• Korean Journal of Mathematics
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• 제19권1호
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• pp.17-24
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• 2011

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0, and $$2(_{2d-2}C_{d-1}-_{2d-2}C_d)f\({\sum_{j=1}^{2d}}x_j\)+{\sum_{{\iota}(j)=0,1,{{\small\sum}_{j=1}^{2d}}{\iota}(j)=d}}\;f\({\sum_{j=1}^{2d}}(-1)^{{\iota}(j)}x_j\)=2(_{2d-1}C_d+_{2d-2}C_{d-1}-_{2d-2}C_d){\sum_{j=1}^{2d}}f(x_j)$$ for all $x_1$, ${\cdots}$, $x_{2d}{\in}X$, then the even mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.

• 대한수학회논문집
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• 제23권3호
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• pp.371-376
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• 2008

In this paper, it is shown that if f satisfies the following functional inequality (0.1) $${\parallel}\sum\limits_{i,j=1}^3\;f{(xi,yj)}{\parallel}{\leq}{\parallel}f(x_1+x_2+x_3,\;y_1+y_2+y_3){\parallel}$$ then f is a bi-additive mapping. We moreover prove that if f satisfies the following functional inequality (0.2) $${\parallel}2\sum\limits_{j=1}^3\;f{(x_j,\;z)}+2\sum\limits_{j=1}^3\;f{(x_j,\;w)-f(\sum\limits_{j=1}^3\;xj,\;z-w)}{\parallel}{\leq}f(\sum\limits_{j=1}^3\;xj,\;z+w){\parallel}$$ then f is an additive-quadratic mapping.

• Korean Journal of Mathematics
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• 제24권1호
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• pp.71-79
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• 2016

In this paper, we determine all positive integers which cannot be represented by a sum of four squares at least 9, and prove that for each N, there are nitely many positive integers which cannot be represented by a sum of four squares at least $N^2$ except $2{\cdot}4^m$, $6{\cdot}4^m$ and $14{\cdot}4^m$ for $m{\geq}0$. As a consequence, we prove that for each $k{\geq} 5$ there are nitely many positive integers which cannot be represented by a sum of k squares at least $N^2$.

• 대한수학회보
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• 제53권4호
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• pp.959-970
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• 2016

In this paper, we generalize the stability for an n-dimensional cubic functional equation in Banach space to set-valued dynamics. Let $n{\geq}2$ be an integer. We define the n-dimensional cubic set-valued functional equation given by $$f(2{{\sum}_{i=1}^{n-1}}x_i+x_n){\oplus}f(2{{\sum}_{i=1}^{n-1}}x_i-x_n){\oplus}4{{\sum}_{i=1}^{n-1}}f(x_i)\\=16f({{\sum}_{i=1}^{n-1}}x_i){\oplus}2{{\sum}_{i=1}^{n-1}}(f(x_i+x_n){\oplus}f(x_i-x_n)).$$ We first prove that the solution of the n-dimensional cubic set-valued functional equation is actually the cubic set-valued mapping in [6]. We prove the Hyers-Ulam stability for the set-valued functional equation.

• 대한수학회보
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• 제45권4호
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• pp.739-748
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• 2008

In this paper, we obtain the general solution of a generalized cubic functional equation, the Hyers-Ulam-Rassias stability, and the stability by using the alternative fixed point for a generalized cubic functional equation $$4f(\sum_{j=1}^{n-1}\;x_j\;+\;mx_n)\;+\;4f(\sum_{j=1}^{n-1}\;x_j+mx_n\;x_j\;-\;mx_n}\;+\;m^2\sum_{j=1}^{n-1}\;(f(2x_j)\;=\;8f(\sum_{j=1}^{n-1}\;x_j)\;+\;4m^2{\sum_{j=1}^{n-1}}\;\(f(x_j+x_n)\;+\;f(x_j-x_n)\)$$ for a positive integer $m\;{\geq}\;1$.

• 호남수학학술지
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• 제30권1호
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• pp.9-20
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• 2008

Let ${X_i{\mid}i{\geq}1}$ be a strictly stationary sequence of associated random variables with mean zero and let ${\sigma}^2=EX_1^2+2\sum\limits_{j=2}^\infty{EX_1}{X_j}$ with 0 < ${\sigma}^2$ < ${\infty}$. Set $S_n={\sum\limits^n_{i=1}^\{X_i}$, the precise asymptotics for ${\varepsilon}^{{\frac{2(r-p)}{2-p}}-1}\sum\limits_{n{\geq}1}n^{{\frac{r}{p}}-{\frac{1}{p}}+{\frac{1}{2}}}P({\mid}S_n{\mid}{\geq}{\varepsilon}n^{{\frac{1}{p}}})$,${\varepsilon}^2\sum\limits_{n{\geq}3}{\frac{1}{nlogn}}p({\mid}Sn{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ and ${\varepsilon}^{2{\delta}+2}\sum\limits_{n{\geq}1}{\frac{(loglogn)^{\delta}}{nlogn}}p({\mid}S_n{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ as ${\varepsilon}{\searrow}0$ are established under the suitable conditions.

• Journal of applied mathematics & informatics
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• 제30권5_6호
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• pp.829-845
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• 2012

In this paper we study general solutions and generalized Hyers-Ulam-Rassias stability of the following function equation $$f(x-\sum^{k}_{i=1}x_i)+(k-1)f(x)+(k-1)\sum^{k}_{i=1}(x_i)=f(x-x_1)+\sum^{k}_{i=2}f(x_i-x)+\sum^{k}_{i=1}\sum^{k}_{j=1,j > i}f(x_i+x_j)$$. for $k{\geq}2$, on non-Archimedean Banach spaces. It will be proved that this equation is equivalent to the so-called quadratic functional equation.

• 한국콘텐츠학회논문지
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• 제2권4호
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• pp.46-50
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• 2002

본 논문은 지문 영상의 전처리중 이진화 수행과정에서 지문 영상의 국부적 밝기 차이에 따른 가장 큰 애로점인 임계치(threshold value) 설정을 대상 지문 영역의 밝기 등에 스스로 적응할 수 있도록 Silt Sum 방법을 응용한 적을 이진화를 수행하였다. 기존의 방법과 비교하여 본 연구에서 제시한 개선된 전처리 방법은 보다 높은 인식 정확도를 제공하며, 이에 따라 실험 결과에서 보는 바와 같이 지문 인식을 위한 특징점 추출 알고리즘에 적용될 수 있다.

• 대한수학회논문집
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• 제32권4호
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• pp.959-970
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• 2017

In this paper, we investigate the transcendental meromorphic solutions with finite order of two different types of difference equations $${\sum\limits_{j=1}^{n}}a_jf(z+c_j)={\frac{P(z,f)}{Q(z,f)}}={\frac{{\sum_{k=0}^{p}}b_kf^k}{{\sum_{l=0}^{q}}d_lf^l}}$$ and $${\prod\limits_{j=1}^{n}}f(z+c_j)={\frac{P(z,f)}{Q(z,f)}={\frac{{\sum_{k=0}^{p}}b_kf^k}{{\sum_{l=0}^{q}}d_lf^l}}$$ that share three distinct values with another meromorphic function. Here $a_j$, $b_k$, $d_l$ are small functions of f and $a_j{\not{\equiv}}(j=1,2,{\ldots},n)$, $b_p{\not{\equiv}}0$, $d_q{\not{\equiv}}0$. $c_j{\neq}0$ are pairwise distinct constants. p, q, n are non-negative integers. P(z, f) and Q(z, f) are two mutually prime polynomials in f.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2003년도 추계학술대회논문집
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• pp.383-388
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• 2003

As an effective way of response evaluation in structural vibration analysis, the phase vector sum(PVS) method used in shaft torsional vibration analysis is introduced. Basic relation of PVS applicable to structural problem is derived and applied to Diesel engine structures. Concepts of forced phase vector sum (FPVS) and significance level (SL) are proposed to visualize the correlation between excitation orders and vibration modes in the SL map. The maximum responses and SL are compared and reviewed to confirm the validity of the method. It is regarded FPVS is adequate to newly evaluate the structural vibration based on excitation information.