• Honam Mathematical Journal
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• v.46 no.2
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• pp.237-248
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• 2024

In this manuscript, the hypersurfaces of non-flat Riemannian 4-space forms are considered. A hypersurface of a 4-dimensional Riemannian space form defined by an isometric immersion 𝐱 : M³ → 𝕄⁴(c) is said to be biconservative if it satisfies the equation (∆²𝐱 )^⊤ = 0, where ∆ is the Laplace operator on M³ and ⊤ stands for the tangent component of vectors. We study an extended version of biconservativity condition on the hypersurfaces of the Riemannian standard 4-space forms. The C-biconservativity condition is obtained by substituting the Cheng-Yau operator C instead of ∆. We prove that C-biconservative hypersurfaces of Riemannian 4-space forms (with some additional conditions) have constant scalar curvature.

• Honam Mathematical Journal
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• v.46 no.1
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• pp.101-113
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• 2024

In this paper, we define the generalized tricomplex numbers and give some algebraic properties of them. By using the matrix representation of generalized tricomplex numbers, we determine a motion on the hypersurface M in eight dimensional generalized linear space ℝ⁸_αβγ and show that this is a homothetic motion. Also, for some special cases of the real numbers α, β and γ, we give some examples of homothetic motions in ℝ⁸ and ℝ⁸₄ and obtain some rotational matrices in these spaces, too.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.279-291
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• 2024

In this paper, we investigate a few strategies to construct Ulrich bundles of small ranks over smooth fourfolds in ℙ⁵, with a focus on the case of special quartic fourfolds. First, we give a necessary condition for Ulrich bundles over a very general quartic fourfold in terms of the rank and the Chern classes. Second, we give an equivalent condition for Pfaffian fourfolds in every degree in terms of arithmetically Gorenstein surfaces therein. Finally, we design a computer-based experiment to look for Ulrich bundles of small rank over special quartic fourfolds via deformation theory. This experiment gives a construction of numerically Ulrich sheaf of rank 4 over a random quartic fourfold containing a del Pezzo surface of degree 5.

• Honam Mathematical Journal
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• v.4 no.1
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• pp.75-84
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• 1982

Let M be a complete and orientable hypersurface of an odd-dimensional sphere $S^{2n+1}$ with quasi-integrable $(f,\;g,\;u,\;{\nu},\;{\lambda})$ -structure. The purpose of the present paper is to prove the following two theorems. (I) If the scalar curvature of M is constant and the function $\lambda$ is not locally constant, then M is a great sphere $S^{2n}$(1) or a product of two spheres with the same dimension $S^{n}(1/\sqrt{2}){\times}S^{n}(1/\sqrt{2})$. (II) Suppose that the sectional curvature of the section $\gamma(u,\;{\nu})$ spanned by u and $\nu$ is constant on M and M is compact. If the second fundamental tensor H of M is positive semi-definite and satisfies trace $$^{t}HH{\leq_-}{2n}$$, then M is a great sphere $S^{2n}$ (1) or a product of two spheres $S^{n}{\times}S^{n}$ or $S^{p}{\times}S^{2n-p}$, p being odd.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.585-604
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• 2012

In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface ${\Sigma}_0$ under the Gauss curvature flow. Let $X:S^n{\times}[0,\;{\infty}){\rightarrow}\mathbb{R}^{n+1}$ be the embeddings of the sphere in $\mathbb{R}^{n+1}$ such that $\Sigma(t)=X(S^n,t)$ is the surface at time t and ${\Sigma}(0)={\Sigma}_0$. As a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate on the interface separating the nonconvex part from the strictly convex side, since one of the curvature will be zero on the interface. By expressing the strictly convex part of the surface near the interface as a graph of a function $z=f(r,t)$ and the non-convex part of the surface near the interface as a graph of a function $z={\varphi}(r)$, we show that if at time $t=0$, $g=\frac{1}{n}f^{n-1}_{r}$ vanishes linearly at the interface, the $g(r,t)$ will become smooth up to the interface for long time before focusing.

• Proceedings of the KOSOMBE Conference
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• v.1997 no.11
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• pp.541-545
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• 1997

This paper presents a modified geometric active contour model or edge detection and segmentation of computed tomography(CT) scan images. The method is based on the level setup approach developed by Osher and Sethian and the modeling of propagation fronts with curvature dependent speeds by Malladi. Based on above algorithms, the geometric active contour is obtained through a particular level set of hypersurface lowing along its gradient force and curvature force. This technique retains the attractive feature which is topological and geometric flexibility of the contour in recovering objects with complex shapes and unknown topologies. But there are limitations in this algorithm which are being not able to separate the object with weak difference from neighbor object. So we use speed limitation filter to overcome those problems. We apply a 2D model to various synthetic cases and the three cases of real CT scan images in order to segment objects with complicated shapes and topologies. From the results, the presented model confirms that it attracts very naturally and efficiently to the desired feature of CT scan images.

• Proceedings of the Korean Vacuum Society Conference
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• 2000.02a
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• pp.25-25
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• 2000

In this talk we discuss the dynamics of hydrogen on the Si(100)-2xl surface. At room temperature the sticking coefficient for molecular hydrogen on this surface is less than 10sup-12. However, hydrogen molecules desorbing from the surface do not have an excess of energy, suggesting at best a small barrier on the exit channel. These observations have led to speculation about the validity of detailed balance in this system. Here we show that this discrepancy can be explained by considering both the surface-molecule co-ordinate and that associated with the Si-Si dimer bond tiltangle. By preparing the surface dimers with a specific tiltangle we demonstrate that the barrier to adsorption is a function of this angle and that the sticking coefficient dramatically increase for certain angles. The adsorption-desopption dynamics can then be described in terms of a common potential energy hypersurface involving both of these co-ordinates. The implications of these observations are also discussed. The dynamics of adsorbed hydrogen atoms on the Si(100) surface is also described. Paired dangling bonds produced following recombinative hydrogen desorption are mobile at elevated temperatures. Pairs of dangling bonds are observed to dissociate, diffuse, and ultimately recombine. At sufficiently elevated temperatures dangling bond exchange reactions are observed. These data are analyzed in terms of an attractive zone and an effective binding interaction between dangling bonds. Insights that this provides into the nature of surface defects and the localized chemistry that occurs on this surface, are also discussed.

• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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• pp.35-43
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• 2009

We say that an isometric immersed hypersurface x : $M^n\;{\rightarrow}\;{\mathbb{R}}^{n+1}$ is of $L_k$-finite type ($L_k$-f.t.) if $x\;=\;{\sum}^p_{i=0}x_i$ for some positive integer p < $\infty$, $x_i$ : $M{\rightarrow}{\mathbb{R}}^{n+1}$ is smooth and $L_kx_i={\lambda}_ix_i$, ${\lambda}_i\;{\in}\;{\mathbb{R}}$, $0{\leq}i{\leq}p$, $L_kf=trP_k\;{\circ}\;{\nabla}^2f$ for $f\;{\in}\'C^{\infty}(M)$, where $P_k$ is the kth Newton transformation, ${\nabla}^2f$ is the Hessian of f, $L_kx\;=\;(L_kx^1,\;{\ldots},\;L_kx^{n+1})$, $x=(x^1,\;{\ldots},\;x^{n+1})$. In this article we study the following(hyper)surfaces in ${\mathbb{R}}^{n+1}$ from the view point of $L_1$-finiteness type: totally umbilic ones, generalized cylinders $S^m(r){\times}{\mathbb{R}}^{n-m}$, ruled surfaces in ${\mathbb{R}}^{n+1}$ and some revolution surfaces in ${\mathbb{R}}^3$.

• Bulletin of the Korean Chemical Society
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• v.21 no.10
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• pp.953-954
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• 2000

The equilibrium geometrical structures of silver atom clusters at their electronic ground states have been theo-retically determined by using the nonrelativistic semiempirical INDO/1 method. The clusters investigated are Agn, Agn+, and Agn- (n = 7 , 8, 9). In order to find the most stable structure, i.e., the global minimum in energy hypersurface, geometry optimization and energy calculation processes have been repeatedly performed for all the possible graphical models by changing the bond parameters (resonance integral values). The heptamers are pentagonal bipyramidal-Ag7(D5h), Ag7+ (D5h), Ag7- (D5h); the octamers are pentagonal bipyramidal with one atom capped-Ag8(D2d), Ag8+ (Cs), Ag8- (D2d); the nonamers are pentagonal bipyramidal with two atoms capped -Ag9(C2v), Ag9+ (C2v), Ag9- (C2v). Our structures are in good agreement with those by ab initio calculations ex-cept for the anionic Ag9- cluster. And it is noted that the INDO/1 method can accurately predict the Ag cluster geometries when a proper set of bond parameters is used.

• The Pure and Applied Mathematics
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• v.30 no.2
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• pp.109-129
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• 2023

In this paper, we introduce a second order linear differential operator ^T□: C^∞ (M) → C^∞ (M) as a natural generalization of Cheng-Yau operator, [8], where T is a (1, 1)-tensor on Riemannian manifold (M, h), and then we show on compact Riemannian manifolds, divT = divT^t, and if divT = 0, and f be a smooth function on M, the condition ^T□ f = 0 implies that f is constant. Hereafter, we introduce T-energy functionals and by deriving variations of these functionals, we define T-harmonic maps between Riemannian manifolds, which is a generalization of L_k-harmonic maps introduced in [3]. Also we have studied fT-harmonic maps for conformal immersions and as application of it, we consider fL_k-harmonic hypersurfaces in space forms, and after that we classify complete fL₁-harmonic surfaces, some fL_k-harmonic isoparametric hypersurfaces, fL_k-harmonic weakly convex hypersurfaces, and we show that there exists no compact fL_k-harmonic hypersurface either in the Euclidean space or in the hyperbolic space or in the Euclidean hemisphere. As well, some properties and examples of these definitions are given.