• 한국인터넷방송통신학회논문지
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• 제14권5호
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• pp.87-93
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• 2014

n/m=qm+r에서 에서 m=7인 단순한 경우에도 주어진 수 n이 m의 배수 판정법은 간단하지가 않다. 만약, m이 두 자리 수 이상이 되면 더욱 복잡해진다. 일반적인 배수 판정법으로 둔켈스 (Dunkels)법이 있지만 n이 컴퓨터로 처리하지 못하는 매우 큰 자리수인 경우 이 방법도 처리할 수 없다. 본 논문은 n과 m의 자리수와 무관하게 n(modm)=0 여부로 n이 m의 배수인지 여부를 검증하는 간단하면서도 정확한 방법을 제안한다. 제안된 방법은 $n=n_1n_2n_3{\cdots}n_k$, $m=m_1m_2{\cdots}m_l$에 대해 $r_1=n_1n_2{\cdots}n_l(mod m)$으로 설정하고, $r_i=r_{i-1}{\times}10+n_i(mod m)$, $i=2,3,{\cdots},k-1+1$로 n의 자리수를 1자리씩 감소시키는 방법을 적용하였다. 제안된 방법을 다양한 n,m 데이터에 적용한 결과 쉽고, 빠르며 정확한 몫과 나머지 값을 구할 수 있음을 보였다.

• 대한수학회지
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• 제54권5호
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• pp.1505-1519
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• 2017

Assume that M is an R-module where R is a commutative ring. A proper submodule N of M is called a weakly 2-absorbing primary submodule of M if $0{\neq}abm{\in}N$ for any $a,b{\in}R$ and $m{\in}M$, then $ab{\in}(N:M)$ or $am{\in}M-rad(N)$ or $bm{\in}M-rad(N)$. In this paper, we extended the concept of weakly 2-absorbing primary ideals of commutative rings to weakly 2-absorbing primary submodules of modules. Among many results, we show that if N is a weakly 2-absorbing primary submodule of M and it satisfies certain condition $0{\neq}I_1I_2K{\subseteq}N$ for some ideals $I_1$, $I_2$ of R and submodule K of M, then $I_1I_2{\subseteq}(N:M)$ or $I_1K{\subseteq}M-rad(N)$ or $I_2K{\subseteq}M-rad(N)$.

• 충청수학회지
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• 제26권3호
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• pp.615-623
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• 2013

In this paper we establish a central limit theorem for weighted sums of $Y_n={\sum_{i=1}^{n}}a_n,_iX_i$, where $\{a_{n,i},\;n{\in}N,\;1{\leq}i{\leq}n\}$ is an array of nonnegative numbers such that ${\sup}_{n{\geq}1}{\sum_{i=1}^{n}}a_{n,i}^2$ < ${\infty}$, ${\max}_{1{\leq}i{\leq}n}a_{n,i}{\rightarrow}0$ and $\{X_i,\;i{\in}N\}$ is a sequence of linear negatively quadrant dependent random variables with $EX_i=0$ and $EX_i^2$ < ${\infty}$. Using this result we will obtain a central limit theorem for partial sums of linear processes.

• 대한수학회지
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• 제41권5호
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• pp.773-793
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• 2004

We consider a system of particles with locations { $X_{i}$ $^{n}$ (t):t$\geq$0,i＝1,…,n} in $R^{d}$ , time-varying weights { $A_{i}$ $^{n}$ (t) : t $\geq$0,i ＝ 1,…,n} and weighted empirical measure processes $V^{n}$ (t)＝1/n$\Sigma$$_{i=1}$$^{n}$ $A_{i}$ $^{n}$ (t)$\delta$ $X_{i}$ $^{n}$ (t), where $\delta$$_{x}$ is the Dirac measure. It is known that there exists the limit of { $V_{n}$ } in the week* topology on M( $R^{d}$ ) under suitable conditions. If { $X_{i}$ $^{n}$ , $A_{i}$ $^{n}$ , $V^{n}$ } satisfies some diffusion equations, applying Ito formula, we prove a central limit type theorem for the empirical process { $V^{n}$ }, i.e., we consider the convergence of the processes η$_{t}$ $^{n}$ ≡ n( $V^{n}$ -V). Besides, we study a characterization of its limit.t.

• 충청수학회지
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• 제22권1호
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• pp.1-5
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• 2009

Let {$X_n,\;n{\geq}1}$ be a sequence of independent identically distributed(i.i.d.) sequence of positive random variables with common absolutely continuous distribution function(cdf) F(x) and probability density function(pdf) f(x) and $E(X^2)<{\infty}$. The random variables $\frac{X_i{\cdot}X_j}{(\Sigma^n_{k=1}X_k)^{2}}$ and $\Sigma^n_{k=1}X_k$ are independent for $1{\leq}i if and only if {$X_n,\;n{\geq}1}$ have gamma distribution.

• 대한수학회지
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• 제39권1호
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• pp.127-135
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• 2002

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n＋1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s}$ denote the set of equimultiple good ideals I in A with h $t_{A}$ I = s, R(I) = A [It] be the Rees algebra of I, and $K_{R(I)}$ denote the canonical module of R(I). Let a I such that $I^{n＋l}$ = a $I^{n}$ for some n$\geq$0 and $\mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}$(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) $K_{R(I)}$(equation omitted)R(I)＋as graded R(I)-modules. (2) $I^2$ = aI and aA : I$\in$ $X^1$$_{A}$._{A}$./.

• 대한수학회보
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• 제52권3호
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• pp.1007-1025
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• 2015

Let $\mathbb{N}_0$ be the set of non-negative integers, and let P(n, l) denote the set of all weak compositions of n with l parts, i.e., $P(n,l)=\{(x_1,x_2,{\cdots},x_l){\in}\mathbb{N}^l_0\;:\;x_1+x_2+{\cdots}+x_l=n\}$. For any element $u=(u_1,u_2,{\cdots},u_l){\in}P(n,l)$, denote its ith-coordinate by u(i), i.e., $u(i)=u_i$. A family $A{\subseteq}P(n,l)$ is said to be t-intersecting if ${\mid}\{i:u(i)=v(i)\}{\mid}{\geq}t$ for all $u,v{\epsilon}A$. A family $A{\subseteq}P(n,l)$ is said to be trivially t-intersecting if there is a t-set T of $[l]=\{1,2,{\cdots},l\}$ and elements $y_s{\in}\mathbb{N}_0(s{\in}T)$ such that $A=\{u{\in}P(n,l):u(j)=yj\;for\;all\;j{\in}T\}$. We prove that given any positive integers l, t with $l{\geq}2t+3$, there exists a constant $n_0(l,t)$ depending only on l and t, such that for all $n{\geq}n_0(l,t)$, if $A{\subseteq}P(n,l)$ is non-trivially t-intersecting, then $${\mid}A{\mid}{\leq}(^{n+l-t-l}_{l-t-1})-(^{n-1}_{l-t-1})+t$$. Moreover, equality holds if and only if there is a t-set T of [l] such that $$A=\bigcup_{s{\in}[l]{\backslash}T}\;A_s{\cup}\{q_i:i{\in}T\}$$, where $$A_s=\{u{\in}P(n,l):u(j)=0\;for\;all\;j{\in}T\;and\;u(s)=0\}$$ and $$q_i{\in}P(n,l)\;with\;q_i(j)=0\;fo\;all\;j{\in}[l]{\backslash}\{i\}\;and\;q_i(i)=n$$.

• 호남수학학술지
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• 제31권3호
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• pp.421-428
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• 2009

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this paper, we showed the following : Let $\mathcal{L}$ be a subspace lattice acting on a Hilbert space $\mathcal{H}$ and let $X_i$ and $Y_i$ be operators in B($\mathcal{H}$) for i = 1, 2, ${\cdots}$. Let $P_i$ be the projection onto $\overline{rangeX_i}$ for all i = 1, 2, ${\cdots}$. If $P_kE$ = $EP_k$ for some k in $\mathbb{N}$ and all E in $\mathcal{L}$, then the following are equivalent: (1) $sup\;\{{\frac{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}:f{\in}H,n{\in}{\mathbb{N}},E{\in}\mathcal{L}}\}$ < ${\infty}$ range $\overline{rangeY_k}\;=\;\overline{rangeX_k}\;=\;\mathcal{H}$, and < $X_kf,\;X_kg$ >=< $Y_kf,\;Y_kg$ > for some k in $\mathbb{N}$ and for all f and g in $\mathcal{H}$. (2) There exists an operator A in Alg$\mathcal{L}$ such that $AX_i$ = $Y_i$ for i = 1, 2, ${\cdots}$ and AA$^*$ = I = A$^*$A.

• Kyungpook Mathematical Journal
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• 제61권4호
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• pp.679-710
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• 2021

We study the tensor product algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m), where P_k(x) is the partition algebra defined by Jones and Martin. We discuss the centralizer of this algebra and corresponding Schur-Weyl dualities and also index the inequivalent irreducible representations of the algebra P_k(x₁) ⊗ P_k(x₂) ⊗ ⋯ ⊗ P_k(x_m) and compute their dimensions in the semisimple case. In addition, we describe the Bratteli diagrams and branching rules. Along with that, we have also constructed the RS correspondence for the tensor product of m-partition algebras which gives the bijection between the set of tensor product of m-partition diagram of P_k(n₁) ⊗ P_k(n₂) ⊗ ⋯ ⊗ P_k(n_m) and the pairs of m-vacillating tableaux of shape [λ] ∈ Γ_k^m, Γ_k^m = {[λ] = (λ₁, λ₂, …, λ_m)|λ_i ∈ Γ_k, i ∈ {1, 2, …, m}} where Γ_k = {λ_i ⊢ t|0 ≤ t ≤ k}. Also, we provide proof of the identity $(n_1n_2{\cdots}n_m)^k={\sum}_{[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ f^[λ]m_k^[λ] where m_k^[λ] is the multiplicity of the irreducible representation of $S{_{n_1}}{\times}S{_{n_2}}{\times}....{\times}S{_{n_m}}$ module indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$, where f^[λ] is the degree of the corresponding representation indexed by ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}$ and ${[{\lambda}]{\in}{\Lambda}^k_{{n_1},{n_2},{\ldots},{n_m}}}=\{[{\lambda}]=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_m){\mid}{\lambda}_i{\in}{\Lambda}^k_{n_i},i{\in}\{1,2,{\ldots},m\}\}$ where ${\Lambda}^k_{n_i}=\{{\mu}=({\mu}_1,{\mu}_2,{\ldots},{\mu}_t){\vdash}n_i{\mid}n_i-{\mu}_1{\leq}k\}$.

• 한국인터넷방송통신학회논문지
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• 제17권6호
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• pp.19-25
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• 2017

본 논문은 비트코인 채굴에 필요한 SHA-256 암호 해시 값(n)을 구성하는 2개의 소수(p,q)를 빠르게 해독하는 소인수분해법을 다룬다. 본 논문에서는 Pollard's Rho 소인수분해 알고리즘의 수행횟수를 월등히 감소시킨 알고리즘을 제안하였다. Rho (${\rho}$) 알고리즘은 $(x_0,y_0)=(2,2)$ 초기치에 대해 $x_i=x^2_{i-1}+1(mod\;n)$과 $y_i=[(y^2_{i-1}+1)^2+1](mod\;n)$을 계산하여 1 < $gcd({\mid}x_i-y_i{\mid},n)$ < n으로 소인수를 구한다. 이 알고리즘은 특정 합성수에 대해서는 소인수 분해에 실패할 수 있다. 제안된 알고리즘은 Pollard Rho 알고리즘에 $(x_0,y_0)=(2^k,2^k)$와 ($2^k,2$), $2{\leq}k{\leq}10$을 적용하였다. 그 결과 모든 합성수에 대해 소인수분해를 할 수 있었으며, Pollard Rho 알고리즘의 수행횟수를 67.94% 감소시켰다.