• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1539-1561
• /
• 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.

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

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.3
• /
• pp.541-549
• /
• 2010

In this paper we derive an integral formula on an (n + 1)-dimensional, compact, minimal contact CR-submanifold M of (n - 1) contact CR-dimension immersed in a unit (2m+1)-sphere $S^{2m+1}$. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

• Communications of the Korean Mathematical Society
• /
• v.24 no.1
• /
• pp.111-125
• /
• 2009

The purpose of this paper is to study n-dimensional QR-submanifolds of (p-1) QR-dimension immersed in a quaternionic projective space $QP^{(n+p)/4}$ of constant Q-sectional curvature 4 and especially to determine such submanifolds under the additional condition concerning with shape operator.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.3
• /
• pp.585-600
• /
• 2011

In this paper we derive an integral formula on an (n + 3)-dimensional, compact, minimal contact three CR-submanifold M of (p-1) contact three CR-dimension immersed in a unit (4m+3)-sphere $S^{4m+3}$. Using this integral formula, we give a sufficient condition concerning the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1641-1650
• /
• 2023

It is our goal in this study to present the structure of isotropic mean Berwald Finsler warped product metrics. We bring out the rich class of warped product Finsler metrics behaviour under this condition. We show that every Finsler warped product metric of dimension n ≥ 2 is of isotropic mean Berwald curvature if and only if it is a weakly Berwald metric. Also, we prove that every locally dually flat Finsler warped product metric is weakly Berwaldian. Finally, we prove that every Finsler warped product metric is of isotropic Berwald curvature if and only if it is a Berwald metric.

• Communications of the Korean Mathematical Society
• /
• v.29 no.1
• /
• pp.131-140
• /
• 2014

In this paper we investigate (n+1)($n{\geq}3$)-dimensional contact CR-submanifolds M of (n-1) contact CR-dimension in a complete simply connected Sasakian space form of constant ${\phi}$-holomorphic sectional curvature $c{\neq}-3$ which satisfy the condition h(FX, Y)+h(X, FY) = 0 for any vector fields X, Y tangent to M, where h and F denote the second fundamental form and a skew-symmetric endomorphism (defined by (2.3)) acting on tangent space of M, respectively.

• Journal of Auto-vehicle Safety Association
• /
• v.6 no.2
• /
• pp.23-28
• /
• 2014

Commercial vehicle ESC assessment for curvature road was conducted. The previous study of ESC activation condition for losing controllability utilizing the test protocols of double lane change and sine with dwell method was conducted without considering the geometric complexity of roadway design. Since critical rollover accidents were frequently observed in the exit ramp zone, variety of curve, slope and bank have been added for analysis conditions. Detailed feature of the ramp including location, dimension and design characteristics have been analyzed from the typical trumpet type ramp design. Analyzing accident data from 2008, two specific ramps have been selected due to their complexity in design and severity in steering operation.

• The Journal of the Korea Contents Association
• /
• v.11 no.8
• /
• pp.123-133
• /
• 2011

In this paper, we propose a new method for toon shading using 3D texture which renders 3d objects in a cartoon style. The conventional toon shading using 1D texture displays shading tone by computing the relative position and orientation between a light vector and surface normal. The 1D texture alone has limits to express the various tone change according to any viewing condition. Therefore Barla et. al. replaces a 1D texture with a 2D texture whose the second dimension corresponds to the view-dependent effects such as level-of-abstraction, depthof-field. The proposed scheme extends 2D texture to 3D texture by adding one dimension with the geometric information of 3D objects such as curvature, saliency, and coordinates. This approach supports two kinds of extensions for cartoon style diversification. First, we support "shape exaggeration effect" to emphasize silhouette or highlight according to the geometric information of 3D objects. Second, we further incorporate "cartoon specific effect", which is examples of screen tone and out focusing frequently appeared in cartoons. We demonstrate the effectiveness of our approach through examples that include a number of 3d objects rendered in various cartoon style.