• Title, Summary, Keyword: Norm

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FUNCTIONS ATTAINING THE SUPREMUM AND ISOMORPHIC PROPERTIES OF A BANACH SPACE

  • D. Acosta, Maria ;Becerra Guerrero, Julio ;Ruiz Galan, Manuel
    • Journal of the Korean Mathematical Society
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    • v.41 no.1
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    • pp.21-38
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    • 2004
  • We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace Μ containing u, it happens that the subset of norm attaining functionals on Μ is second Baire category in $M^{*}$ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to $\ell$$_1$, where the norm is the restriction of a Luxembourg norm on $L_1$. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.m.

An Efficient Implementation of Hybrid $l^1/l^2$ Norm IRLS Method as a Robust Inversion (강인한 역산으로서의 하이브리드 $l^1/l^2$ norm IRLS 방법의 효율적 구현기법)

  • Ji, Jun
    • Geophysics and Geophysical Exploration
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    • v.10 no.2
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    • pp.124-130
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    • 2007
  • Least squares ($l^2$ norm) solutions of seismic inversion tend to be very sensitive to data points with large errors. The $l^1$ norm minimization gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) method gives efficient approximate solutions of these $l^1$ norm problems. I propose an efficient implementation of the IRLS method for a hybrid $l^1/l^2$ minimization problem that behaves as $l^2$ norm fit for small residual and $l^1$ norm fit for large residuals. The proposed algorithm shows more robust characteristics to the decision of the threshold value than the l1 norm IRLS inversion does with respect to the threshold value to avoid singularity.

On the Use of Adaptive Weights for the F-Norm Support Vector Machine

  • Bang, Sung-Wan;Jhun, Myoung-Shic
    • The Korean Journal of Applied Statistics
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    • v.25 no.5
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    • pp.829-835
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    • 2012
  • When the input features are generated by factors in a classification problem, it is more meaningful to identify important factors, rather than individual features. The $F_{\infty}$-norm support vector machine(SVM) has been developed to perform automatic factor selection in classification. However, the $F_{\infty}$-norm SVM may suffer from estimation inefficiency and model selection inconsistency because it applies the same amount of shrinkage to each factor without assessing its relative importance. To overcome such a limitation, we propose the adaptive $F_{\infty}$-norm ($AF_{\infty}$-norm) SVM, which penalizes the empirical hinge loss by the sum of the adaptively weighted factor-wise $L_{\infty}$-norm penalty. The $AF_{\infty}$-norm SVM computes the weights by the 2-norm SVM estimator and can be formulated as a linear programming(LP) problem which is similar to the one of the $F_{\infty}$-norm SVM. The simulation studies show that the proposed $AF_{\infty}$-norm SVM improves upon the $F_{\infty}$-norm SVM in terms of classification accuracy and factor selection performance.

A Mixed Norm Image Restoration Algorithm Using Multi Regularization Parameters (다중 정규화 매개 변수를 이용한 혼합 norm 영상 복원 방식)

  • Choi, Kwon-Yul;Kim, Myoung-Jin;Hong, Min-Cheol
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.32 no.11C
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    • pp.1073-1078
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    • 2007
  • In this paper, we propose an iterative mixed norm image restoration algorithm using multi regularization parameters. A functional which combines the regularized $l_2$ norm functional and the regularized $l_4$ norm functional is proposed to efficiently remove arbitrary noise. The smoothness of each functional is determined by the regularization parameters. Also, a regularization parameter is used to determine the relative importance between the regularized $l_2$ norm functional and the regularized $l_4$ norm functional using kurtosis. An iterative algorithm is utilized for obtaining a solution and its convergence is analyzed. Experimental results demonstrate the capability of the proposed algorithm.

THE RIESZ THEOREM IN FUZZY n-NORMED LINEAR SPACES

  • Kavikumar, J.;Jun, Young-Bae;Khamis, Azme
    • Journal of applied mathematics & informatics
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    • v.27 no.3_4
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    • pp.541-555
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    • 2009
  • The primary purpose of this paper is to prove the fuzzy version of Riesz theorem in n-normed linear space as a generalization of linear n-normed space. Also we study some properties of fuzzy n-norm and introduce a concept of fuzzy anti n-norm.

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THE GROUP OF HAMILTONIAN HOMEOMORPHISMS IN THE L-NORM

  • Muller, Stefan
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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    • pp.1769-1784
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    • 2008
  • The group Hameo (M, $\omega$) of Hamiltonian homeomorphisms of a connected symplectic manifold (M, $\omega$) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the $L^{(1,{\infty})}$-Hofer norm (and not the $L^{\infty}$-Hofer norm) on the space of Hamiltonian paths (see below for the definitions). A justification for this choice was given in [7]. In this article we study the $L^{\infty}$-case. In view of the fact that the Hofer norm on the group Ham (M, $\omega$) of Hamiltonian diffeomorphisms does not depend on the choice of the $L^{(1,{\infty})}$-norm vs. the $L^{\infty}$-norm [9], Y.-G. Oh and D. McDuff (private communications) asked whether the two notions of Hamiltonian homeomorphisms arising from the different norms coincide. We will give an affirmative answer to this question in this paper.

A Study on the Housing Adjustment in the First Half of Cho-Sun Dynasty - with special perspectives of microsociological approach - (조선전반기 가족의 주거조절에 관한 연구 - 미시사회학적 접근으로 -)

  • Hong, Hyung-Ock
    • Journal of architectural history
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    • v.2 no.1
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    • pp.25-41
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    • 1993
  • This study was made to analyze the housing adjustment phenomenon in the first half of Chosun Dynasty by applying Microsociological approach. By reviewing the housing adjustment theory of Morris and Winter, research model for the period was developed in terms of socioeconomic characteristics, normative housing deficit (=cultural norm-housing condition+family norm), constraints, behavioral propensities, and housing adjustment mechanism with the following results : 1. In the first half of Chosun Dynasty the size of the house, the house site and decorating items were specified by law (cultural norm) according to the social status. Although the law was constraints for the housing phenomenon, it was not applied universally. Frequantly the law was violated by the upper class. By the middle of the Dynasty the family norm became more important for the housing phenomenon than the cultural norm. 2. Efforts were made to practice the Confucianism as a cultural norm in the first half of Chosun Dynasty At that time Husband-Living-in-Wife's-House was more popular than Wife-Living-in-Husband's-House. Because the customs were against the Confucianism, the latter was encouraged by law. But it did not change. Instead a compromised system became popular in the middle of the Dynasty. The house shrine was practiced to increase the symbolism of the family, which, in turn, exerted influences in deciding the housing site (cultural norm). These cultural norm was not accepted as the family norm untill the second half of the Dynasty. These trends forced the man and woman use separate areas of the house, and formulated a hierarchic positions within a house. 3. It was shown that the settlement of Confucianism as a family norm was closely related to the popularization of the Ondol system in the house, which functioned as one of the behavioral propensities to encourage diversity of space for many purposes. Though the Ondol system was accepted as a useful heating system earlier, this became more popular in the middle of the Dynasty because the housing pattern with Ondol fitted very well with a large family system with patriarchism. Ondol system for one or two rooms substituted Ondol for all rooms in the second half of the Dynasty. 4. From the beginning of the Dynasty housing adjustment of the family was determined by the social status and by law (cultural norm). Within this cultural norm each family decided its adjustment mechanism according to its economic ability (family norm). Family norm was more important factor than the cultural norm to determine the micro-space pattern in the house. But this period witnessed the formations of new conditions by the ruling class's efforts to implement new ethics for hierarchy and sexual discrimination. According to these conditions the Confucianism overruled the family norm in the later period.

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Robust inversion of seismic data using ${\ell}^1/{\ell}^2$ norm IRLS method (${\ell}^1/{\ell}^2$ norm IRLS 방법을 사용한 강인한 탄성파자료역산)

  • Ji Jun
    • 한국지구물리탐사학회:학술대회논문집
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    • pp.227-232
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    • 2005
  • Least squares (${\ell}^2-norm$) solutions of seismic inversion tend to be very sensitive to data points with large errors. The ${\ell}^p-norm$ minimization for $1{\le}p<2$ gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions of these ${\ell}^p-norm$ problems. I propose a simple way to implement the IRLS method for a hybrid ${\ell}^1/{\ell}^2$ minimization problem that behaves as ${\ell}^2$ fit for small residual and ${\ell}^1$ fit for large residuals. Synthetic and a field-data examples demonstrates the improvement of the hybrid method over least squares when there are outliers in the data.

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