}^1(\mathbb{Q}_p^d)$, we define the Hardy-Littlewood maximal function of f on $\mathbb{Q}_p^d$ by $$M_pf(x)=sup\frac{1}{\gamma{\in}\mathbb{Z}|B_{\gamma}(x)|H}{\int}_{B\gamma(x)}|f(y)|dy$$, where |E|$_H$ denotes the Haar measure of a measurable subset E of $\mathbb{Q}_p^d$ and $B_\gamma(x)$ denotes the p-adic ball with center x ${\in}\;\mathbb{Q}_p^d$ and radius $p^\gamma$. If 1 < q $\leq\;\infty$, then we prove that $M_p$ is a bounded operator of $L^q(\mathbb{Q}_p^d)$ into $L^q(\mathbb{Q}_p^d)$; moreover, $M_p$ is of weak type (1, 1) on $L^1(\mathbb{Q}_p^d)$, that is to say, |{$x{\in}\mathbb{Q}_p^d:|M_pf(x)|$>$\lambda$}|$_H{\leq}\frac{p^d}{\lambda}||f||_{L^1(\mathbb{Q}_p^d)},\;\lambda$ > 0 for any f ${\in}L^1(\mathbb{Q}_p^d)$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.1
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• pp.169-176
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• 2010

Let $k={\mathbb{F}}_q(T)$ and ${\mathbb{A}}={\mathbb{F}}_q[T]$. In this paper, we obtain the the criterion for the parity of the ideal class number h(${\mathcal{O}}_K$) of the real biquadratic function field $K=k(\sqrt{P_1},\;\sqrt{P_2})$, where $P_1$, $P_2{\in}{\mathbb{A}}$ be two distinct monic primes of even degree.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1357-1365
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• 2013

We find a lower bound on the number of real/inert imagi-nary/ramified imaginary quadratic extensions of the function field $\mathbb{F}_q(t)$ whose ideal class groups have an element of a fixed order, where $q$ is a power of 2.

• 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.

• Journal of Korean Society of Forest Science
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• v.89 no.1
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• pp.18-23
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• 2000

Leaves of various, 3 to 5-year-old Quercus hybrids were intermediate in size between their parental species. The petiole length was the smallest in the hybrids of Q. dentata ${\times}$ Q. crispula $F_1$ and was intermediate in the hybrids of Q. aliena ${\times}$ Q. serrata $F_1$ and Q. dentata ${\times}$ Q. aliena $F_1$ between their parents. The number of serration in hybrids was close to their mother tree's in most of crossing combinations. The serration depth and the ratio between longitudinal and transverse length of leaves were intermediate between the values of their parental species.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.39-44
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• 1996

Throughout this paper $Z^p, Q_p$ and C_p$ will denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of $Q_p$, respectively. Let $v_p$ be the normalized exponential valuation of $C_p$ with $$\mid$p$\mid$_p = p^{-v_p (p)} = p^{-1}$. We set $p^* = p$ for any prime p > 2 $p^* = 4 for p = 2$.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.463-478
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• 2009

Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let ${\tau}{\in}k{\cap}$ℌ, q = exp(${\pi}i{\tau}$). And let n, t be positive integers with $1{\leq}t{\leq}n-1$. Then $q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $_{1{\psi}1}$ summation. These are also related to Rogers-Ramanujan continued fractions.

• Journal of the Korean Institute of Electrical and Electronic Material Engineers
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• v.20 no.11
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• pp.959-964
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• 2007

To investigate the influence of partial discharge(PD) distribution characteristics due to various defects on the power cable joints interface, we used the K-means clustering method. As the result of PD number(n) distribution analyzing on $\Phi-n$ graph, the phase angle($\Phi$) of cluster centroid shifted to $0^{\circ}\;and\;180^{\circ}$ increasing with applying voltage. It was confirmed that the PD quantify(q) and euclidean distance of centroid were increased with applying voltage from the centroid distribution analyzing of $\Phi-q$ plane. The dispersion degree was increased with calculated standard deviation of the $\Phi-q$ cluster centroid. The PD number and mean value on $\Phi-q$ graph were some different by electric field concentration with defect types.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.125-140
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• 2006

Let $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ be the number of general4-regular graphs on n labelled vertices with $l_1+2l_2$ loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with five types of properties to determine the asymptotic behavior of $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ and hence that of g(2n), the total number of general 4-regular graphs where $l_1,\;l_2,\;d,\;t\;and\;q\;=\;o(\sqrt{n})$, respectively. We show that almost all general 4-regular graphs are 2-connected. Moreover, we determine the asymptotic numbers of general 4-regular graphs with given connectivities.