• Journal of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.1083-1100
• /
• 2011

We show that some class of spacelike maximal surfaces and timelike minimal surfaces match smoothly across the singular curve of the surfaces. Singular Bj$\"{o}$rling representation formulae for generalized spacelike maximal surfaces and for generalized timelike minimal surfaces play important roles in the explanation of this phenomenon.

• Journal for History of Mathematics
• /
• v.21 no.1
• /
• pp.45-78
• /
• 2008

Theory of minimal surfaces has always been in the center of differential geometry. The most difficult part in minimal surfaces is how to find meaningful examples. In this paper we survey the history of search for minimal surfaces. We also introduce examples of recently emerging maximal surfaces in the Lorentz-Minkowski space and compare the processes in the search for the minimal and the maximal surfaces.

• East Asian mathematical journal
• /
• v.35 no.1
• /
• pp.109-115
• /
• 2019

The purpose of this paper is to investigate various kinds of degeneracy of maximal surfaces in ${\mathbb{L}}^n$ in view of the generalized Gauss map.

• East Asian mathematical journal
• /
• v.37 no.1
• /
• pp.131-140
• /
• 2021

The purpose of this paper is to investigate various kinds of degeneracy of maximal surfaces in ��4 in view of the generalized Gauss map.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.4
• /
• pp.1003-1019
• /
• 2021

In this paper, we introduce a class of maximal functions along twisted surfaces in ℝⁿ×ℝ^m of the form {(𝜙(|v|)u, 𝜑(|u|)v) : (u, v) ∈ ℝⁿ×ℝ^m}. We prove L^p bounds when the kernels lie in the space L^q (𝕊^n-1×𝕊^m-1). As a consequence, we establish the L^p boundedness for such class of operators provided that the kernels are in L log L(𝕊^n-1×𝕊^m-1) or in the Block spaces B^0,0_q (𝕊^n-1×𝕊^m-1) (q > 1).

• Journal for History of Mathematics
• /
• v.23 no.4
• /
• pp.31-46
• /
• 2010

Theory of minimal surfaces has always been in the center of differential geometry. The most difficult part in minimal surfaces is how to find meaningful examples. In this paper we survey the history of search for minimal surfaces. We also introduce examples of recently emerging maximal surfaces in the Lorentz-Minkowski space and compare the processes in the search for the minimal and the maximal surfaces.

• Journal of the Korean Mathematical Society
• /
• v.58 no.1
• /
• pp.69-90
• /
• 2021

In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ △γ(ℝ+) and Ω ∈ ����β(Sn-1) for some γ > 1 and β > 1. Here Ω ∈ ����β(Sn-1) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

• ETRI Journal
• /
• v.21 no.3
• /
• pp.41-48
• /
• 1999

We have calculated differential reflection coefficient for isolated well structure of micro-scale, etched on dielectric surface. The differential reflection coefficient is computed using Green's second integral theorem. The purpose of our computation is to find a class of well profiles which give maximal diffusive scattering. To have such a maximal effect, we have concluded that the waist radius of Gaussian beam and its wavelength should be comparable to the well width and that well depth has to be larger than a wavelength. Exact calculation of differential reflection coefficients of dielectric surface with isolated structure on it may be used for the examination of dielectric surfaces and also in making simple but efficient diffuser.

• Honam Mathematical Journal
• /
• v.38 no.1
• /
• pp.151-167
• /
• 2016

In Minkowskian 3-spacetime $L^3$ we find timelike or spacelike surface of revolution for the given Gauss curvature K = -1, 0, 1 and mean curvature H = 0. In fact, we set up the surface of revolution with the time axis for z-axis to be able to draw those surfaces on standard pictures in Minkowskian 3-spacetime $L^3$.

• Korean Journal of Computational Design and Engineering
• /
• v.14 no.5
• /
• pp.297-305
• /
• 2009

Automatic generation of the mid-surfaces of a CAD model is becoming a useful function in that it can help increase the efficiency of engineering analysis as far as it does not affect the result seriously. Several methods had been proposed previously to automatically generate the mid-surfaces, but they often failed to generate the mid-surfaces of complex CAD models. Due to the inherent difficulty of this mid-surface generation problem, it may not be possible to come up with a complete and general method to solve this problem. Since a method that can handle a specific case may not work for different cases, it seems that developing case-specific methods ends up with solving only a fraction of the problem. In this paper, therefore, we propose a method to generate mid-surfaces based on a divide-and-conquer paradigm. This method first decomposes a complex CAD model into simple volumes. The mid-surfaces of the simple volumes are automatically generated by the existing methods, and then they are converted into the mid-surfaces of the original CAD model.