• 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).

• Journal for History of Mathematics
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• v.31 no.4
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• pp.183-196
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• 2018

In this study, we propose an approximate method to make Jisuguimundo with magic number 93 to 109. In this method, for two numbers p, q with a relationship of M = 2p+q, we use eight pairs of two numbers with sum p and five pairs of two numbers with sum q. Such numbers must be between 1 and 30. Instead of determining all positions of thirty numbers, this method shows that Jisuguimundo with magic number 93 to 109 can be made by determining positions of thirteen numbers $a_i$(i = 1, 2, ${\cdots}$, 8), $b_5$, $c_i$(i = 1, 2, 3, 4). Method 1 is used to make Jisuguimundo with magic number 93 to 108, and method 2 is used to make Jisuguimundo with magic number 109.

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1249-1268
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• 2011

For imaginary quadratic number fields F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$, where ${\varepsilon}{\in}${-1,-2} and distinct primes $p_i{\equiv}1$ mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})4$ and binary quadratic forms such as $p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$, where $h+(2p_1)$ is the narrow class number of $\mathbb{Q}(\sqrt{2p_1})$. The results are also very useful for numerical computations.

• The Korean Journal of Ecology
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• v.26 no.4
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• pp.155-163
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• 2003

In order to analyze the succession process from a pine forest to an oak forest, the tree growth of Pinus densiflora and Quercus mongolica ssp. crispula was monitored in a permanent quadrat for 23 years. The measurements were carried out for the stem diameter (DBH) of Pinus densiflora between 1977 and 1999 and for the height of Quercus mongolica ssp. crispula saplings between 1998 and 2000. The floristic composition and the locations of the individual P. densiflora and Q. mongolica ssp. crispula trees and saplings in the quadrat were recorded. P densiflora and Q. mongolica ssp. crispula individuals were randomly distributed within the quadrat. The relative growth rates (RGR) of DBH in P. densiflora were 0.085 $yr^{-1}$ for large trees and 0.056 $yr^{-1}$ for small trees in 1977. The RGR of height for Q. mongolica ssp. crispula was 0.122 $yr^{-1}$. The growth curve for DBH of P. densiflora was approximated by the logistic equation: $$DBH(t) = 30 {[1+1.16exp(-0.13 t)]}^{-1}$$ where DBH (t) the DBH (cm) in year t and t is the number of years since 1977. The growth in height of P. densiflora and Q. mongolica ssp. crispula was described by following equations: $$H (t) = 20.2 {[1+0.407exp(-0.137 t)]}^{-1} (P. densiflora)$$ $$H (t) = 30 {[1+20.7exp(-0.122 t)}^{-1} (Q. mongolica ssp. crispula)$$ Where H (t) is the tree height (m) in year t and t is the number of years since 1977 in P. densiflora and 1998 in Q. mongolica ssp. crispula. With these equations we predicted that the height of Q. mongolica ssp. crispula increases from 2 m in 1999 to 20 m in 2029. Therefore, Q. mongolica ssp. crispula and P. densiflora will be approximately the same height in 2029. The years required for succession from a pine forest to an oak forest are expected 33 with the range between 23 and 44 years.

• Journal of Environmental Science International
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• v.11 no.6
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• pp.537-541
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• 2002

This study aimed to obtain the relative formula with the unit treatment cost according to the treatment of a sewage plant in the service area under highway. The following results were obtained. The correlative formula connected to amount of sewage(Q)generation was as follows ; between an annual amount of sale(C) showed Q=19.113$.$C$\^$0.9294/, and between the number of users(P) showed Q=2${\times}$10$\^$-8/ $.$P$^2$- 0.0298$.$P + 75,666. The correlative formula connected to the treatment cost was as follows , according to the amount of sewage generation showed S= 3${\times}$10$\^$-6/$.$Q 0-0.2266$.$Q+29,895, according to the elimination of BOD(E) showed S=6${\times}$10$\^$-5/$.$E$^2$-0.6717$.$E + 27,744, according to the annual amount of sale showed S=0.0005 C$^2$-4.8013$.$C + 35,118, with the number, of persons(P) using the service area showed S= 2${\times}$10$\^$-8/ $.$P$^2$- 0.046$.$P + 48,803.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1181-1188
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• 2010

q-Volkenborn integrals ([8]) and fermionic invariant q-integrals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25]) studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by $${\varsigma}E,q,\varepsilon(s)=[2]q \sum\limits_{n=0}^\infty\frac{(-1)^n\epsilon^nq^{sn}}{[n]_q}$$ where 0 < q < 1, $\mathfrak{R}$(s) > 1, $\varepsilon{\in}T_p$, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.361-370
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• 2021

Let 𝕂 be a complete ultrametric algebraically closed field and 𝓐 (𝕂) be the 𝕂-algebra of entire function on 𝕂. For any p-adic entire functions f ∈ 𝓐 (𝕂) and r > 0, we denote by |f|(r) the number sup {|f (x)| : |x| = r} where |·|(r) is a multiplicative norm on 𝓐 (𝕂). In this paper we study some growth properties of composite p-adic entire functions on the basis of their relative (p, q)-𝜑 order where p, q are any two positive integers and 𝜑 (r) : [0, +∞) → (0, +∞) is a non-decreasing unbounded function of r.

• Journal of IKEEE
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• v.15 no.4
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• pp.354-362
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• 2011

One of the unsolved problems in Artificial Neural Networks is related to the capacity of a neural network. This paper presents a CoreNet which has a multi-leveled input and a multi-leveled output as a 2-layered artificial neural network. I have suggested an equation for calculating the capacity of the CoreNet, which has a p-leveled input and a q-leveled output, as $a_{p,q}=\frac{1}{2}p(p-1)q^2-\frac{1}{2}(p-2)(3p-1)q+(p-1)(p-2)$. With an odd value of p and an even value of q, (p-1)(p-2)(q-2)/2 needs to be subtracted further from the above equation. The simulation model 1(3)-1(6) has 3 levels of an input and 6 levels of an output with no hidden layer. The simulation result of this model gives, out of 216 possible functions, 80 convergences for the number of implementable function using the cot(x) input leveling method. I have also shown that, from the simulation result, the two diverged functions become implementable by precalculating the weight values. The simulation result and the precalculation of the weight values give the same result as the above equation in the total number of implementable functions.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• The Transactions of the Korea Information Processing Society
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• v.7 no.6
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• pp.1778-1784
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• 2000

We consider the weighted shortest path updating problem, that is, the problem to reconstruct the weighted shortest paths in response to topology change of the network. This appear proposes a distributed algorithms that reconstructs the weighted shortest paths after several processors and links are added and deleted. its message complexity and ideal-time complexity are O(p$^2$+q+n') and O(p$^2$+q+n') respectively, where n' is the number of processors in the network after the topology change, q is the number of added links, and p is the total number of processors in he biconnected components (of the network before the topology change) including the deleted links or added links.