• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.91-97
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• 1991

This paper studies the structure and continuity of derivations of the Banach algebra $C^n(I)$ of n times continuously differentiable functions on an interval I into Banach $C^n(I)$-modules.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.93-100
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• 2001

Author proves weak type estimates of the maximal function associated with the Bochner-Riesz means while it is claimed p=2n/(n+1+$2\delta) and 0<\delta\leq(n-1)/2$ that the maximal function is bounded on L^p-{rad}$.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.497-501
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• 1996

Let B denote the unit ball in $C^n(n \geq 1)$, and $\upsilon$ the 2n-dimensional Lebesgue measure on B normalized so that $\upsilon(B) = 1$, while $\sigma$ is the normalized surface measure on the boundary S of B.

• East Asian mathematical journal
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• v.19 no.2
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• pp.221-232
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• 2003

An identity for n-time differentiable functions of a real variable in terms of multiple integrals and applications for Ostrowski type inequalities are given.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1071-1085
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• 2016

For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.269-277
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• 2008

Let $f=(f_n)$ be a nonnegative submartingale such that ${\parallel}f{\parallel}{\infty}{\leq}1\;and\;g=(g_n)$ be a martingale, adapted to the same filtration, satisfying $${\mid}d_{gn}{\mid}{\leq}{\mid}df_n{\mid},\;n=0,\;1,\;2,\;....$$ The paper contains the proof of the sharp inequality $$\limits^{sup}_ n\;\mathbb{E}{\Phi}({\mid}g_n{\mid}){\leq}{\Phi}(1)$$ for a class of convex increasing functions ${\Phi}\;on\;[0,\;{\infty}]$, satisfying certain growth condition. As an application, we show a continuous-time version for stochastic integrals and a related estimate for smooth functions on Euclidean domain.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.927-934
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• 2011

Invertible transformations over n-bit words are essential ingredients in many cryptographic constructions. When n is large (e.g., n = 64) such invertible transformations are usually represented as a composition of simpler operations such as linear functions, S-P networks, Feistel structures and T-functions. Among them we study T-functions which are probably invertible and are very useful in stream ciphers. In this paper we study some conditions on a T-function h(x) such that f(x) = x + h(x) has a single cycle on ${\mathbb{Z}}_{2^n}$.

• The KIPS Transactions:PartB
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• v.11B no.6
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• pp.717-722
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• 2004

This paper presents fuzzy rules to predict diagnosis of Wisconsin breast cancer using neural network with weighted fuzzy membership functions (NNWFM). NNWFM is capable of self-adapting weighted membership functions to enhance accuracy in prediction from the given clinical training data. n set of small, medium, and large weighted triangular membership functions in a hyperbox are used for representing n set of featured input. The membership functions are randomly distributed and weighted initially, and then their positions and weights are adjusted during learning. After learning, prediction rules are extracted directly from the enhanced bounded sums of n set of weighted fuzzy membership functions. Two number of prediction rules extracted from NNWFM outperforms to the current published results in number of rules and accuracy with 99.41%.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1105-1127
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• 2013

Let C[0, $t$] denote the function space of real-valued continuous paths on [0, $t$]. Define $X_n\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}\;:\;C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\ldots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\ldots},x(t_n),x(t_{n+1}))$, respectively, where $0=t_0 <; t_1 <{\ldots} < t_n < t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form $fr((v_1,x),{\ldots},(v_r,x)){\int}_{L_2}_{[0,t]}\exp\{i(v,x)\}d{\sigma}(v)$ for $x{\in}C[0,t]$, where $\{v_1,{\ldots},v_r\}$ is an orthonormal subset of $L_2[0,t]$, $f_r{\in}L_p(\mathbb{R}^r)$, and ${\sigma}$ is the complex Borel measure of bounded variation on $L_2[0,t]$. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.239-252
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• 2021

The prime target of this paper is to compare some relative (k, n) Nevanlinna defects with relative (k, n) Valiron defects from the view point of integrated moduli of logarithmic derivative of entire and meromorphic functions where k and n are any two non-negative integers.