• Journal of the Korean Institute of Intelligent Systems
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• v.15 no.6
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• pp.754-758
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• 2005

There have been several tipical methods being used tomeasure the fuzziness (entropy) of fuzzy sets. Pedrycz is the original motivation of this paper. Recently, Wang and Chiu [FSS103(1999) 443-455] and Pedrycz [FSS 64(1994) 21-30] showed the relationship(addition, subtraction, multiplication) between the entropies of the resultant fuzzy number and the original fuzzy numbers of same type. In this paper, using Lebesgue-Stieltjes integral, we generalize results of Wang and Chiu [FSS 103(1999) 443-455] concerning entropy arithmetic operations without the condition of same types of fuzzy numbers. And using this results and trade-off relationship between information energy and entropy, we study more properties of information energy of fuzzy numbers.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.773-778
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• 2009

Recently, Xue et al. [Computers and Mathematics with Applications 55 (2008) 1215-1224] proposed a formula for the expected value of a function of a fuzzy variable based on the assumption that the fuzzy variable has a continuous membership function. In conclusion, they remained the case where the membership function of the fuzzy variable is discontinuous for the future research, and then expected to get similar results. Thus this note is to propose a new formula for the expected value of a function of a general fuzzy variable which is not restricted on having a continuous membership function. Furthermore, we give an example which cannot be applied to the formula that Xue et al. proposed. We also use the same example given by Xue et al. to show how to apply the new formula.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.269-278
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• 2008

Let ${\partial}D$ be the boundary of the open unit disk D in the complex plane and $L^p({\partial}D)$ the class of all complex, Lebesgue measurable function f for which $\{\frac{1}{2\pi}{\int}_{-\pi}^{\pi}{\mid}f(\theta){\mid}^pd\theta\}^{1/p}<{\infty}$. Let P be the orthogonal projection from $L^p({\partial}D)$ onto ${\cap}_{n<0}$ ker $a_n$. For $f{\in}L^1({\partial}D)$, ${\hat{f}}(z)=\frac{1}{2\pi}{\int}_{-\pi}^{\pi}P_r(t-\theta)f(\theta)d{\theta}$ is the harmonic extension of f. Let ${\hat{P}}$ be the composition of P with the harmonic extension. In this paper, we will show that if $1, then ${\hat{P}}:L^p({\partial}D){\rightarrow}H^p(D)$ is bounded. In particular, we will show that ${\hat{P}}$ is unbounded on $L^{\infty}({\partial}D)$.

• Kyungpook Mathematical Journal
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• v.50 no.4
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• pp.545-556
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• 2010

A good amount of work has been done on degree of approximation of functions belonging to Lip${\alpha}$, Lip($\xi$(t),r) and W($L_r,\xi(t)$) and classes using Ces$\`{a}$ro, N$\"{o}$rlund and generalised N$\"{o}$rlund single summability methods by a number of researchers ([1], [10], [8], [6], [7], [2], [3], [4], [9]). But till now, nothing seems to have been done so far to obtain the degree of approximation of functions using (N,$p_n$)(C, 1) product summability method. Therefore the purpose of present paper is to establish two quite new theorems on degree of approximation of function $f\;\in\;Lip({\alpha},r)$ class and $f\;\in\;W(L_r,\;\xi(t))$ class by (N, $p_n$)(C, 1) product summability means of its Fourier series.