• The Pure and Applied Mathematics
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• v.25 no.1
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• pp.1-5
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• 2018

It is proved that 'maximum' is actually attained in the following risk measure representation $${\rho}_m(X)={max \atop Q{\in}Q_m}E_Q[-X]$$.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.561-569
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• 2018

In this paper, we introduce some notions of orthogonality in the setting of Finsler $C^*$-modules and investigate their relations with the Birkhoff-James orthogonality. Suppose that ($E,{\rho}$) and ($F,{\rho}^{\prime}$) are Finsler modules over $C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, respectively, and ${\varphi}:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ is a *-homomorphism. A map ${\Psi}:E{\rightarrow}F$ is said to be a ${\varphi}$-morphism of Finsler modules if ${\rho}^{\prime}({\Psi}(x))={\varphi}({\rho}(x))$ and ${\Psi}(ax)={\varphi}(a){\Psi}(x)$ for all $a{\in}{\mathcal{A}}$ and all $x{\in}E$. We show that each ${\varphi}$-morphism of Finsler $C^*$-modules preserves the Birkhoff-James orthogonality and conversely, each surjective linear map between Finsler $C^*$-modules preserving the Birkhoff-James orthogonality is a ${\varphi}$-morphism under certain conditions. In fact, we state a version of Wigner's theorem in the framework of Finsler $C^*$-modules.

• The Pure and Applied Mathematics
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• v.25 no.1
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• pp.49-57
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• 2018

We have introduced two new notions of convexity and closedness in functional analysis. Let X be a real normed space, then $C({\subseteq}X)$ is functionally convex (briefly, F-convex), if $T(C){\subseteq}{\mathbb{R}}$ is convex for all bounded linear transformations $T{\in}B$(X, R); and $K({\subseteq}X)$ is functionally closed (briefly, F-closed), if $T(K){\subseteq}{\mathbb{R}}$ is closed for all bounded linear transformations $T{\in}B$(X, R). By using these new notions, the Alaoglu-Bourbaki-Eberlein-${\check{S}}muljan$ theorem has been generalized. Moreover, we show that X is reflexive if and only if the closed unit ball of X is F-closed. James showed that for every closed convex subset C of a Banach space X, C is weakly compact if and only if every $f{\in}X^{\ast}$ attains its supremum over C at some point of C. Now, we show that if A is an F-convex subset of a Banach space X, then A is bounded and F-closed if and only if every element of $X^{\ast}$ attains its supremum over A at some point of A.

• The Bulletin of The Korean Astronomical Society
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• v.35 no.2
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• pp.38.1-38.1
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• 2010

We present the results of the survey for submillimeter galaxies in the MS0451 (04h 54m 10.8s, -03d 00m 57.0s) at z = 0.55 and PKS1138-262 (11h 40m 48.25s, -26d 29m 10.1s) at z = 2.16 with the 1.1mm bolometer array AzTEC at the James Clerk Maxwell Telescope. The samples were centered on a prominent large-scale structure overdensity. Submillimeter galaxies seem to be starburst galaxies at high redshift ($z\;\geq\;1$) with high starformation rates ($\sim1000M\odot\;yr^{-1}$) or active galactic nuclei (AGN). We have obtained AzTEC images using the AzTEC data reduction pipeline with the IDL language. Through a bayes' theorem, we determined the extragalaxy catalogue, containing the false-detection rate, completeness, flux deboosting correction, and the source positional uncertainty in this region. We compared the catalogue with HST, DSS, 2MASS observations.

• Journal of the Korean Data and Information Science Society
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• v.26 no.3
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• pp.653-659
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• 2015

The Pythagorean formula for baseball postulated by James (1982) indicates the winning percentage as a function of runs scored and runs allowed. However sometimes, the Pythagorean formula gives a less accurate estimate of winning percentage. We use the records of team vs team historic win loss records of Korean professional baseball clubs season from 2005 and 2014. Using assumption that the difference between winning percentage and pythagorean expectation are affected by unusual distribution of runs scored and allowed, we suppose that difference depends on mean, standard deviation, and coefficient of variation of runs scored per game and runs allowed per game, respectively. In conclusion, the discrepancy is mainly related to the coefficient of variation and standard deviation for run allowed per game regardless of run scored per game.

• Restorative Dentistry and Endodontics
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• v.47 no.1
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• pp.1.1-1.13
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• 2022

Objectives: This study assessed the shape and anatomical relationship of the mental foramen (MF) to mandibular posterior teeth in an Indian sub-population. Materials and Methods: In total, 475 existing cone-beam computed tomography records exhibiting 950 MFs and including the bilateral presence of mandibular premolars and first molars were assessed. Images were evaluated 3-dimensionally to ascertain the position, shape, and anatomical proximity of MFs to mandibular teeth. The position and shape of MFs were measured and calculated. The Pythagorean theorem was used to calculate the distance between the root apex of the mandibular teeth and the MF. Results: MFs exhibited a predominantly round shape (left: 67% and right: 65%) followed by oval (left: 30% and right: 31%) in both males and females and in different age groups. The root apices of mandibular second premolars (left: 71% and right: 62%) were closest to the MF, followed by distal to the first premolars and mesial to the second premolars. The mean vertical distance between the MF and the nearest tooth apex calculated on sagittal sections was 2.20 mm on the right side and 2.32 mm on the left side; no significant difference was found according to sex or age. The distance between the apices of the teeth and the MF was ≥ 4 mm (left; 4.09 ± 1.27 mm and right; 4.01 ± 1.15 mm). Conclusions: These findings highlight the need for clinicians to be aware of the location of the MF in treatment planning and while performing non-surgical and surgical endodontic procedures.