• Title/Summary/Keyword: Cotton tensor

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CLASSIFICATION OF (k, 𝜇)-ALMOST CO-KÄHLER MANIFOLDS WITH VANISHING BACH TENSOR AND DIVERGENCE FREE COTTON TENSOR

  • De, Uday Chand;Sardar, Arpan
    • Communications of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.1245-1254
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    • 2020
  • The object of the present paper is to characterize Bach flat (k, 𝜇)-almost co-Kähler manifolds. It is proved that a Bach flat (k, 𝜇)-almost co-Kähler manifold is K-almost co-Kähler manifold under certain restriction on 𝜇 and k. We also characterize (k, 𝜇)-almost co-Kähler manifolds with divergence free Cotton tensor.

BACH ALMOST SOLITONS IN PARASASAKIAN GEOMETRY

  • Uday Chand De;Gopal Ghosh
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.763-774
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    • 2023
  • If a paraSasakian manifold of dimension (2n + 1) represents Bach almost solitons, then the Bach tensor is a scalar multiple of the metric tensor and the manifold is of constant scalar curvature. Additionally it is shown that the Ricci operator of the metric g has a constant norm. Next, we characterize 3-dimensional paraSasakian manifolds admitting Bach almost solitons and it is proven that if a 3-dimensional paraSasakian manifold admits Bach almost solitons, then the manifold is of constant scalar curvature. Moreover, in dimension 3 the Bach almost solitons are steady if r = -6; shrinking if r > -6; expanding if r < -6.

A Study on the Characteristics of Plant Fiber Materials for Diffusion Tensor Imaging Phantom (확산텐서영상 팬텀 제작을 위한 식물섬유 재료의 특성에 관한 연구)

  • Lee, Jung-Hoon
    • Journal of radiological science and technology
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    • v.43 no.6
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    • pp.475-480
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    • 2020
  • The purpose of this study was to reconstruct diffusion tensor tractography (DTT) using stem of garlic and asparagus for in vitro phantom of diffusion tensor imaging (DTI), and to compare and evaluate the fractional anisotropy (FA) value and the apparent diffusion coefficient (ADC) value to determine whether it can be used as materials for in vitro phantoms. Among various plant fibers such as stem of garlic, palmae, cotton, asparagus, etc., stem of garlic and asparagus, which are considered to be the most suitable for making phantoms, and whose shape is considered to be the most suitable for making phantoms, were selected and tests were conducted. Holes were made in a plastic bucket at an angle of 0°, 30°, 60°, 90°, and 120°, then tubes were inserted. In the tube, asparagus and stem of garlic were inserted as far in as possible, and the inserted tube was inserted into the center of the heat bathed gelatin to harden. We were able to reproduce DTT images in asparagus and stem of garlic. Fiber tissues of asparagus and stem of garlic did not show complete connectivity, but the reconstructed images of DTT showed good connectivity. The FA values of asparagus in the tubes were 0.198 at 0° (straight), 0.207 at 30°, 0.187 at 60°, 0.231 at 90°, and 0.204 at 120°. In addition, the FA values of stem of garlic in the tubes were 0.235 at 0°, 0.236 at 30°, 0.216 at 60°, 0.218 at 90°, and 0.257 at 120°. The ADC values of asparagus in the tubes were 1.545 at 0°, 1.677 at 30°, 1.629 at 60°, 1.535 at 90°, and 1.725 at 120°. In addition, the ADC values of stem of garlic in the tubes were 1.252 at 0°, 1.396 at 30°, 1.698 at 60°, 1.756 at 90°, and 1.466 at 120°. For the best expressed DTT reconstruction image, it showed the longest connectivity in the straight line as we hypothesized. In addition, when comparing the FA values and ADC values of fiber tissues of stem of garlic and asparagus, FA value was generally higher in stem of garlic and ADC value was slightly higher in asparagus.