• 대한수학회지
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• 제60권2호
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• pp.359-374
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• 2023

In this paper we give the first steps toward the study of the Harbourne-Hirschowitz condition and the anticanonical orthogonal property for regular surfaces. To do so, we consider the Kodaira dimension of the surfaces and study the cases based on the Enriques-Kodaira classification.

• 대한수학회지
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• 제38권5호
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• pp.1069-1105
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• 2001

Regular maps and hypermaps are cellular decompositions of closed surfaces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familar examples of regular maps. The gret dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [59] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [19], and more recently in Coxeter’s and Moser’s book [25] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck\`s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas, Many of these links are nicely surveyed in the recent papers of Jones [55] and Jones and Singerman [54]. The presented survey paper is based on the talk given by the author at the conference “Mathematics in the New Millenium”held in Seoul, October 2000. The idea was, on one hand side, to show the relationship of (regular) maps and hypermaps to the above mentioned fields of mathematics. On the other hand, we wanted to stress some ideas and results that are important for understanding of the nature of these interesting mathematical objects.

• 대한수학회논문집
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• 제18권1호
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• pp.1-20
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• 2003

본 요약논문에서는 단순 연결된 리만곡면들의 isometry군, 그 군의 이산부분군을 이용한 리 만곡면들의 tessellation 그리고 regular map에 대해 소개하고 그 응용과 상호연관성들에 대해 살펴본다. 그리고, 여러가지 관점에서의 regular map의 분류에 대해 소개하고, 최근까지 연구되어진 바에 대해 정리해 보고자 한다.

• 대한수학회지
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• 제43권5호
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• pp.991-1018
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• 2006

Hypermaps are cellular embeddings of hypergraphs in compact and connected surfaces, and are a generalisation of maps, that is, 2-cellular decompositions of closed surfaces. There is a well known correspondence between hypermaps and co-compact subgroups of the free product $\Delta=C_2*C_2*C_2$. In this correspondence, hypermaps correspond to conjugacy classes of subgroups of $\Delta$, and hypermap coverings to subgroup inclusions. Towards the end of [9] the authors studied regular hypermaps with extra symmetries, namely, G-symmetric regular hypermaps for any subgroup G of the outer automorphism Out$(\Delta)$ of the triangle group $\Delta$. This can be viewed as an extension of the theory of regularity. In this paper we move in the opposite direction and restrict regularity to normal subgroups $\Theta$ of $\Delta$ of finite index. This generalises the notion of regularity to some non-regular objects.

• 호남수학학술지
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• 제28권1호
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• pp.83-111
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• 2006

After introducing Tropical Algebraic Varieties, we give a polyhedral description of tropical hypersurfaces. Using TOPCOM and GAP, we show that there exist 59 types of two dimensional tropical quadric surfaces. We also show a criterion for a quadric hypersurface to be non-degenerate in terms of a tropical rank.

• 한국표면공학회지
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• 제33권4호
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• pp.261-265
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• 2000

Geometrically and chemically well defined Pt/alumina model catalysts have been prepared. To this end, we fabricated electrochemically alumina supports in which pores of constant size, length and shape were regularly distributed over a wide area of the surface. Platinum particles were dispersed on the pore surfaces via organometallic chemical vapor deposition technique using (trimethyl) methylcyclopentadienylplatinum (IV) as a precursor. The chemical composition of the alumina plane surfaces was examined by Auger electron spectroscopy and the adsorption characteristics of the platinum particles were studied by thermal desorption spectroscopy. A variety of industrial catalytic problems are now open for further investigation utilizing the Pt/alumina model catalysts.

• 대한수학회보
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• 제54권1호
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• pp.159-175
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• 2017

We solve the $Bj{\ddot{o}}rling$ problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve ${\gamma}$ and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to ${\gamma}$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains ${\gamma}$ and the unit normal of which on ${\gamma}$ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.

• International Journal of Naval Architecture and Ocean Engineering
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• 제10권3호
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• pp.361-366
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• 2018

Ice crushing occurs in many situations that involve a sliding frictional component such as sports involving ice-contact, ice interaction with ship hulls, and ice-on-ice sliding/crushing within glaciers and between interacting sea ice floes. Ice crushing-friction tests were conducted in the lab at $-10^{\circ}C$ using a set of acrylic ice-crushing platens that included a flat smooth surface and a variety of high-roughness surfaces with regular arrays of small prominences. The experiments were part of Phase II tests of the Blade Runners technology for reducing ice-induced vibration. Ice was crushed against the platens where the ice movement had both a vertical and a horizontal component. High-speed imaging through the platens was used to observe the ice contact zone as it evolved during the tests. Vertical crushing rates were in the range 10-30 mm/s and the horizontal sliding rates were in the range 4.14-30 mm/s. Three types of freshwater ice were used. Friction coefficients were extraordinarily low and were proportional to the ratio of the tangential sliding rate and the normal crushing rate. For the rough surfaces all of the friction coefficient variation was determined by the fluid dynamics of a slurry that flowed through channels that developed between leeward-facing facets of the prominences and the moving ice. The slurry originated from a highly-lubricating self-generating squeeze film of ice particles and melt located between the encroaching intact ice and the surfaces.

• 대한수학회지
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• 제53권4호
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• pp.725-735
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• 2016

A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces.

• 한국진공학회:학술대회논문집
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• 한국진공학회 1998년도 제14회 학술발표회 논문개요집
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• pp.147-148
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• 1998