• Journal for History of Mathematics
• /
• v.15 no.1
• /
• pp.147-168
• /
• 2002

In this paper, we introduce the arrangement of hyperplanes and the graph theory. In particular, we explain how to study the 4-color problem by using characteristic polynomials of the arrangement of hyperplanes. The 4-color problem was appeared in 1852 at first and Appel and Haken proved it by using computer in 1976. The arrangement of hyperplanes induced from a graph is called a graphic arrangement. Graphic arrangement is a subarrangement of Braid arrangement. Thus the chromatic function of a graph is equal to the characteristic polynomial of a graphic arrangement. If we use this result, we can apply the theory of the arrangement of hyperplanes to the study for the chromatic functions.

• Honam Mathematical Journal
• /
• v.43 no.4
• /
• pp.679-689
• /
• 2021

The first author suggested an exact volume formula of the hypercubes [0, 1]ⁿ clipped by several hyperplanes expressed directly in terms of linear coefficients of the hyperplanes. However, it requires awkward assumptions to apply the formula to various situations. We suggest a concrete method to overcome those restrictions for two or three hyperplanes using 𝜖-perturbation, which gives an exact value applicable for any kind of arrangement of hyperplanes with no consideration.

• Honam Mathematical Journal
• /
• v.31 no.1
• /
• pp.25-29
• /
• 2009

In this paper, we classify essential arrangements $\mathcal{A}$ of hyperplanes in $\mathbb{R}^3$ with ${\mid}\mathcal{A}{\mid}$ = 5 and give a full description of each case in terms of freeness.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.5
• /
• pp.1281-1289
• /
• 2016

Richard Stanley suggested the problem of finding combinatorial proofs of formulas for counting regions of certain hyperplane arrangements defined by hyperplanes of the form $x_i=0$, $x_i=x_j$, and $x_i=2x_j$ that were found using the finite field method. We give such proofs, using embroidered permutations and linear extensions of posets.

• Proceedings of the Korean Institute of Intelligent Systems Conference
• /
• 1998.06a
• /
• pp.634-639
• /
• 1998

A methodology based on the concept of variable string length GA(VGA) is developed for determining automatically the number of hyperplanes and their appropriate arrangement for modeling the class boundaries of a given training data set in RN. The genetic operators and fitness functionare newly defined to take care of the variability in chromosome length. Experimental results on different artificial and real life data sets are provided.

• Journal of the Korean Mathematical Society
• /
• v.57 no.6
• /
• pp.1407-1433
• /
• 2020

Parts I-IV showed that the number of ways to place q nonattacking queens or similar chess pieces on an n × n chessboard is a quasipolynomial function of n whose coefficients are essentially polynomials in q. For partial queens, which have a subset of the queen's moves, we proved complete formulas for these counting quasipolynomials for small numbers of pieces and other formulas for high-order coefficients of the general counting quasipolynomials. We found some upper and lower bounds for the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. Here we discover more about the counting quasipolynomials for partial queens, both familiar and strange, and the nightrider and its subpieces, and we compare our results to the empirical formulas found by Kotššovec. We prove some of Kotššovec's formulas and conjectures about the quasipolynomials and their high-order coefficients, and in some instances go beyond them.