• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1143-1158
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• 2006

We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space using an analytic continuation argument. In this paper we show this method can be further generalized to find the volume of a domain with smooth boundary with suitable regularity in dimension 2 and 3. We also discuss that this volume is invariant under the group of hyperbolic isometries and that this regularity condition is sharp.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1359-1380
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• 2018

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.

• The Mathematical Education
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• v.21 no.3
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• pp.7-11
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• 1983

A measurable function f defined on a measurable subset A of the real line R is called pth power summable on A if │f│$^{p}$ is integrable on A and the set of all pth power summable functions on A is denoted by L$^{p}$ (A). For each member f in L$^{p}$ (A), we define ∥f∥$_{p}$ =(equation omitted) For real numbers p and q where (equation omitted) and (equation omitted), we discuss the Holder's inequality ∥fg∥$_1$<∥f∥$_{p}$ ∥g∥$_{q}$ , f$\in$L$^{p}$ (A), g$\in$L$^{q}$ (A) and the Minkowski inequality ∥＋g∥$_{p}$ <∥f∥$_{p}$ ＋∥g∥$_{p}$ , f,g$\in$L$^{p}$ (A). In this paper also discuss that L$_{p}$ (A) becomes a metric space with the metric $\rho$ : L$^{p}$ (A) $\times$L$^{p}$ (A) longrightarrow R where $\rho$(f,g)=∥f-g∥$_{p}$ , f,g$\in$L$^{p}$ (A). Then, in this paper prove the Riesz-Fischer theorem, i.e., the space L$^{p}$ (A) is complete and that the space L$^{p}$ (A) is separable.

• Honam Mathematical Journal
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• v.45 no.2
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• pp.231-247
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• 2023

In this study, we mean that timelike q-helices are curves whose q-frame fields make a constant angle with a non-zero fixed axis. We present the necessary and sufficient conditions for timelike curves via the q-frame to be q-helices in Lorentz-Minkowski 3-space. Then we find some results of the relations between q-helices and Darboux q-helices. Furthermore, we portray Darboux q-helices as special curves whose Darboux vector makes a constant angle with a non-zero fixed axis by choosing the curve as one of the types of q-helices, and also the general case.

• Journal of the Korean earth science society
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• v.40 no.4
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• pp.353-381
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• 2019

The general world model for homogeneous and isotropic universe has been proposed. For this purpose, we introduce a global and fiducial system of reference (world reference frame) constructed on a (4+1)-dimensional space-time, and assume that the universe is spatially a 3-dimensional hypersurface embedded in the 4-dimensional space. The simultaneity for the entire universe has been specified by the global time coordinate. We define the line element as the separation between two neighboring events on the expanding universe that are distinct in space and time, as viewed in the world reference frame. The information that determines the kinematics of the geometry of the universe such as size and expansion rate has been included in the new metric. The Einstein's field equations with the new metric imply that closed, flat, and open universes are filled with positive, zero, and negative energy, respectively. The curvature of the universe is determined by the sign of mean energy density. We have demonstrated that the flat universe is empty and stationary, equivalent to the Minkowski space-time, and that the universe with positive energy density is always spatially closed and finite. In the closed universe, the proper time of a comoving observer does not elapse uniformly as judged in the world reference frame, in which both cosmic expansion and time-varying light speeds cannot exceed the limiting speed of the special relativity. We have also reconstructed cosmic evolution histories of the closed world models that are consistent with recent astronomical observations, and derived useful formulas such as energy-momentum relation of particles, redshift, total energy in the universe, cosmic distance and time scales, and so forth. The notable feature of the spatially closed universe is that the universe started from a non-singular point in the sense that physical quantities have finite values at the initial time as judged in the world reference frame. It has also been shown that the inflation with positive acceleration at the earliest epoch is improbable.