• Communications of the Korean Mathematical Society
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• v.17 no.3
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• pp.541-550
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• 2002

In this Paper, We look at 3 Simple function L assigning to an integer n the smallest positive integer n such that any product of n consecutive numbers is divisible by n. Investigated are the interesting properties of the function. The function L(n) is completely determined by L(p$\^$k/), where p$\^$k/ is a factor of n, and satisfies L(m$.$n) $\leq$ L(m)+L(n), where the equality holds for infinitely many cases.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.365-371
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• 2016

Let f be a transcendental meromorphic function in the complex plane $\mathbb{C}$, and a be a nonzero constant. We give a quantitative estimate of the characteristic function T(r, f) in terms of $N(r,1/(f^2(f^{\prime})^n-a))$, which states as following inequality, for positive integers $n{\geq}2$, $$T(r,f){\leq}\(3+{\frac{6}{n-1}}\)N\(r,{\frac{1}{af^2(f^{\prime})^n-1}}\)+S(r,f)$$.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.765-779
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• 2021

The fundamental goal of this paper is to investigate some inequalities involving the special beta function in n variables. Our theoretical results obtained here are extensions and refinements for some inequalities already discussed in the literature.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1535-1549
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• 2020

This paper deals with the inverse of tails of Hurwitz zeta function. More precisely, for any positive integer s ≥ 2 and 0 ≤ a < 1, we give an algorithm for finding a simple form of f_s,a(n) such that $$\lim_{n{\rightarrow}{\infty}}\{\({\sum\limits_{k=n}^{\infty}}{\frac{1}{(k+a)^s}}\)^{-1}-f_{s,a}(n)\}=0$$. We show that f_s,a(n) is a polynomial in n-a of order s-1. All coefficients of f_s,a(n) are represented in terms of Bernoulli numbers.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.927-944
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• 2022

Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝⁿ in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.26 no.2
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• pp.315-318
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• 2016

In this paper, we present a new preimage attack on MJH, a double-block-length block cipher-based hash function. Currently, the best attack requires $O(2^{3n/2})$ queries for the 2n-bit MJH hash function based on an n-bit block cipher, while our attack requires $O(n2^n)$ queries and the same amount of memory, significantly improving the query complexity compared to the existing attack.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.75-80
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• 2021

For a locus with two alleles (IA and IB), the frequencies of the alleles are represented by $$p=f(I^A)={\frac{2N_{AA}+N_{AB}}{2N} },\;q=f(I^B)={\frac{2N_{BB}+N_{AB}}{2N}}$$ where NAA, NAB and NBB are the numbers of IA IA, IA IB and IB IB respectively and N is the total number of populations. The frequencies of the genotypes expected are calculated by using p2, 2pq and q2. So in this paper, we consider the method of whether some genotypes is in Hardy-Weinburg equilibrium. Also we calculate the probability generating function for the offspring number of genotype produced by a mating of the ith male and jth female under a diploid model of N population with N1 males and N2 females. Finally, we have conditional joint probability generating function of genotype frequencies.

• Honam Mathematical Journal
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• v.44 no.1
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• pp.135-145
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• 2022

This paper aims to investigate characterizations on parameters k₁, k₂, k₃, k₄, k₅, l₁, l₂, l₃, and l₄ to find relation between the class of 𝓗(k, l, m, n, o) hypergeometric functions defined by $$5_F_4\[{\array{k_1,\;k_2,\;k_3,\;k_4,\;k_5\\l_1,\;l_2,\;l_3,\;l_4}}\;:\;z\]=\sum\limits_{n=2}^{\infty}\frac{(k_1)_n(k_2)_n(k_3)_n(k_4)_n(k_5)_n}{(l_1)_n(l_2)_n(l_3)_n(l_4)_n(1)_n}z^n$$. We need to find k, l, m and n that lead to the necessary and sufficient condition for the function zF([W]), G = z(2 - F([W])) and $H_1[W]=z^2{\frac{d}{dz}}(ln(z)-h(z))$ to be in 𝓢^*(2^-r), r is a positive integer in the open unit disc 𝒟 = {z : |z| < 1, z ∈ ℂ} with $$h(z)=\sum\limits_{n=0}^{\infty}\frac{(k)_n(l)_n(m)_n(n)_n(1+\frac{k}{2})_n}{(\frac{k}{2})_n(1+k-l)_n(1+k-m)_n(1+k-n)_nn(1)_n}z^n$$ and $$[W]=\[{\array{k,\;1+{\frac{k}{2}},\;l,\;m,\;n\\{\frac{k}{2}},\;1+k-l,\;1+k-m,\;1+k-n}}\;:\;z\]$$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1425-1432
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• 2014

The Euler zeta function is defined by ${\zeta}_E(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^8}$. The purpose of this paper is to find formulas of the Euler zeta function's values. In this paper, for $s{\in}\mathbb{N}$ we find the recurrence formula of ${\zeta}_E(2s)$ using the Fourier series. Also we find the recurrence formula of $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2_{n-1})^{2s-1}}$, where $s{\geq}2({\in}\mathbb{N})$.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.49-59
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• 2021

Recently, Andrews introduced partition functions ����(n) and ${\bar{\mathcal{EO}}}$(n) where the function ����(n) denotes the number of partitions of n in which every even part is less than each odd part and the function ${\bar{\mathcal{EO}}}$(n) denotes the number of partitions enumerated by ����(n) in which only the largest even part appears an odd number of times. In this paper we obtain some congruences modulo 2, 4, 10 and 20 for the partition function ${\bar{\mathcal{EO}}}$(n). We give a simple proof of the first Ramanujan-type congruences ${\bar{\mathcal{EO}}}$ (10n + 8) ≡ 0 (mod 5) given by Andrews.