• Journal of applied mathematics & informatics
• /
• v.29 no.1_2
• /
• pp.135-143
• /
• 2011

As a zero set of some multivariate splines, the piecewise algebraic variety is a kind of generalization of the classical algebraic variety. This paper presents an algorithm for isolating real roots of the zero-dimensional piecewise algebraic variety which is based on interval evaluation and the interval zeros of univariate interval polynomials in Bernstein form. An example is provided to show the proposed algorithm is effective.

• Journal of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.469-479
• /
• 1996

Toric variety is a normal algebraic variety containing algebraic torus $T_N$ as an open dense subset with an algebraic action of $T_N$ which is an extension of the group law of $T_N$. A toric variety can be described in terms of a certain collection, which is called a fan, of cones. From this fact, the properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the relations among the generators. That is, we can translate the diffcult algebrogeometric properties of toric varieties into very simple properties about the combinatorics of cones in affine spaces over the reals.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.1
• /
• pp.299-318
• /
• 2017

Sextic moment problems with an infinite algebraic variety are still widely open. We study the problem with a single cubic column relation associated to 3 parallel lines in which the variety is infinite. It turns out that this specific column relation has a strong connection with moment problems that have a symmetric algebraic variety. We present more concrete solutions to some sextic moment problems with a symmetric variety.

• Communications of the Korean Mathematical Society
• /
• v.10 no.4
• /
• pp.949-961
• /
• 1995

Let f be a smooth G map from a nonsingular real algebraic G variety to an equivariant Grassmann variety. We use some G vector bundle theory to find a necessary and sufficient condition to approximate f by an entire rational G map. As an application we algebraically approximate a smooth G map between G spheres when G is an abelian group.

• East Asian mathematical journal
• /
• v.19 no.2
• /
• pp.317-335
• /
• 2003

In this paper, we characterize SL(2, $\mathbb{C}$)-representations of an n-periodic link $\tilde{L}$ in terms of SL(2, $\mathbb{C}$)-representations of its quotient link L and express the SL(2, $\mathbb{C}$)-representation variety R($\tilde{L}$) of $\tilde{L}$ as the union of n affine algebraic subsets which have the same dimension. Also, we show that the dimension of R($\tilde{L}$) is bounded by the dimensions of affine algebraic subsets of the SL(2, $\mathbb{C}$)-representation variety R(L) of its quotient link L.

• Communications of the Korean Mathematical Society
• /
• v.11 no.3
• /
• pp.807-814
• /
• 1996

Let $G = Z_2$. Let X be any compact connected orientable nonsingular real algebraic variety of dim X = k = odd with the trivial G action, and let Y be the unit sphere $S^{2n-k}$ with the antipodal action of G. Then we prove that any G invariant entire rational map $f : x \times Y \to S^{2n}$ is G homotopically trivial. We apply this result to prove that any entire rational map $g : X \times RP^{2n-k} \to S^{2n}$ is homotopically trivial.

• Honam Mathematical Journal
• /
• v.28 no.1
• /
• pp.83-111
• /
• 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.

• Research in Mathematical Education
• /
• v.15 no.4
• /
• pp.311-325
• /
• 2011

In this paper I propose that teaching students the most efficient method of problem solving may curtail students' creativity. Instead it is important to arm students with a variety of problem solving heuristics. It is the students' responsibility to decide which heuristic will solve the problem. The chosen heuristic is the one which is meaningful to the students.

• Journal of the Korean Mathematical Society
• /
• v.58 no.2
• /
• pp.451-471
• /
• 2021

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (ℂ∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.

• Journal of the Chungcheong Mathematical Society
• /
• v.28 no.2
• /
• pp.251-259
• /
• 2015

Let $\mathbb{F}^d_q$ be a d-dimensional vector space over a finite field $\mathbb{F}^d_q$ with q elements. We endow the space $\mathbb{F}^d_q$ with a normalized counting measure dx. Let ${\sigma}$ be a normalized surface measure on an algebraic variety V contained in the space ($\mathbb{F}^d_q$, dx). We define the restricted averaging operator AV by $A_Vf(X)=f*{\sigma}(x)$ for $x{\in}V$, where $f:(\mathbb{F}^d_q,dx){\rightarrow}\mathbb{C}$: In this paper, we initially investigate $L^p{\rightarrow}L^r$ estimates of the restricted averaging operator AV. As a main result, we obtain the optimal results on this problem in the case when the varieties V are any nondegenerate algebraic curves in two dimensional vector spaces over finite fields. The Fourier restriction estimates for curves on $\mathbb{F}^2_q$ play a crucial role in proving our results.