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

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.977-998
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• 2003

Let k be an imaginary quadratic field, η the complex upper half plane, and let $\tau$ $\in$ η $textsc{k}$, p = $e^{{\pi}i{\tau}}$. In this article, using the infinite product formulas for g2 and g3, we prove that values of certain infinite products are transcendental whenever $\tau$ are imaginary quadratic. And we derive analogous results of Berndt-Chan-Zhang ([4]). Also we find the values of (equation omitted) when we know j($\tau$). And we construct an elliptic curve E : $y^2$ = $x^3$ ＋ 3 $x^2$ ＋ {3-(j/256)}x ＋ 1 with j = j($\tau$) $\neq$ 0 and P = (equation omitted) $\in$ E.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.735-747
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• 2018

Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.471-478
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• 2006

In this paper we prove that if AX=b is a partial sign-solvable linear system with A being sign non-singular matrix and if ${\alpha}=\{j:\;x_j\;is\;sign-determined\;by\; Ax=b\}, then $A_{\alpha}X_{\alpha}=b_{\alpha}$ is a sign-solvable linear system, where $A_{\alpha}$ denotes the submatrix of A occupying rows and columns in o and xo and be are subvectors of x and b whose components lie in ${\alpha}$. For a sign non-singular matrix A, let $A_l,\;...,A_{\kappa}$ be the fully indecomposable components of A and let ${\alpha}_i$ denote the set of row numbers of $A_r,\;r=1,\;...,\;k$. We also show that if $A_x=b$ is a partial sign-solvable linear system, then, for $r=1,\;...,\;k$, if one of the components of xor is a fixed zero solution of Ax=b, then so are all the components of x_{{\alpha}r}$.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.37-43
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• 2006

We prove that $M_1\longrightarrow^f\;M_2$ is an injective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ if and only if $M_1\;and\;M_2$ are injective left R-modules, $M_1\longrightarrow^f\;M_2$ is isomorphic to a direct sum of representation of the types $E_l{\rightarrow}0$ and $M_1\longrightarrow^{id}\;M_2$ where $E_l\;and\;E_2$ are injective left R-modules. Then, we generalize the result so that a representation$M_1\longrightarrow^{f_1}\;M_2\; \longrightarrow^{f_2}\;\cdots\;\longrightarrow^{f_{n-1}}\;M_n$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\cdots}{\rightarrow}{\bullet}$ is an injective representation if and only if each $M_i$ is an injective left R-module and the representation is a direct sum of injective representations.

• Journal of Advanced Navigation Technology
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• v.4 no.1
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• pp.78-92
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• 2000

This study will be regarded significant in terms that empirical analysis was used to prove "Aviation Safety", a variable which had not been regarded as a airline choice factor within Korea air transport market so far, has an effect on the aviation consumers' airline preference change and choice after recent frequent aviation accidents. On releasing this paper. I wish, it can be another opportunity for Korean two national flag airlines to reappraise and reinforce the significance of "aviation safety" and set forth immediate vigorous efforts to support the government's aviation safety improvement countermeasures The study was performed to provide data and information for this.

• Kyungpook Mathematical Journal
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• v.45 no.1
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• pp.33-43
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• 2005

Clones are sets of operations which are closed under composition and contain all projections. Identities of clones of term operations of a given algebra correspond to hyperidentities of this algebra, i.e., to identities which are satisfied after any replacements of fundamental operations by derived operations ([7]). If any identity of an algebra is satisfied as a hyperidentity, the algebra is called solid ([3]). Solid algebras correspond to free clones. These connections will be extended to so-called generalized clones, to strong hyperidentities and to strongly solid varieties. On the basis of a generalized superposition operation for terms we generalize the concept of a unitary Menger algebra of finite rank ([6]) to unitary Menger algebras with infinitely many nullary operations and prove that strong hyperidentities correspond to identities in free unitary Menger algebras with infinitely many nullary operations.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.149-165
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• 2018

A. Bejancu [2] was the first who instigated the new concept in differential geometry, i.e., CR-submanifolds. On the other hand, CR-submanifolds were generalized by B. Y. Chen [7] as slant submanifolds. Further, he gave the notion of a Kaehlerian slant submanifold as a proper slant submanifold. This article has two objectives. For the first objective, we derive Simons' type formula for a minimal Kaehlerian slant submanifold in a complex space form. Then, by applying this formula, we give a complete classification of a minimal Kaehlerian slant submanifold in a complex space form and also obtain its some immediate consequences. The second objective is to prove some results about semi-parallel submanifolds.

• East Asian mathematical journal
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• v.24 no.4
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• pp.317-328
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• 2008

In this paper we introduce ${\pi}G{\alpha}$-LC sets, ${\pi}G{\alpha}-LC^*$ sets and ${\pi}G{\alpha}-LC^{**}$ sets and different notions of generalizations of continuous functions in topological space and discuss some of their properties. Further we prove pasting lemma for ${\pi}G{\alpha}-LC^{**}$ continuous functions and ${\pi}G{\alpha}-LC^{**}$ irresolute functions.

• Bulletin of the Korean Chemical Society
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• v.35 no.3
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• pp.805-810
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• 2014

The shape invariant potentials are proved to be exactly solvable, i.e. the wave functions and energies of a particle moving under the influence of the shape invariant potentials can be algebraically determined without any approximations. It is well known that the SWKB quantization is exact for all shape invariant potentials though the SWKB quantization itself is approximate. This mystery has not been mathematically resolved yet and may not be solved in a concrete fashion even in the future. Therefore, in the present work, to understand (not prove) the mystery an attempt of deriving exactly solvable potentials directly from the SWKB quantization has been made. And it turns out that all the derived potentials are shape invariant. It implicitly explains why the SWKB quantization is exact for all known shape invariant potentials. Though any new potential has not been found in this study, this brute-force derivation of potentials helps one understand the characteristics of shape invariant potentials.