• Journal of Korean Ophthalmic Optics Society
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• v.20 no.2
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• pp.133-140
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• 2015

Purpose: It was investigated whether two different stabilization designs of toric contact lenses changed the rotational axis and degree of toric lenses according to body posture and gaze direction in the present study. Methods: Toric soft contact lenses with Lo-Torque$^{TM}$ design and ASD design (accelerated stabilized design) were fitted on 52 eyes aged in 20s-30s. Then, rotational degree was measured at the five gaze directions including front gaze and the lying position. Results: When gazing the front and vertical directions in the upright posture, lens was much rotated to nasal side for the Lo-Torque$^{TM}$ design and temporal side for the ASD design. When gazing horizontal direction, both design lenses were rotated against to the gaze direction. Rotation degree was the smallest at superior direction gaze and the largest at nasal gaze. In case of the rotation degree less than $5^{\circ}$, Lo-Torque$^{TM}$ design was more frequent when gazing front and vertical directions, and ASD design was more frequent when gazing horizontal direction. In addition, the lens with Lo-Torque$^{TM}$ design was lesser rotation degree than with ASD design immediately after lying. On the other hand, the lens with ASD design was lesser rotation degree than with Lo-Torque$^{TM}$ design 1 minute later after lying. Conclusions: This study confirmed that axis rotation of the lens induced by gaze direction and posture was different according to axis stabilization design during wearing toric soft contact lens.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.761-765
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• 2012

We prove that for a toric manifold (respectively, a quasitoric manifold) M, any graded ring isomorphism $H^*(M){\rightarrow}H^*({\Pi}_{i=1}^{m}\mathbb{C}P^{ni})$ can be realized by a diffeomorphism (respectively, a homeomorphism) ${\Pi}_{i=1}^{m}\mathbb{C}P^{ni}{\rightarrow}M$.

• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.331-360
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• 2017

Unknotting numbers for torus knots and links are well known. In this paper, we present a new approach to determine the position of unknotting number crossing changes in a toric braid such that the closure of the resultant braid is equivalent to the trivial knot or link. Further we give unknotting numbers of more than 600 knots.

• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.1011-1018
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• 2022

In this work, magnetic geodesics over the space of Kähler potentials are studied through a variational method for a generalized Landau-Hall functional. The magnetic geodesic equation is calculated in this setting and its relation to a perturbed complex Monge-Ampère equation is given. Lastly, the magnetic geodesic equation is considered over the special case of toric Kähler potentials over toric Kähler manifolds.

• Journal of Korean Ophthalmic Optics Society
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• v.18 no.4
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• pp.449-456
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• 2013

Purpose: The present study aimed to investigate the effects of corneal eccentricity and shape on the rotational pattern of toric soft lens by the postural change of lens wearers. Methods: The corneal eccentricity of 41 eyes (aged 20s) having -1.0 D with-the-rule corneal astigmatism (WRCA) was measured, and then toric soft lenses were fitted with the amount of total astigmatism. In lying and straight postures, the rotation of toric soft lenses was recorded by a camera attached to slitlamp and analyzed. Results: Most toric soft lens designed with accelerated stabilization rotated to the temporal direction, which was the lying position direction, regardless of corneal eccentricity, and some lenses rotated to the nasal direction for high corneal eccentricity and corneal type of asymmetric bowtie. There was no correlation between the amount of rotation and corneal eccentricity right after of contact lens wearing in straight and lying posture, however, the amount of rotation was the greater for the cornea with the higher eccentricity after the subjects laying down for some period. The speed of lens rotation started to decrease after the subjects laying down, but the speed was not different according to corneal eccentricity difference. The amount of lens rotation for symmetric and asymmetric bowtie-typed corneas increased more than it for oval-typed cornea, and it was same even with time elapsing. The speed of lens rotation in lying posture was the slowest in asymmetric bowtie-typed cornea compared with other corneal types. Conclusions: From the present study, it was revealed that the rotational pattern of toric soft lens was affected by corneal eccentricity and corneal shape when the wearer's posture changed. Thus, it should be considered for the development of the fitting guideline and the design of toric soft lens.

• Journal of Korean Ophthalmic Optics Society
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• v.16 no.4
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• pp.439-444
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• 2011

Purpose: To represent the shape of toric corea in the elliptical function for the determination of curvature distribution and lacrimal thickness between cornea and contact lens when the lens is fitted. Methods: Topography measurements of corneal curvature and curvature equation derived from the assumed elliptical function were evaluated using the Excel program which included the necessary equation derived. Results: Mathematical expressions for the cornea whose ribbon shaped-topography image, in which the center does not coincide with the corneal apex, can be determined. Conclusions: For the application where the higher accuracy on the cornea is not required, such as higher order aberration, the cornea cal be expressed in the simple elliptical function.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.853-868
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• 2015

Let G be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible G-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e., closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e., each lattice point appears precisely once in the formula, with coefficient +1. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.

• Journal of Korean Ophthalmic Optics Society
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• v.19 no.2
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• pp.189-198
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• 2014

Purpose: The present has analyzed the correlation between the direction of lens and the amount of rotation upon soft toric contact lens fitting after classifying the corneal astigmatism. Methods: Soft toric contact lens was fitted on 114 with-the-rule astigmatic eyes with total astigmatism of at least -0.75 D in their 20s and 30s according to the fitting guideline of the manufacturer and the correlation between the astigmatic degree and the rotational direction/amount of rotation was analyzed by when keeping the eyes on the front and by changing the direction of gaze. As for re-orientation movement. The speed of lens re-orientation and total amount of lens rotation was compared and analyzed by corneal astigmatism after mis-location of lens of $45^{\circ}$ to temporal and nasal direction, respectively. Results: The positive correlations were shown between corneal astigmatism and the direction of lens rotation and between corneal astigmatism and the amount of lens rotation. Meanwhile, the amount of lens rotation was different by the direction of gaze however, there was no correlation with corneal astigmatism. The speed of lens re-orientation was fastest in the group of high astigmatic degree when the lens was mis-located to both temporal and nasal directions. Conclusions: For optimal axis stabilization of toric soft lens, it is proposed that the adjustment of fitting guideline considering corneal astigmatism is necessary since the current fitting guideline is only based on total astigmatism.

• Journal of Korean Ophthalmic Optics Society
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• v.16 no.3
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• pp.265-272
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• 2011

Purpose: The clinical usefulness of rotation evaluation using objective refraction in toric soft lenses fitting was investigated. Methods: Toric soft lenses were fitted for 32 subjects (64 eyes; mean age of 24.69 ${\pm}$ 1.65 years) with astigmatism and both eyes of each subject were fitted with toric soft lenses. Objective refraction-based lenses rotation was evaluated from refraction and over-refraction data by indirect calculating technique. These calculated data were compared with the measured data from slit lamp with direct measuring technique. Results: Orientation of toric soft lenses around zero position (within ${\pm}$ 5$^{\circ}$ vertical line) was investigated. The orientations to the direction of nose of measured and calculated values were 69.78% and 63.64%, respectively, which showed similar values between two techniques. Agreement frequency between measured and calculated values in the magnitude of lenses rotation 54.69% and 82.82% for 10$^{\circ}$ and 20$^{\circ}$ of vertical line, respectively. The 95% limits of agreement between calculation and measurement were from -10.08$^{\circ}$ to 12.65$^{\circ}$, and mean difference was 1.29$^{\circ}$ within ${\pm}$ 10$^{\circ}$. The result showed there was no significant difference (p = 0.1984) and high correlation (r = 0.56, p = 0.0004) between two techniques. But the 95% limits of agreement was widen in ${\pm}$ 20$^{\circ}$ of vertical line. The magnitude of lens rotation between two methods was 9.66 ${\pm}$ 6.16$^{\circ}$, 16.17 ${\pm}$ 12.38$^{\circ}$ and 10.58 ${\pm}$ 12.02$^{\circ}$ for normal, loose and tight fitted conditions. Conclusions: From the results with smaller difference between two techniques, it was found that higher availability of subjective over-refraction data can be used as a supplementary tool for subjective refraction. An application using objective refraction with direct measuring could be provide high success in prescription on toric soft lenses.

• Journal of Korean Ophthalmic Optics Society
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• v.6 no.2
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• pp.65-70
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• 2001

We obtained the sum of two curvature ($C_x+C_y$) in toric lens which two toroidal surface is the right angle each other. $$C_x+C_y=\frac{x^2+y^2}{2r_1}+\frac{x^2}{2}(\frac{1}{r_2}-\frac{1}{r_1})$$ and the sum of two curvature ($C_a+C_b$) in toric lens about the cross angle. $$(C_a+C_b)=\frac{x^2cos^2{\alpha}_1}{2r_1}+\frac{x^2cos^2{\alpha}_2}{2r_2}+\frac{y^2sin^2{\alpha}_1}{2r_1}+\frac{y^2sin^2{\alpha}_2}{2r_2}$$ and claculated the parameter S, C, ${\theta}$ of a combination power in toric lens of the cross angle including surface curvature ($C_x$, $C_y$) values. $$S=(n-1)\[\frac{C_x}{x^2}+\frac{C_y}{y^2}\]-\frac{C}{2},\;C=-\frac{2(n-1)}{sin2{\theta}}\[\frac{C_x}{x^2}+\frac{C_y}{y^2}\]$$ $${\theta}=\frac{1}{2}tan^{-1}\[-\frac{{C_xy^2sin2{\theta}_1}+{C_yx^2sin2{\theta}_2}}{{C_xy^2cos2{\theta}_1}+{C_yx^2cos2{\theta}_2}}\]$$.