• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.129-142
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• 2003

In this paper we study the chromatic sum functions for rooted nonseparable simple maps on the plane. The chromatic sum function equation for such maps is obtained . The enumerating function equation of such maps is derived by the chromatic sum equation of such maps. From the chromatic sum equation of such maps, the enumerating function equation of rooted nonseparable simple bipartite maps on the plane is also derived.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.183-196
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• 2005

In this paper we study the chromatic sum functions for rooted nonseparable near-triangular maps on the projective plane. A chromatic sum equation for such maps is obtained.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.159-172
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• 2004

A map is singular if each edge is on the same face on a surface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper po-vides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enumerating functions of such maps are also derived.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.1
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• pp.269-276
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• 2015

This paper proves the Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture of the vertex coloring problem, which is so far unresolved. The Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture states that "the union of k copies of k-cliques intersecting in at most one vertex pairwise is k-chromatic." i.e., x(G)=k. In a bid to prove this conjecture, this paper employs a method in which it determines the number of intersecting vertices and that of cliques that intersect at one vertex so as to count a vertex of the minimum degree ${\delta}(G)$ in the Minimum Independent Set (MIS) if both the numbers are even and to count a vertex of the maximum degree ${\Delta}(G)$ in otherwise. As a result of this algorithm, the number of MIS obtained is x(G)=k. When applied to $K_k$-clique sum intersecting graphs wherein $3{\leq}k{\leq}8$, the proposed method has proved to be successful in obtaining x(G)=k in all of them. To conclude, the Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture implying that "the k-number of $K_k$-clique sum intersecting graph is k-chromatic" is proven.

• Korean Journal of Optics and Photonics
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• v.11 no.6
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• pp.390-394
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• 2000

An optical system for iris pattern recognition of the human eye is designed. The lens system is triplet type, and characterized to minimize longitudinal chromatic aberration and Petzval sum. The distance from object to image is 200 mm-300 mm and the effective focal length is 50 mm. Performance of the imaging system is assessed by calculating the ray-fan and spot diagram for Fraunhofer C, d and F line for object height 0 mm, 4 mm and 6 mm. Furthermore, MTFs are calculated. All of the spot sizes are less than 0.05 mm in diameter. The MTF values are higher than 0.5 in the spatial frequency range up to 20 lines/mm for all of the object heights. ights.

• Korean Journal of Optics and Photonics
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• v.12 no.3
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• pp.240-249
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• 2001

A high speed ultra-compact lens with a diffractive optical element (DOE) is designed, which can be applied to mobile communication devices such as IMT2000, PDA, notebook computer, etc. The designed hybrid lens has sufficiently high performance of less than f/2.2, compact size of 3.3 mm (1st surf. to image), and wide field angle of more than 30 deg. compared with the specifications of a single lens. By proper choice of the aspheric and DOE surface which has very large negative dispersion, we can correct chromatic and high order aberrations through the optimization technique. From Seidel third order aberration theory and Sweatt modeling, the initial data and surface configurations, that is, the combination condition of the DOE and the aspherical surface are obtained. However, due to the consideration of diffraction efficiency of a DOE, we can choose only four cases as the optimization input, and present the best solution after evaluating and comparing those four cases. On the other hand, we also report dramatic improvement in optical performance by inserting another refractive lens (so-called, field flattener), that keeps the refractive power of an original DOE lens and makes the petzval sum zero in the original DOE lens system. ystem.