• Korean Journal of Mathematics
• /
• v.26 no.4
• /
• pp.545-559
• /
• 2018

In this paper, we study the position vector of a developable q-slant ruled surface in the Euclidean 3-space $E^3$ in means of the Frenet frame of a q-slant ruled surface. First, we determinate the natural representations for the striction curve and ruling of a q-slant ruled surface. Then we obtain general parameterization of a developable q-slant ruled surface with respect to the conical curvature of the surface. Finally, we introduce some examples for the obtained result.

• Journal of the Korean Mathematical Society
• /
• v.60 no.5
• /
• pp.1023-1041
• /
• 2023

In this paper, motivated by the work of Q. S. Zhang in [25], we derive optimal Li-Yau gradient bounds for positive solutions of the f-heat equation on closed manifolds with Bakry-Émery Ricci curvature bounded below.

• Journal of the Korean Mathematical Society
• /
• v.61 no.2
• /
• pp.255-277
• /
• 2024

In this paper, we introduce the notion of stress-energy tensor Q of the traceless Ricci tensor for Riemannian manifolds (Mⁿ, g), and investigate harmonicity of Riemannian curvature tensor and Weyl curvature tensor when (M, g) satisfies some geometric structure such as critical point equation or vacuum static equation for smooth functions.

• Kyungpook Mathematical Journal
• /
• v.47 no.3
• /
• pp.381-392
• /
• 2007

In 1906, da Rios, a student of Leivi-Civita, wrote a master's thesis modeling the motion of a vortex in a viscous fluid by the motion of a curve propagating in $R^3$, in the direction of its binormal with a speed equal to its curvature. Much later, in 1971 Hasimoto showed the equivalence of this system with the non-linear Schr$\ddot{o}$dinger equation (NLS) $$q_t=i(q_{ss}+\frac{1}{2}{\mid}q{\mid}^2q$$. In this paper, we use the same idea as Terng used in her lecture notes but different technique to extend the above relation to the case of $R^3$, and obtained an analogous equation that $$q_t=i[q_{ss}+(\frac{1}{2}{\mid}q{\mid}^2+1)q]$$.

• Honam Mathematical Journal
• /
• v.31 no.3
• /
• pp.381-398
• /
• 2009

Let $G\;=\;O(k){\times}O(k){\times}O(q)$ and let $M^n$ be a closed G-invariant minimal hypersurface with constant scalar curvature in $S^{n+1}$. Then we obtain a theorem: If $M^n$ has 2 distinct principal curvatures at some point p, then the square norm of the second fundamental form of $M^n$, S = n.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.327-332
• /
• 2002

In this paper we prove that if M is a J-invariant sub-manifold of codimension 2 in a complex projective space $P_{n+1}(C)$, and the second fundamental tensor is cyclic-parallel or M has harmonic curvature, then M is locally complex quadric Q$_n$(C) or P$_n$(C).

• Journal of Korean Ophthalmic Optics Society
• /
• v.17 no.2
• /
• pp.223-232
• /
• 2012

Purpose: Validating a new research method to determine posterior corneal curvature and asphericity(Q) in vivo, based on measurements of anterior corneal topography and corneal thickness. Methods: Anterior corneal topographic data, derived from the Medmont E300 corneal topographer, and total corneal thickness data measured along the horizontal corneal meridian using the Holden-Payor optical pachometer, were used to calculate the anterior and posterior corneal apical radii of curvature and Q. To calculate accurate total corneal thickness the local radius of anterior corneal curvature, and an exact solution for the relationship between real and apparent thickness were taken into consideration. This method differs from previous approach. An elliptical curve for anterior and posterior cornea were calculated by using best fit algorism of the anterior corneal topographic data and derived coordinates of the posterior cornea respectively. For validation of the calculations of the posterior corneal topography, ten polymethyl methacrylate (PMMA) lenses and right eyes of five adult subjects were examined. Results: The mean absolute accuracy (${\pm}$standard deviation(SD)) of calculated posterior apical radius and Q of ten PMMA lenses was $0.053{\pm}0.044mm$ (95% confidence interval (CI) -0.033 to 0.139), and $0.10{\pm}0.10$ (95% CI -0.10 to 0.31) respectively. The mean absolute repeatability coefficient (${\pm}SD$) of the calculated posterior apical radius and Q of five human eyes was $0.07{\pm}0.06mm$ (95% CI -0.05 to 0.19) and $0.09{\pm}0.07$ (95% CI -0.05 to 0.23), respectively. Conclusions: The result shows that acceptable accuracy in calculations of posterior apical radius and Q was achieved. This new method shows promise for application to the living human cornea.

• Publications of The Korean Astronomical Society
• /
• v.30 no.2
• /
• pp.309-313
• /
• 2015

If the present universe is slightly open then pre-inflation curvature would appear as a cosmic dark-flow component of the CMB dipole moment. We summarize current cosmological constraints on this cosmic dark flow and analyze the possible constraints on parameters characterizing the pre-inflating universe in an inflation model with a present-day very slightly open ${\Lambda}CDM$ cosmology. We employ an analytic model to show that for a broad class of inflation-generating effective potentials, the simple requirement that the observed dipole moment represents the pre-inflation curvature as it enters the horizon allows one to set upper and lower limits on the magnitude and wavelength scale of pre-inflation fluctuations in the inflaton field and the curvature parameter of the pre-inflation universe, as a function of the fraction of the total initial energy density in the inflaton field. We estimate that if the current CMB dipole is a universal dark flow (or if it is near the upper limit set by the Planck Collaboration) then the present constraints on ${\Lambda}CDM$ cosmological parameters imply rather small curvature ${\Omega}_k{\sim}0.1$ for the pre-inflating universe for a broad range of the fraction of the total energy in the inflaton field at the onset of inflation. Such small pre-inflation curvature might be indicative of open-inflation models in which there are two epochs of inflation.

• Journal of the Korean Institute of Intelligent Systems
• /
• v.20 no.5
• /
• pp.624-630
• /
• 2010

As electrocardiogram(ECG) signals are generally sampled with a frequency of over 200Hz, a method to compress diagnostic information without losing data is required to store and transmit them efficiently. In this paper, an ECG signal compression method, which uses feature points based on curvature, is proposed. The feature points of P, Q, R, S, T waves, which are critical components of the ECG signal, have large curvature values compared to other vertexes. Thus, these vertexes are extracted with the proposed method, which uses local extremum of curvatures. Furthermore, in order to minimize reconstruction errors of the ECG signal, extra vertexes are added according to the iterative vertex selection method. Through the experimental results on the ECG signals from MIT-BIH Arrhythmia database, it is concluded that the vertexes selected by the proposed method preserve all feature points of the ECG signals. In addition, they are more efficient than the AZTEC(Amplitude Zone Time Epoch Coding) method.

• The Pure and Applied Mathematics
• /
• v.28 no.2
• /
• pp.143-153
• /
• 2021

In this paper, we study some curvature properties of Sasakian 3-manifolds associated to Ƶ-tensor. It is proved that if a Sasakian 3-manifold (M, g) satisfies one of the conditions (1) the Ƶ-tensor is of Codazzi type, (2) M is Ƶ-semisymmetric, (3) M satisfies Q(Ƶ, R) = 0, (4) M is projectively Ƶ-semisymmetric, (5) M is Ƶ-recurrent, then (M, g) is of constant curvature 1. Several consequences are drawn from these results.