• Honam Mathematical Journal
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• v.34 no.3
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• pp.423-438
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• 2012

In this paper, we discuss numbers of downwards and upwards in generalized paper folding sequences. We compute the exact number of downwards and upwards in $R^n_p$ and $(R_pR_q)^n$ by using the properties of recursive sequences where n, p and q are natural numbers with $p{\geq}2$ and $q{\geq}2$.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.413-421
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• 2017

In this paper, we study the generalizations of Euler numbers and polynomials by using the q-extension with p-adic integral on $\mathbb{Z}_p$. We call these: the generalized q-${\omega}$-Euler numbers $E^{({\alpha})}_{n,q,{{\omega}}(a)$ and polynomials $E^{({\alpha})}_{n,q,{\omega}}(x;a)$. We investigate some elementary properties and relations for $E^{({\alpha})}_{n,q,{{\omega}}(a)$ and $E^{({\alpha})}_{n,q,{\omega}}(x;a)$.

• Journal of the Korea Society of Computer and Information
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• v.19 no.7
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• pp.71-76
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• 2014

There is to be virtually impossible to solve the very large digits of prime number p and q from composite number n=pq using integer factorization in typical public-key cryptosystems, RSA. When the public key e and the composite number n are known but the private key d remains unknown in an asymmetric-key RSA, message decryption is carried out by first obtaining ${\phi}(n)=(p-1)(q-1)=n+1-(p+q)$ and then using a reverse function of $d=e^{-1}(mod{\phi}(n))$. Integer factorization from n to p,q is most widely used to produce ${\phi}(n)$, which has been regarded as mathematically hard. Among various integer factorization methods, the most popularly used is the congruence of squares of $a^2{\equiv}b^2(mod\;n)$, a=(p+q)/2,b=(q-p)/2 which is more commonly used then n/p=q trial division. Despite the availability of a number of congruence of scares methods, however, many of the RSA numbers remain unfactorable. This paper thus proposes an algorithm that directly and immediately obtains ${\phi}(n)$. The proposed algorithm computes $2^k{\beta}_j{\equiv}2^i(mod\;n)$, $0{\leq}i{\leq}{\gamma}-1$, $k=1,2,{\ldots}$ or $2^k{\beta}_j=2{\beta}_j$ for $2^j{\equiv}{\beta}_j(mod\;n)$, $2^{{\gamma}-1}$ < n < $2^{\gamma}$, $j={\gamma}-1,{\gamma},{\gamma}+1$ to obtain the solution. It has been found to be capable of finding an arbitrarily located ${\phi}(n)$ in a range of $n-10{\lfloor}{\sqrt{n}}{\rfloor}$ < ${\phi}(n){\leq}n-2{\lfloor}{\sqrt{n}}{\rfloor}$ much more efficiently than conventional algorithms.

• Journal of IKEEE
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• v.18 no.4
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• pp.593-602
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• 2014

One of the purposes of an artificial neural netowrk(ANNet) is to implement the largest number of functions as possible with the smallest number of nodes and layers. This paper presents a CoreNet which has a multi-leveled input value and a multi-leveled output value with a 2-layered ANNet, which is the basic structure of an ANNet. I have suggested an equation for calculating the capacity of the CoreNet, which has a p-leveled input and a q-leveled output, as $a_{p,q}={\frac{1}{2}}p(p-1)q^2-{\frac{1}{2}}(p-2)(3p-1)q+(p-1)(p-2)$. I've applied this CoreNet into the simulation model 1(5)-1(6), which has 5 levels of an input and 6 levels of an output with no hidden layers. The simulation result of this model gives, the maximum 219 convergences for the number of implementable functions using the cot(${\sqrt{x}}$) input leveling method. I have also shown that, the 27 functions are implementable by the calculation of weight values(w, ${\theta}$) with the multi-threshold lines in the weight space, which are diverged in the simulation results. Therefore the 246 functions are implementable in the 1(5)-1(6) model, and this coincides with the value from the above eqution $a_{5,6}(=246)$. I also show the implementable function numbering method in the weight space.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.233-242
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• 1998

Let $k = Q(\sqrt{m})$ be a real quadratic field. In this paper, the following theorems on p-divisibility of the class number h of k are studied for each prime pp. Theorem 1. If the discriminant of k has at least three distinct prime divisors, then 2 divides h. Theorem 2. If an odd prime p divides h, then p divides $B_{a,\chi\omega^{-1}}$, where $\chi$ is the nontrivial character of k, and $\omega$ is the Teichmuller character for pp. Theorem 3. Let $h_n$ be the class number of $k_n$, the nth layer of the $Z_p$-extension $k_\infty$ of k. If p does not divide $B_{a,\chi\omega^{-1}}$, then $p \notmid h_n$ for all $n \geq 0$.

• Honam Mathematical Journal
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• v.38 no.1
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• pp.169-212
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• 2016

Entire functions have been investigated so popularly to have been divided into a large number of specialized subjects. Even the limited subject of entire functions is too vast to be dealt with in a single volume with any approach to completeness. Here, in this paper, we choose to investigate certain interesting results associated with the comparative growth properties of iterated entire functions using (p, q)-th order and (p, q)-th lower order, in a rather comprehensive and systematic manner.

• The Pure and Applied Mathematics
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• v.3 no.1
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• pp.47-52
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• 1996

We consider the third order linear homogeneous differential equation L$_3$(y) = y(equation omitted) ＋ P($\chi$)y' ＋ Q($\chi$)y = 0 (E) P($\chi$) $\geq$ 0, Q($\chi$) ＞ 0 and P($\chi$)/Q($\chi$) is nondecreasing on [${\alpha}$, $\infty$) for some real number ${\alpha}$. (1) In this paper we discuss the distribution of zeros of solutions and a condition of oscillatory for equation (E).(omitted)

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.407-416
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• 2013

Let E be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ occurring in the $p$-primary part of the Tate-Shafarevich group of E over $\mathbb{Q}$.

• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.509-527
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• 2013

A Krawtchouk polynomial is introduced as the classical Mac-Williams identity, which can be expressed in weight-enumerator-free form of a linear code and its dual code over a Hamming scheme. In this paper we find a new explicit expression for the $p$-number and the $q$-number, which are more generalized notions of the Krawtchouk polynomial in the P-polynomial schemes by using an extended version of a discrete Green's function. As corollaries, we obtain a new expression of the Krawtchouk polynomial over the Hamming scheme and the Eberlein polynomial over the Johnson scheme. Furthermore, we find another version of the MacWilliams identity over a Hamming scheme.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.509-520
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• 2003

Let p be a prime number, $Q_{p}$ the field of p-adic numbers, K a finite extension of $Q_{p}$, $\bar{K}}$ a fixed algebraic closure of K and $C_{p}$ the completion of K with respect to the p-adic valuation. Let E be a closed subfield of $C_{p}$, containing K. Given elements $t_1$...,$t_{r}$ $\in$ E for which the field K($t_1$...,$t_{r}$) is dense in E, we construct integral bases of E over K.