• Title/Summary/Keyword: Curvature-Weighted

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COMPARISON THEOREMS IN FINSLER GEOMETRY WITH WEIGHTED CURVATURE BOUNDS AND RELATED RESULTS

  • Wu, Bing-Ye
    • Journal of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.603-624
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    • 2015
  • We first extend the notions of weighted curvatures, including the weighted flag curvature and the weighted Ricci curvature, for a Finsler manifold with given volume form. Then we establish some basic comparison theorems for Finsler manifolds with various weighted curvature bounds. As applications, we obtain some McKean type theorems for the first eigenvalue of Finsler manifolds, some results on weighted curvature and fundamental group for Finsler manifolds, as well as an estimation of Gromov simplicial norms for reversible Finsler manifolds.

PERELMAN TYPE ENTROPY FORMULAE AND DIFFERENTIAL HARNACK ESTIMATES FOR WEIGHTED DOUBLY NONLINEAR DIFFUSION EQUATIONS UNDER CURVATURE DIMENSION CONDITION

  • Wang, Yu-Zhao
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1539-1561
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    • 2021
  • We prove Perelman type 𝒲-entropy formulae and differential Harnack estimates for positive solutions to weighed doubly nonlinear diffusion equation on weighted Riemannian manifolds with CD(-K, m) condition for some K ≥ 0 and m ≥ n, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.

SOME INTEGRAL INEQUALITIES FOR THE LAPLACIAN WITH DENSITY ON WEIGHTED MANIFOLDS WITH BOUNDARY

  • Fanqi Zeng
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.2
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    • pp.325-338
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    • 2023
  • In this paper, we derive a Reilly-type inequality for the Laplacian with density on weighted manifolds with boundary. As its applications, we obtain some new Poincaré-type inequalities not only on weighted manifolds, but more interestingly, also on their boundary. Furthermore, some mean-curvature type inequalities on the boundary are also given.

Determination of Curvature Radius of Magnetic Tool Using Weighted Magnetic Flux Density in Magnetic Abrasive Polishing (자속밀도 가중치에 의한 자유곡면 자기연마 공구곡률 선정)

  • Son, Chul-Bae;Ryu, Man-Hee;Kwak, Jae-Seob
    • Journal of the Korean Society of Manufacturing Process Engineers
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    • v.12 no.3
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    • pp.69-75
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    • 2013
  • During the magnetic abrasive polishing of a curved surface, the improvement in surface roughness varies with the maximum value and distribution of magnetic flux density. Thus, in this study, the magnetic flux density on the curved surface was simulated according to curvature radii of magnetic tool. As a result of the simulation, the 14.5mm of the magnetic tool had a higher maximum magnetic flux density and it showed a large weighted magnetic flux density. The weighted magnetic flux density means the highest value for the magnetic flux density in the curvature of the magnetic tool. From the experimental verification, the better improvement in surface roughness was observed on wider area at the 14.5mm radius of the magnetic tool than other radii.

COMPLETE NONCOMPACT SUBMANIFOLDS OF MANIFOLDS WITH NEGATIVE CURVATURE

  • Ya Gao;Yanling Gao;Jing Mao;Zhiqi Xie
    • Journal of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.183-205
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    • 2024
  • In this paper, for an m-dimensional (m ≥ 5) complete non-compact submanifold M immersed in an n-dimensional (n ≥ 6) simply connected Riemannian manifold N with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of M, the norm of its weighted mean curvature vector |Hf| and the weighted real-valued function f, we can obtain: • several one-end theorems for M; • two Liouville theorems for harmonic maps from M to complete Riemannian manifolds with nonpositive sectional curvature.

RULED SURFACES IN E3 WITH DENSITY

  • Altin, Mustafa;Kazan, Ahmet;Karadag, H.Bayram
    • Honam Mathematical Journal
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    • v.41 no.4
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    • pp.683-695
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    • 2019
  • In the present paper, we study curves in 𝔼3 with density $e^{ax^2+by^2}$, where a, b ∈ ℝ not all zero constants and give the parametric expressions of the curves with vanishing weighted curvature. Also, we create ruled surfaces whose base curves are the curve with vanishing weighted curvature and the ruling curves are Smarandache curves of this curve. Then, we give some characterizations about these ruled surfaces by obtaining the mean curvatures, Gaussian curvatures, distribution parameters and striction curves of them.

CURVATURE-WEIGHTED SURFACE SIMPLIFICATION ALGORITHM USING VERTEX-BASED GEOMETRIC FEATURES

  • CHOI, HAN-SOO;GWON, DALHYEON;HAN, HEEJAE;KANG, MYUNGJOO
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.24 no.1
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    • pp.23-37
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    • 2020
  • The quadratic error metric (QEM) algorithm has been frequently used for simplification of triangular surface models that utilize the vertex-pair algorithm. Simplified models obtained using such algorithms present the advantage of smaller storage capacity requirement compared to the original models. However, a number of cases exist where significant features are lost geometrically, and these features can generally be preserved by utilizing the advantages of the curvature-weighted algorithm. Based on the vertex-based geometric features, a method capable of preserving the geometric features better than the previous algorithms is proposed in this work. To validate the effectiveness of the proposed method, a simplification experiment is conducted using several models. The results of the experiment indicate that the geometrically important features are preserved well when a local feature is present and that the error is similar to those of the previous algorithms when no local features are present.

VANISHING PROPERTIES OF p-HARMONIC ℓ-FORMS ON RIEMANNIAN MANIFOLDS

  • Nguyen, Thac Dung;Pham, Trong Tien
    • Journal of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1103-1129
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    • 2018
  • In this paper, we show several vanishing type theorems for p-harmonic ${\ell}$-forms on Riemannian manifolds ($p{\geq}2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of $N^{n+m}$ with flat normal bundle, we prove that any p-harmonic ${\ell}$-form on M is trivial if N has pure curvature tensor and M satisfies some geometric conditions. Then, we obtain a vanishing theorem on Riemannian manifolds with a weighted $Poincar{\acute{e}}$ inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds M and point out that there is no nontrivial p-harmonic ${\ell}$-form on M provided that the Ricci curvature has suitable bound.