• Journal of applied mathematics & informatics
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• 제11권1_2호
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• pp.109-123
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• 2003

A tree is called starlike if it has exactly one vertex of degree greate. than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) : {1,2, …, n} and let H = {$H_1$, $H_2$, …, $H_{n}$} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of $H_{i}$ with the i-th vertex of G. In particular, if H is the family of the paths $P_k_1,P_k_2,...P_k_2$ with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G($k_1,k_2,...k_n$). For any $(x_1,x_2,...,x_n)\;\in\;{I_*}^n$, where $I_{*}$ = : {0,1}, let G$(x_1,x_2,...,x_n)$ be the subgraph of G which is obtained by deleting the vertices $i_1,i_2,...i_j\;\in\;V(G)\;(O\leq j\leq n)$, provided that $x_i_1=x_i_2=...=x_i_j=o.\;Let \;G[x_1,x_2,...x_n]$ be characteristic polynomial of G$(x_1,x_2,...,x_n)$, understanding that G[0,0,...,0] $\equiv$1. We prove that $G[k_1,k_2,...,k_n]-\sum_{x\in In}[{\prod_{\imath=1}}^n\;P_k_i+x_i-2(\lambda)](-1)...G[x_1,x_2,...,X_n]$ where x=($x_1,x_2,...,x_n$);G[$k_1,k_2,...,k_n$] and $P_n(\lambda)$ denote the characteristic polynomial of G($k_1,k_2,...,k_n$) and $P_n$, respectively. Besides, if G is a graph with $\lambda_1(G)\;\geq1$ we show that $\lambda_1(G)\;\leq\;\lambda_1(G(k_1,k_2,...,k_n))<\lambda_1(G)_{\lambda_1}^{-1}(G}$ for all positive integers $k_1,k_2,...,k_n$, where $\lambda_1$ denotes the largest eigenvalue.

• 충청수학회지
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• 제33권3호
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• pp.319-326
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• 2020

In this paper, we establish a general solution of the following functional equation $$mf\({\sum\limits_{k=1}^{n}}x_k\)+{\sum\limits_{t=1}^{m}}f\({\sum\limits_{k=1}^{n-i_t}}x_k-{\sum\limits_{k=n-i_t+1}^{n}}x_k\)=2{\sum\limits_{t=1}^{m}}\(f\({\sum\limits_{k=1}^{n-i_t}}x_k\)+f\({\sum\limits_{k=n-i_t+1}^{n}}x_k\)\)$$ where m, n, t, i_t ∈ ℕ such that 1 ≤ t ≤ m < n. Also, we study Hyers-Ulam-Rassias stability for the generalized quadratic functional equation with n-variables and m-combinations form in quasi-𝛽-normed spaces and then we investigate its application.

• 충청수학회지
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• 제31권4호
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• pp.397-409
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• 2018

In this paper, we investigate the generalized Hyers-Ulam stability of the functional equation $$f\({\sum\limits_{i=1}^{n}}x_i\)+{\sum\limits_{1{\leq}i<j{\leq}n}}f(x_i-x_j)-n{\sum\limits_{i=1}^{n}f(x_i)=0$$ for integer values of n such that $n{\geq}2$, where f is a mapping from a vector space V to a Banach space Y.

• 한국정보과학회논문지:시스템및이론
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• 제29권9호
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• pp.514-519
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• 2002

단백질의 구조를 예측하는 과정에 사용될 수 있는 다음 문제를 고려한다. 길이가 n이고 원소가 모두 양수인 두 배열 A, B와 양수 M이 주어질 때, A[i]＋…A[j]＋B[k]＋…B[ι]=M이 되는 부배열 쌍 A[i]＋…A[j],$1{\leq}i{\leq}j{\leq}n$과 B[k], …, B[l], $1{\leq}k{\leq}l{\leq}n$을 모두 찾으시오. 본 논문에서는 이 문제를 $Ο(n^2log n+K)$ 시간에 Ο(n) 메모리를 사용하여 해결하는 알고리즘을 제시한다. 단, K는 찾은 부배열 쌍의 수이다. 기존의 결과는$Ο(n^2log +Klog n)$ 시간과 Ο(n) 메모리였다.

• 대한수학회보
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• 제42권4호
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• pp.679-690
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• 2005

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, f($x_1,{\cdots},\;x_n$) a multilinear polynomial in n non-commuting variables over K, a $\in$ R. Supppose that, for any $x_1,{\cdots},\;x_n\;\in\;I,\;a[d(f(x_1,{\cdots},\;x_n)),\;f(x_1,{\cdots},\;x_n)]$ = 0. If $[f(x_1,{\cdots},\;x_n),\;x_{n+1}]x_{n+2}$ is not an identity for I and $$S_4(I,\;I,\;I,\;I)\;I\;\neq\;0$$, then aI = ad(I) = 0.

• 자연과학논문집
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• 제8권2호
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• pp.5-7
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• 1996

{X,$X_n$,n$\geq$1}은 독립이고 평균이 영으로 같은 확률분포를 갖는 확률변수 열이고, {$a_ni$, 1$\leq$i$\leq$n,n$\geq$1}은 수열일 때 가중합 $sum_{i=1}^n$$a_{ni}$$X_i$가 0에 확률 1로 수렴할 충분조건을 제시한다.

• 대한수학회보
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• 제32권2호
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• pp.329-336
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• 1995

Let $A_1, A_2, \ldots, A_n$ be a sequence of events on a given probability space and let $m_n$ be the number of those A's which occur. Put $S_{0,n} = 1$ and $$ S_{k,n} = \Sigma P(A_i_1 \cap A_i_2 \cap \cdots \cap A_i_k), (a \leq k)$$ where the summation is over all subscripts satisfying $1 \let i_1 < i_2 < \cdots < i_k \leq n$.

• Kyungpook Mathematical Journal
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• 제45권2호
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• pp.293-299
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• 2005

Let ${\cal{L}}$ be a commutative subspace lattice on a Hilbert space ${\cal{H}}$ and X and Y be operators on ${\cal{H}}$. Let $${\cal{M}}_X=\{{\sum}{\limits_{i=1}^n}E_{i}Xf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}$$ and $${\cal{M}}_Y=\{{\sum}{\limits_{i=1}^n}E_{i}Yf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}.$$ Then the following are equivalent. (i) There is an operator A in $Alg{\cal{L}}$ such that AX = Y, Ag = 0 for all g in ${\overline{{\cal{M}}_X}}^{\bot},A^*A=AA^*$ and every E in ${\cal{L}}$ reduces A. (ii) ${\sup}\;\{K(E, f)\;:\;n\;{\in}\;{\mathbb{N}},f_i\;{\in}\;{\cal{H}}\;and\;E_i\;{\in}\;{\cal{L}}\}\;<\;\infty,\;{\overline{{\cal{M}}_Y}}\;{\subset}\;{\overline{{\cal{M}}_X}}$and there is an operator T acting on ${\cal{H}}$ such that ${\langle}EX\;f,Tg{\rangle}={\langle}EY\;f,Xg{\rangle}$ and ${\langle}ET\;f,Tg{\rangle}={\langle}EY\;f,Yg{\rangle}$ for all f, g in ${\cal{H}}$ and E in ${\cal{L}}$, where $K(E,\;f)\;=\;{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Y\;f_{i}{\parallel}/{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Xf_{i}{\parallel}$.

• Journal of Pharmaceutical Investigation
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• 제9권4호
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• pp.1-7
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• 1979

The analytical method of phenobarbital and mephobarbital in the pharmaceutical preparation were investigated by the partition chromatographic method. The influence of 13 kinds of excipients and various kinds of ingredients such as 8 kinds of antipyretics and 5 kinds of antacids on the partition chromatographic analysis of phenobarbital and mephobarbital in the preparation were investigated. The results were not affected by materials with the exception of caffeine, aspirin and sulpyrin.

• 대한수학회보
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• 제52권2호
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• pp.531-540
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• 2015

Let (R, m) be a Noetherian local ring, I an ideal of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{&gt;k}(I,A)$ the supremum of length of A-cosequence in dimension > k in I defined by Nhan-Hoang [8]. It is shown that for all $t{\leq}r$ the sets $$(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/I^n,A)))_{{\geq}k}\;and\\(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/(a_1^{n_1},{\cdots},a_l^{n_l}),A)))_{{\geq}k}$$ are independent of the choice of $n,n_1,{\cdots},n_l$ for any system of generators ($a_1,{\cdots},a_l$) of I.