• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.495-505
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• 2018

The uniqueness problems on entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study an entire function f(z) that shares a nonzero polynomial a(z) with $f^{(1)}(z)$, together with its linear differential polynomials of the form: $L=L(f)=a_1(z)f^{(1)}(z)+a_2(z)f^{(2)}(z)+{\cdots}+a_n(z)f^{(n)}(z)$, where the coefficients $a_k(z)(k=1,2,{\ldots},n)$ are rational functions and $a_n(z){\not{\equiv}}0$.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.255-260
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• 2006

Let $m$ and $k$ be any positive integers, let $\mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(\mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $\mathbb{Z}_k$ and let ${\phi}_m(k)$ the order of $G_m(\mathbb{Z}_k)$. In this paper, ${\phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(\mathbb{Z}_{st})$ is isomorphic to $G_m(\mathbb{Z}_s){\times}G_m(\mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${\phi}_m(p^n)=p^{m^2}{\cdot}{\phi}_m(p^{n-1})=p{^{2m}}^2{\cdot}{\phi}_m(p^{n-2})={\cdots}=p^{{(n-1)m}^2}{\cdot}{\phi}_m(p)$, and so ${\phi}_m(k)={\phi}_m(p_1^n1){\cdot}{\phi}_m(p_2^{n2}){\cdots}{\phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.115-126
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• 1995

Let A be the class of univalent functions f(z)=z＋${\alpha}$$_2$z$^2$＋${\alpha}$$_3$z$^3$＋…(1.1) which are analytic in the unit disk $\Delta$= {z：│z│＜1}. Let S＊(p) be the subclass of A composing of functions which are starlike of order $\rho$. A function f(z) belonging to the class A is said to be starlike of order $\rho$ ($\rho$(equation omitted) 0) if and only if z$\^$-l/ f(z) (equation omitted) 0 (z$\in$$\Delta$) and (equation omitted (1.2).(omitted)

• The Pure and Applied Mathematics
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• v.12 no.1
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• pp.57-63
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• 2005

Let ∑(p)(p ∈ N) be the class of functions f(z) = z/sup -p/ + α/sub 1-p/ z/sup 1-p/ + α/sub 2-p/z/sup 2-p/ + ... analytic in 0 < ｜z｜ < 1 and let M(p, λ, μ)(0 < λ≤ 2 and 2λ(λ - 1) ≤ μ ≤ λ²) denote the class of functions f(z) ∈ ∑(p) which satisfy (equation omitted). The object of the present paper is to derive some properties of functions in the class M(p, λ, μ).

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.927-931
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• 1994

Let G be a cyclic group of order 2 and let $S^1$ denote the unit circle in $R^2$ with the standard metric. We consider smooth G-vector bundles over $S^1$ when G acts on $S^1$ by reflection. Then the fixed point set of G on $S^1$ is two points ${z_0, z_1}$. Let $E$\mid$_{z_0} and E$\mid$_{z_1}$ be the fiber G-representation spaces at $z_0$ and $z_1$ respectively. We associate an orthogonal G-representation $\rho_i : G \to O(n)$ to $E$\mid$_{z_i}, i = 0, 1$. Let det $p\rho_i(g), g \neq 1$, be denoted by det $E$\mid$_{z_i}$ since det $\rho_i(g)$ is independent of choice of $\rho_i$.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1209-1251
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• 2015

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$, and more generally, the crystallographic groups of $Sol_1^4$. The maximal compact subgroup of Isom($Sol_1^4$) is $D_4={\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$. We shall exhibit an infra-solvmanifold of $Sol_1^4$ whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; the trivial group, ${\mathbb{Z}}_2$ (5 families), ${\mathbb{Z}}_4$, ${\mathbb{Z}}_2{\times}{\mathbb{Z}}_2$ (5 families), and ${\mathbb{Z}}_4{\rtimes}{\mathbb{Z}}_2$ (2 families).

• Bulletin of the Korean Chemical Society
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• v.23 no.1
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• pp.35-40
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• 2002

Photoreaction of guest organic anions in layered organic-inorganic hybrid materials was investigated. The layered hybrids were synthesized by an anion-exchange reaction of $[LiAl_2(OH)_6]Cl{\cdot}yH_2O$ layered double hydroxide with aqueous (Z,Z)- and (E,E)-muconates under inert atmospheric condition, to give new organicinorganic hybrids of $[LiAl_2(OH)_6]_2[(Z,Z)-C_6H_4O_4]{\cdot}zH_2O$ and $[LiAl_2(OH)_6]_2[(E,E)-C_6H_4O_4]{\cdot}H_2O$, respectively. The basal spacings calculated by XRPD of intercalates indicate that muconate anions have almost vertical arrangements against the host $[LiAl_2(OH)_6]^+$ lattices in the interlayer of organic-inorganic hybrid materials. When UV light was irradiated on the suspension of $[LiAl_2(OH)_6]_2[(Z,Z)-C_6H_4O_4]{\cdot}zH_2O$, the (Z,Z)-muconate anions of the gallery space of hybrids were polymerized in the aqueous media while it was isomerized into more stable (E,E)-muconate in the methanollic suspension in the presence of catalytic amount of molecular iodine. All the products were characterized using elemental analysis, TGA, XRPD, FT-IR, $^1H$ NMR and $^{13}C$ CP-MAS NMR.

• Korean Journal of Materials Research
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• v.21 no.12
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• pp.644-649
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• 2011

[ $A_{3-2x/3}Al_{1-z}In_{z}O_4F:Eu_x^{3+}$ ](A = Ca, Sr, Ba, x = -0.15, z = 0, 0.1) oxyfluoride phosphors were simply prepared by the solid-state method at $1050^{\circ}C$ in air. The phosphors had the bright red photoluminescence (PL) spectra of an $A_{3-2x/3}Al_{1-z}In_{z}O_4F$ for $Eu^{3+}$ activator. X-ray diffraction (XRD) patterns of the obtained red phosphors were exhibited for indexing peak positions and calculating unit-cell parameters. Dynamic excitation and emission spectra of $Eu^{3+}$ activated red oxyfluoride phosphors were clearly monitored. Red and blue shifts gradually occurred in the emission spectra of $Eu^{3+}$ activated $A_3AlO_4F$ oxyfluoride phosphors when $Sr^{2+}$ by $Ca^{2+}$ and $Ba^{2+}$ ions were substituted, respectively. The concentration quenching as a function of $Eu^{3+}$ contents in $A_{3-2x/3}AlO_4F:Eu^{3+}$ (A = Ca, Sr, Ba) was measured. The interesting behaviors of defect-induced $A_{3-2x/3}Al_{1-z}In_{z}O_{4-{\alpha}}F_{1-{\delta}}$ phosphors with $Eu^{3+}$ activator are discussed based on PL spectra and CIE coordinates. Substituting $In^{3+}$ into the $Al^{3+}$ position in the $A_{3-2x/3}AlO_4F:Eu^{3+}$ oxyfluorides resulted in the relative intensity of the red emitted phosphors noticeably increasing by seven times.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1651-1672
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• 2023

In 2010, Li-Ye [13, Theorem 0.1] proved that P(ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), Γ"(z)) ≢ 0 in ℂ, where m is a non-negative integer, and P(u₀, u₁, . . . , u_m, v₀, v₁, v₂) is any non-trivial polynomial in its arguments with coefficients in the field ℂ. Later on, Li-Ye [15, Theorem 1] proved that P(z, Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z), ζ(z)) ≢ 0 in z ∈ ℂ for any non-trivial distinguished polynomial P(z, u₀, u₁, . . ., u_n, v) with coefficients in a set L_δ of the zero function and a class of nonzero functions f from ℂ to ℂ ∪ {∞} (cf. [15, Definition 1]). In this paper, we prove that P(z, ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z)) ≢ 0 in z ∈ ℂ, where m and n are two non-negative integers, and P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is any non-trivial polynomial in the m + n + 2 variables u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n with coefficients being meromorphic functions of order less than one, and the polynomial P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is a distinguished polynomial in the n + 1 variables v₀, v₁, . . . , v_n. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.

• The Korean Journal of Pesticide Science
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• v.17 no.2
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• pp.140-143
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• 2013

We investigated optimal condition for synthesis of (E)-2-hexenyl (E)-2-hexenoate (1) and (E)-2-hexenyl (Z)-3-hexenoate (2), the pheromone components of Riptortus pedestris, by Steglich esterification. The reaction with 1.1-1.5 equivalent of dicyclohexylcarbodiimide (DCC), 1.5-2.0 equivalent of (E)-2-hexenol, and 0.1 equivalent 4-dimethylaminopyrinde (DMAP) to (E)-2-hexenoic acid in toluene or (Z)-3-hexenoic acid in dichloromethane led 1 and 2 in 76-78% and 87-91% yield, respectively.