• 호남수학학술지
• /
• 제36권4호
• /
• pp.787-794
• /
• 2014

We prove three convex hull theorems on triangles and circles. Given a triangle ${\triangle}$ and a point p, let ${\triangle}^{\prime}$ be the triangle each of whose vertices is the intersection of the orthogonal line from p to an extended edge of ${\triangle}$. Let ${\triangle}^{{\prime}{\prime}}$ be the triangle whose vertices are the centers of three circles, each passing through p and two other vertices of ${\triangle}$. The first theorem characterizes when $p{\in}{\triangle}$ via a distance duality. The triangle algorithm in [1] utilizes a general version of this theorem to solve the convex hull membership problem in any dimension. The second theorem proves $p{\in}{\triangle}$ if and only if $p{\in}{\triangle}^{\prime}$. These are used to prove the third: Suppose p be does not lie on any extended edge of ${\triangle}$. Then $p{\in}{\triangle}$ if and only if $p{\in}{\triangle}^{{\prime{\prime}}$.

• 대한수학회보
• /
• 제60권6호
• /
• 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.

• Kyungpook Mathematical Journal
• /
• 제63권1호
• /
• pp.123-129
• /
• 2023

For an ordered set W = {w₁, w₂, . . . , w_k} of vertices and a vertex v in a connected graph G, the k-vector r(v|W) = (d(v, w₁), d(v, w₂), . . . , d(v, w_k)) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, G ⊙ H of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(G ⊙ K₁), characterize all graphs G with dim(G ⊙ K₁) = dim(G), and prove that dim(G ⊙ K₁) = n - 1 if and only if G is the complete graph K_n or the star graph K_1,n-1.

• 대한수학회보
• /
• 제44권3호
• /
• pp.507-516
• /
• 2007

Let M be a complete Riemannian manifold and L be a $Schr\"{o}dinger$ operator on M. We prove that if M has finitely many L-nonparabolic ends, then the space of bounded L-harmonic functions on M has the same dimension as the sum of dimensions of the spaces of bounded L-harmonic functions on the L-nonparabolic end, which vanish at the boundary of the end.

• 대한수학회보
• /
• 제50권1호
• /
• pp.143-159
• /
• 2013

This paper is concerned with the long time behavior for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension $n{\leq}3$. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that the equations possesses a global attractor in $H^k({\Omega})^4$ ($k{\geq}0$) space.

• 대한수학회논문집
• /
• 제20권3호
• /
• pp.521-529
• /
• 2005

We prove that the dimension of the space of positive (bounded, respectively) solutions for the Schrodinger operator whose potential q is nonnegative on a graph with q-regular ends is equal to the number of ends (q-nonparabolic ends, respectively).

• Communications for Statistical Applications and Methods
• /
• 제18권3호
• /
• pp.357-365
• /
• 2011

This paper investigates a penalized likelihood method for estimating the parameter of normal mixtures in multivariate settings with full covariance matrices. The proposed model estimates the number of components through the addition of a penalty term to the usual likelihood function and the construction of a penalized likelihood function. We prove the consistency of the estimator and present the simulation results on the multi-dimensional nor-mal mixtures up to the 8-dimension.

• 대한수학회지
• /
• 제39권1호
• /
• pp.91-102
• /
• 2002

Let M and N be CR manifolds with nondegenerate Levi forms of hypersurface type of dimension 2m ＋ 1 and 2n ＋ 1, respectively, where 1 $\leq$ m $\leq$ n. Let f : M longrightarrow N be a CR mapping. Under a generic assumption we construct a complete system of finite order for the infinitesimal deformations of f. In particular, we prove the space of infinitesimal deformations of f forms a finite dimensional Lie algebra.

• 대한수학회논문집
• /
• 제35권1호
• /
• pp.43-61
• /
• 2020

We extend the work of A. Beauville on rank 2 vector bundles on a smooth curve in several directions. We give families of examples with large dimension, add new existence and non-existence results and prove the existence of indecomposable limits with arbitrary rank. To construct the large dimensional families we use the examples of limits of rank 2 trivial bundles on ℙ² and ℙ³ due to C. Banica. We also consider a more flexible notion: limits of trivial bundles on nearby curves.

• 대한수학회논문집
• /
• 제20권4호
• /
• pp.731-749
• /
• 2005

In this paper, we study the two dimensional g-Navier­Stokes equations on some unbounded domain ${\Omega}\;{\subset}\;R^2$. We prove the existence of the global attractor for the two dimensional g-Navier­Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.