• 대한수학회보
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• 제42권2호
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• pp.337-358
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• 2005

Let M be a real hypersurface with almost contact metric structure $(\phi,\;\xi,\;\eta,\;g)$ in a nonflat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ commutes with both the structure tensor $\phi$ and the Ricc tensor S, then M is a Hopf hypersurface in $M_n(c)$ provided that the mean curvature of M is constant or $g(S\xi,\;\xi)$ is constant.

• 대한수학회보
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• 제56권5호
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• pp.1219-1233
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• 2019

Let $E{\rightarrow}M$ be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection $D^E$. R. Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when $D^E$ is flat. We study also the Einstein property on E proving, among other results, that if $k{\geq}2$ and the base manifold is Einstein with positive constant scalar curvature, then there is a 1-parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat.

• 대한수학회보
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• 제58권6호
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• pp.1315-1325
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• 2021

In this paper, we firstly prove some general inequalities for the Neumann eigenvalues for domains contained in a Euclidean n-space ℝⁿ. Using the general inequalities, we can derive some new Neumann eigenvalues estimates which include an upper bound for the (k + 1)^th eigenvalue and a new estimate for the gap of the consecutive eigenvalues. Moreover, we give sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator with Navier boundary condition on compact Riemannian manifolds with boundary and positive Ricci curvature.

• 대한수학회보
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• 제61권2호
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• pp.433-450
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• 2024

Using a Reilly type integral formula due to Li and Xia [23], we prove several geometric inequalities for affine connections on Riemannian manifolds. We obtain some general De Lellis-Topping type inequalities associated with affine connections. These not only permit to derive quickly many well-known De Lellis-Topping type inequalities, but also supply a new De Lellis-Topping type inequality when the 1-Bakry-Emery Ricci curvature is bounded from below by a negative function. On the other hand, we also achieve some Lichnerowicz type estimate for the first (nonzero) eigenvalue of the affine Laplacian with the Robin boundary condition on Riemannian manifolds.

• Restorative Dentistry and Endodontics
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• 제49권2호
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• pp.12.1-12.10
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• 2024

Objectives: This study evaluated the impact of different methods of irrigant agitation on smear layer removal in the apical third of curved mesial canals of 3 dimensionally (D) printed mandibular molars. Materials and Methods: Sixty 3D-printed mandibular second molars were used, presenting a 70° curvature and a Vertucci type II configuration in the mesial root. A round cavity was cut 2 mm from the apex using a trephine of 2 mm in diameter, 60 bovine dentin disks were made, and a smear layer was formed. The dentin disks had the adaptation checked in the apical third of the teeth with wax. The dentin disks were evaluated in environmental scanning electron microscope before and after the following irrigant agitation methods: G1(PIK Ultrasonic Tip), G2 (Passive Ultrasonic Irrigation with Irrisonic- PUI), G3 (Easy Clean), G4 (HBW Ultrasonic Tip), G5 (Ultramint X Ultrasonic tip), and G6 (conventional irrigation-CI) (n = 10). All groups were irrigated with 2.5% sodium hypochlorite and 17% ethylenediaminetetraacetic acid. Results: All dentin disks were 100% covered by the smear layer before treatment, and all groups significantly reduced the percentage of the smear layer after treatment. After the irrigation protocols, the Ultra-X group showed the lowest coverage percentage, statistically differing from the conventional, PIK, and HBW groups (p < 0.05). There was no significant difference among Ultramint X, PUI-Irrisonic, and Easy Clean (p > 0.05). None of the agitation methods could remove the smear layer altogether. Conclusions: Ultramint X resulted in the most significant number of completely clean specimens.