• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.537-542
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• 2013

• Investigative Magnetic Resonance Imaging
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• v.20 no.2
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• pp.95-104
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• 2016

Purpose: To evaluate the knee joint after double-bundle anterior cruciate ligament (ACL) reconstruction with three-dimensional (3D) isotropic magnetic resonance (MR) image, and to directly compare the ACL graft findings on 3D MR with the clinical results. Materials and Methods: From January 2009 to December 2014, we retrospectively reviewed MRIs of 39 patients who had reconstructed ACL with double bundle technique. The subjects were examined using 3D isotropic proton-density sequence and routine two-dimensional (2D) sequence on 3.0T scanner. The MR images were qualitatively evaluated for the intraarticular curvature, graft tear, bony impingement, intraosseous tunnel cyst, and synovitis of anteromedial and posterolateral bundles (AMB, PLB). In addition anterior tibial translation, PCL angle, PCL ratio were quantitatively measured. KT arthrometric values were reviewed for anterior tibial translation as positive or negative. The second look arthroscopy results including tear and laxity were reviewed. Results: Significant correlations were found between an AMB tear on 3D-isotropic proton density MR images and arthroscopic proven AMB tear or laxity (P < 0.05). Also, a significant correlation was observed between increased PCL ratio on 3D isotropic MRI and the arthroscopic findings such as tear, laxities of grafts (P < 0.05). KT arthrometric results were found to be significantly correlated with AMB tears (P < 0.05) and tibial tunnel cysts (P < 0.05). Conclusion: An AMB tear on 3D-isotropic MRI was correlated with arthroscopic results qualitatively and quantitatively. 3D isotropic MRI findings can aid the evaluation of ACL grafts after double bundle reconstruction.

• 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.