• Title/Summary/Keyword: n-norm

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Grand Average in MEG and Crude Estimation of Anatomical Site (뇌자도에서 전체 평균과 이를 이용한 해부학적 위치 추정)

  • Kwon H.;Kim K.;Kim J. M.;Lee Y. H.;Park Y. K.
    • Journal of Biomedical Engineering Research
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    • v.25 no.6
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    • pp.575-580
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    • 2004
  • In this work, a method is presented to find an anatomical site of a current source crudely in a standard brain using grand average of MEG data. Minimum norm estimation algorithm and truncated singular value decomposition were applied to calculate the distributed sources that can reproduce the measured signals. Grand average over all subjects was obtained from the transformed signals, which would be detected in a standard sensor plane by the obtained distributed current sources. In the simulation study, it was shown that the localized dipole using the grand average is consistent with the mean location of localized dipoles of all subjects within several mm even with large inter-individual differences of sensor positions. This result suggests that the mean location of low level signal source can be estimated as a dipole source in grand average and it was confirmed in the localization of the current source of N100m. when the localized dipole is registered on a standard brain. This result also suggests that the activity region obtained from grand average can be crudely estimated on a standard brain using the source location of the N100m as a reference point.

WEAK AND STRONG CONVERGENCE OF MANN'S-TYPE ITERATIONS FOR A COUNTABLE FAMILY OF NONEXPANSIVE MAPPINGS

  • Song, Yisheng;Chen, Rudong
    • Journal of the Korean Mathematical Society
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    • v.45 no.5
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    • pp.1393-1404
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    • 2008
  • Let K be a nonempty closed convex subset of a Banach space E. Suppose $\{T_{n}\}$ (n = 1,2,...) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ${\cap}_{n=1}^{\infty}$ F$\(T_n){\neq}{\phi}$. For $x_0{\in}K$, define $x_{n+1}={\lambda}_{n+1}x_{n}+(1-{\lambda}_{n+1})T_{n+1}x_{n},n{\geq}0$. If ${\lambda}_n{\subset}[0,1]$ satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n=0$, we proved that $\{x_n\}$ weakly converges to some $z{\in}F\;as\;n{\rightarrow}{\infty}$ in the framework of reflexive Banach space E which satisfies the Opial's condition or has $Fr{\acute{e}}chet$ differentiable norm or its dual $E^*$ has the Kadec-Klee property. We also obtain that $\{x_n\}$ strongly converges to some $z{\in}F$ in Banach space E if K is a compact subset of E or there exists one map $T{\in}\{T_{n};n=1,2,...\}$ satisfy some compact conditions such as T is semi compact or satisfy Condition A or $lim_{n{\rightarrow}{\infty}}d(x_{n},F(T))=0$ and so on.

Optimal Rates of Convergence in Tensor Sobolev Space Regression

  • Koo, Ja-Yong
    • Journal of the Korean Statistical Society
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    • v.21 no.2
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    • pp.153-166
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    • 1992
  • Consider an unknown regression function f of the response Y on a d-dimensional measurement variable X. It is assumed that f belongs to a tensor Sobolev space. Let T denote a differential operator. Let $\hat{T}_n$ denote an estimator of T(f) based on a random sample of size n from the distribution of (X, Y), and let $\Vert \hat{T}_n - T(f) \Vert_2$ be the usual $L_2$ norm of the restriction of $\hat{T}_n - T(f)$ to a subset of $R^d$. Under appropriate regularity conditions, the optimal rate of convergence for $\Vert \hat{T}_n - T(f) \Vert_2$ is discussed.

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CRITICAL METRICS ON NEARLY KAEHLERIAN MANIFOLDS

  • Pak, Jin-Suk;Yoo, Hwal-Lan
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.9-13
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    • 1992
  • In this paper, we consider the function related with almost hermitian structure on a copact complex manifold. More precisely, on a 2n-diminsional complex manifold M admitting 2-form .ohm. of rank 2n everywhere, assume that M admits a metric g such that g(JX, JY)=g(X,Y), that is, assume that g defines an hermitian structure on M admitting .ohm. as fundamental 2-form-the 'almost complex structure' J being determined by g and .ohm.:g(X,Y)=.ohm.(X,JY). We consider the function I(g):=.int.$_{M}$ $N^{2}$d $V_{g}$, where N is the norm of Nijenhuis tensor N defined by (J,g). by (J,g).

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SUMMABILITY IN MUSIELAK-ORLICZ HARDY SPACES

  • Jun Liu;Haonan Xia
    • Journal of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1057-1072
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    • 2023
  • Let 𝜑 : ℝn × [0, ∞) → [0, ∞) be a growth function and H𝜑(ℝn) the Musielak-Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called 𝜃-summability is considered for multi-dimensional Fourier transforms in H𝜑(ℝn). Precisely, with some assumptions on 𝜃, the authors first prove that the maximal operator of the 𝜃-means is bounded from H𝜑(ℝn) to L𝜑(ℝn). As consequences, some norm and almost everywhere convergence results of the 𝜃-means, which generalizes the well-known Lebesgue's theorem, are then obtained. Finally, the corresponding conclusions of some specific summability methods, such as Bochner-Riesz, Weierstrass and Picard-Bessel summations, are also presented.

QUADRATIC MAPPINGS ASSOCIATED WITH INNER PRODUCT SPACES

  • Lee, Sung Jin
    • Korean Journal of Mathematics
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    • v.19 no.1
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    • pp.77-85
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    • 2011
  • In [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $${\sum_{i=1}^{n}}\left\|x_i-{\frac{1}{n}}{\sum_{j=1}^{n}}x_j \right\|^2={\sum_{i=1}^{n}}{\parallel}x_i{\parallel}^2-n\left\|{\frac{1}{n}}{\sum_{i=1}^{n}}x_i \right\|^2$$ holds for all $x_1$, ${\cdots}$, $x_n{\in}V$. Let V, W be real vector spaces. It is shown that if an even mapping $f:V{\rightarrow}W$ satisfies $$(0.1)\;{\sum_{i=1}^{2n}f}\(x_i-{\frac{1}{2n}}{\sum_{j=1}^{2n}}x_j\)={\sum_{i=1}^{2n}}f(x_i)-2nf\({\frac{1}{2n}}{\sum_{i=1}^{2n}}x_i\)$$ for all $x_1$, ${\cdots}$, $x_{2n}{\in}V$, then the even mapping $f:V{\rightarrow}W$ is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.

FUNCTIONAL EQUATIONS ASSOCIATED WITH INNER PRODUCT SPACES

  • Park, Choonkil;Huh, Jae Sung;Min, Won June;Nam, Dong Hoon;Roh, Seung Hyeon
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.4
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    • pp.455-466
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    • 2008
  • In, [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $$n{\left\|{\frac{1}{n}}{\sum\limits_{i=1}^{n}}x_i{\left\|^2+{\sum\limits_{i=1}^{n}}\right\|}{x_i-{\frac{1}{n}}{\sum\limits_{j=1}^{n}x_j}}\right\|^2}={\sum\limits_{i=1}^{n}}{\parallel}x_i{\parallel}^2$$ holds for all $x_1,{\cdots},x_{n}{\in}V$. Let V,W be real vector spaces. It is shown that if a mapping $f:V{\rightarrow}W$ satisfies $$(0.1){\hspace{10}}nf{\left({\frac{1}{n}}{\sum\limits_{i=1}^{n}}x_i \right)}+{\sum\limits_{i=1}^{n}}f{\left({x_i-{\frac{1}{n}}{\sum\limits_{j=1}^{n}}x_i}\right)}\\{\hspace{140}}={\sum\limits_{i=1}^{n}}f(x_i)$$ for all $x_1$, ${\dots}$, $x_{n}{\in}V$ $$(0.2){\hspace{10}}2f\(\frac{x+y}{2}\)+f\(\frac{x-y}{2} \)+f\(\frac{y}{2}-x\)\\{\hspace{185}}=f(x)+f(y)$$ for all $x,y{\in}V$. Furthermore, we prove the generalized Hyers-Ulam stability of the functional equation (0.2) in real Banach spaces.

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STABILITY OF FUNCTIONAL EQUATIONS ASSOCIATED WITH INNER PRODUCT SPACES: A FIXED POINT APPROACH

  • Park, Choonkil;Hur, Jae Sung;Min, Won June;Nam, Dong Hoon;Roh, Seung Hyeon
    • Korean Journal of Mathematics
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    • v.16 no.3
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    • pp.413-424
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    • 2008
  • In [21], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $$n{\parallel}\frac{1}{n}\sum\limits_{i=1}^{n}x_i{\parallel}^2+\sum\limits_{i=1}^{n}{\parallel}x_i-\frac{1}{n}\sum\limits_{j=1}^{n}x_j{\parallel}^2=\sum\limits_{i=1}^{n}{\parallel}x_i{\parallel}^2$$ holds for all $x_1,{\dots},x_n{\in}V$. We consider the functional equation $$nf(\frac{1}{n}\sum\limits^n_{i=1}x_i)+\sum\limits_{i=1}^{n}f(x_i-\frac{1}{n}\sum\limits_{j=1}^{n}x_j)=\sum\limits_{i=1}^nf(x_i)$$ Using fixed point methods, we prove the generalized Hyers-Ulam stability of the functional equation $$(1)\;2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})=f(x)+f(y)$$.

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A GENERALIZATION OF A RESULT OF CHOA ON ANALYTIC FUNCTIONS WITH HADAMARD GAPS

  • Stevic Stevo
    • Journal of the Korean Mathematical Society
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    • v.43 no.3
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    • pp.579-591
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    • 2006
  • In this paper we obtain a sufficient and necessary condition for an analytic function f on the unit ball B with Hadamard gaps, that is, for $f(z)\;=\;{\sum}^{\infty}_{k=1}\;P_{nk}(z)$ (the homogeneous polynomial expansion of f) satisfying $n_{k+1}/n_{k}{\ge}{\lambda}>1$ for all $k\;{\in}\;N$, to belong to the weighted Bergman space $$A^p_{\alpha}(B)\;=\;\{f{\mid}{\int}_{B}{\mid}f(z){\mid}^{p}(1-{\mid}z{\mid}^2)^{\alpha}dV(z) < {\infty},\;f{\in}H(B)\}$$. We find a growth estimate for the integral mean $$\({\int}_{{\partial}B}{\mid}f(r{\zeta}){\mid}^pd{\sigma}({\zeta})\)^{1/p}$$, and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space $H_{p,q,{\alpha}$(B) and weighted Bergman space on polydisc $A^p_{^{\to}_{\alpha}}(U^n)$ are also given.

Characteristic Evaluation of Exposed Dose with NORM added Consumer Product based on ICRP Reference Phantom (ICRP 기준팬텀 기반의 천연방사성핵종이 포함된 가공제품 사용으로 인한 피폭선량 특성 평가)

  • Yoo, Do Hyeon;Lee, Hyun Cheol;Shin, Wook-Geun;Choi, Hyun Joon;Min, Chul Hee
    • Journal of Radiation Protection and Research
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    • v.39 no.4
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    • pp.159-167
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    • 2014
  • In Korea, July 2012, the law as called 'Act on Safety Control of Radioactive Rays Around Living Environment' was implemented to control the consumer product containing Naturally Occurring Radioactive Material (NORM), but, there are no appropriate database and effective dose calculation system. The aim of this study was to develop evaluation technique of the exposure dose with the use of the consumer products containing NORM and to understand the characteristics of the exposed dose according to the radiation type and energy. For the evaluate of exposure dose, the ICRP reference phantom was simulated by the MCNPX code based on Monte Carlo method, and the minimum, medium, maximum energy of alphas, betas, gammas from the representative NORM of Uranium decay series were used as the source term in the simulation. The annual effective doses were calculated by the exposure scenario of the consumer product usage time and position. Short range of the alpha and beta rays are mostly delivered the dose to the skin. On the other hand, the gamma rays mostly delivered the similar dose to all of the organs. The results of the annual effective dose with $1Bq{\cdot}g^{-1}$ radioactive stone-bed and 10% radioactive concentration were employed with the usage time of 7 hours 50 minute per day, the maximum annual effective dose of alphas, betas, gammas were calculated 0.0222, 0.0836, $0.0101mSv{\cdot}y^{-1}$, respectively.