• Bulletin of the Korean Chemical Society
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• 제9권3호
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• pp.149-152
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• 1988

A mixture of (Z,Z)-3,13-octadecadien-1-yl acetate(1) and its (E,Z)-isomer(2), the sex pheromone of the cherry tree borer, Synanthedon hector Butler was synthesized. (Z)-11-Octadecen-1-al(3) was prepared from 1,10-decandiol. The Wittig reaction the above aldehyde3 with carboethoxymethylenetriphenylphosphorane, or the Wadsworth-Emmons reaction of the above aldehyde3 with the anion of triethylphosphonoacetate gave ethyl (Z,Z)-2,13-octadecadienoate and its (E,Z)-isomer. Deconjugative protonation of ethyl (Z,Z)-2,13-octadecadienoate and its (E,Z)-isomer with potassium hexamethyldisilazide followed by aqueous ammonium chloride work-up afforded stereoselectiv디y ethyl (E,Z)-3,13-octadecadienoate and its (Z,Z)-isomer, respectively, of which stereoselectivity was adjusted to give the product in the required ratio. Exposure of the above deconjugated ester to excess lithium aluminium hydride resulted in formation of the penultimate (Z,Z)-3,13-octadecadien-1-ol and its (E,Z)-isomer. Acetylation of the desired alcohols afford the final products, (Z,Z)-3,13-octadecadien-1-yl acetate(1) and its (E,Z)-isomer(2).

• 대한수학회보
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• 제57권4호
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• pp.839-850
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• 2020

For functions f(z) = z + a₂z² + a₃z³ + ⋯ belonging to particular classes, this paper finds sharp bounds for the initial coefficients a₂, a₃, a₄, as well as the sharp estimate for the second order Hankel determinant H₂(2) = a₂a₄ - a₂³. Two classes are treated: first is the class consisting of f(z) = z + a₂z² + a₃z³ + ⋯ in the unit disk 𝔻 satisfying $$\|\(\frac{z}{f(z)}\)^{1+{\alpha}}\;f^{\prime}(z)-1\|<{\lambda},\;0<{\alpha}<1,\;0<{\lambda}{\leq}1.$$ The second class consists of Bazilevič functions f(z) = z+a₂z²+a₃z³+⋯ in 𝔻 satisfying $$Re\{\(\frac{f(z)}{z}\)^{{\alpha}-1}\;f^{\prime}(z)\}>0,\;{\alpha}>0.$$

• Korean Journal of Mathematics
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• 제31권3호
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• pp.363-372
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• 2023

In this paper, we consider two functional equations: (1) h(𝓕(x, y, z) + 2x + y + z) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) + 2x + y) + h(xy) + yh(x + z) + 2h(z), (2) h(𝓕(x, y, z) - y + z + 2e) + 2h(x + y) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) - y + 2e) + 2h(x + y + z) + h(xy) + yh(x + z), without any regularity assumption for all x, y, z in a unital algebra A, where 𝓕 : A³ → A is defined by 𝓕(x, y, z) := h(x + y + z) - h(x + y) - h(z) for all x, y, z ∈ A. Also, we find general solutions of these equations in unital algebras. Finally, we prove the Hyers-Ulam stability of (1) and (2) in unital Banach algebras.

• Kyungpook Mathematical Journal
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• 제64권1호
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• pp.185-196
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• 2024

In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), b_i(z) for i = 0, 1, 2 and d_j (z) for j = 0, ..., 4 are given functions, △_cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the b_i(z) and the d_j(z) are polynomials, and d₄(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d₀(z) + 1, or, deg a(z) = deg d₀(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

• 한국약용작물학회지
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• 제10권3호
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• pp.162-166
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• 2002

산초에서 시기에 따라 차이가 크게 나타나는 주요성분은 hexanal, (E)-2-hexenal, ${\alpha}-pinene$, myrcene, (Z)-ocimene+limonene, $(E)-{\beta}-ocimene$, linalool, citronellal, estragole 이었으며, 특히 estragole은 모든 지역에서 공통적으로 나타났다. 초피에서 시기에 따라 차이를 보이는 성분은 hexanal, (Z)-3-hexenol, n-hexanol, ${\alpha}-pinene$, limonene, 1,8-cineole, citronellal, estragole, citronellol 이었으며, 이중 (Z)-3-hexenol ${\alpha}-pinene$, limonene, citronellal이 모든 지역에서 공통으로 나타났다. 수목원에서 월별 변화를 보이는 성분을 관찰한 결과. 산초는 ${\alpha}-pinene$, myrcene, (Z)-3-hexenyl acetate, ${\alpha}-phellandrene$, (Z)-ocimene+limonene, ${\beta}-phellandrene$, linalool, geranyl acetate 이였으며, 초피는 hexanal, (Z)-3-hexenol, (E)-2-hexenal, n-hexanol, ${\alpha}-pinene$, (Z)-ocimene, limonene, citronellal, geranyl acetate, ${\beta}-caryophyllene$ 이였다. 그러나 지역별 채집한 개체에 나타났던 estragole이 수목원에 식재된 산초나무 잎에서 미량으로만 존재한 이유는 열매를 맺지 못할 정도의 어린나무였기 때문이라고 생각되어 estragole 성분은 열매를 맺는 시기에 증가되는 성분이라 사료되었다.

• 대한수학회지
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• 제52권4호
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• pp.727-749
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• 2015

We obtain the conditions on ${\beta}$ so that $1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3$ implies p(z) ${\prec}$ (2+z)/(2-z), $1+(1-{\alpha})z$, $(1+(1-2{\alpha})z)/(1-z)$, ($0{\leq}{\alpha}$<1), exp(z) or ${\sqrt{1+z}}$. Similar results are obtained by considering the expressions $1+{\beta}zp^{\prime}(z)/p(z)$, $1+{\beta}zp^{\prime}(z)/p^2(z)$ and $p(z)+{\beta}zp^{\prime}(z)/p(z)$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition ${\mid}log(zf^{\prime}(z)/f(z)){\mid}$ < 1 or ${\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}$ < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0$.

• 호남수학학술지
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• 제2권1호
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• pp.13-18
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• 1980

모든 기호(記號)는 G.E Bredon의 저(著) Sheaf Theory의 기호(記號)를 따른다. A가 torsion free이며 elementary sheaf이라 하자. 그리고 L을 injective L-module이라 하자 $dim_{\varphi}X<{\infty}$이라면 support의 $family{\varphi}$와 locally subset z에 대하여 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{\varphi}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-p}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ 이며 support의 family c와 compact subset z에 대하여도 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{c}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-y}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ A가 elementary이면 locally closed z와 z에서 closed인 $z^{\prime}$ 그리고 $z^{\prime\prime}=z-z^{\prime}$에 대하여 exact sequence ⋯⋯${\rightarrow}H^{z^{\prime}}_{p}\;(X:A){\rightarrow}H^{z}_{p}(X:A){\rightarrow}H^{z^{\prime\prime}}_{p}\;(X:A){\rightarrow}$⋯⋯ 가 존재(存在)한다.

• 대한수학회보
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• 제57권4호
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• pp.1061-1073
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• 2020

In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^n+Qd(z,f)=\sum\limits_{i=1}^{3}pi(z)e^{{\alpha}_i(z)}$$ are studied by using Nevanlinna theory, where n ≥ 5 is an integer, Q_d(z, f) is a differential polynomial in f of degree d ≤ n - 4 with rational functions as its coefficients, p₁(z), p₂(z), p₃(z) are non-vanishing rational functions, and α₁(z), α₂(z), α₃(z) are nonconstant polynomials such that α'₁(z), α'₂(z), α'₃(z) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.

• 호남수학학술지
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• 제34권1호
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• pp.45-54
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• 2012

In this paper, we prove stabilities of multiplicative functional equations in three variables such as $r(\frac{x+y+z}{3})-r(x+y+z)$=$\frac{2r(\frac{x+y}{2})r(\frac{y+z}{2})r(\frac{z+x}{2})}{r(\frac{x+y}{2})r(\frac{y+z}{2})+r(\frac{y+z}{2})r(\frac{z+x}{2})+r(\frac{z+x}{2})r(\frac{x+y}{2})}$ and $r(\frac{x+y+z}{3})+r(x+y+z)$=$\frac{4r(\frac{x+y}{2})r(\frac{y+z}{2})r(\frac{z+x}{2})}{r(\frac{x+y}{2})r(\frac{y+z}{2})+r(\frac{y+z}{2})r(\frac{z+x}{2})+r(\frac{z+x}{2})r(\frac{x+y}{2})}$.

• 디지털콘텐츠학회 논문지
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• 제8권1호
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• pp.85-91
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• 2007

본 논문에서는 포인트 샘플링을 이용한 음함수 곡면에서 나타나는 앨리어싱을 줄이기 위한 새로운 하이브리드 앤티앨리어싱 방법을 제안한다. 하이브리드 앤티앨리어싱 방법은 하나의 음함수 곡면에 대해 Z-버퍼에서 나타나는 픽셀 값들의 차이를 구하여 세가지 형태의 Z-버퍼를 각각 사용하는 방법이다. 차이의 수준을 정하여 차이가 심할 때는 멀티 Z-버퍼를 사용하고 차이가 중간 정도일 때는 더블 Z-버퍼를 사용하며 차이가 경미할 때는 원래의 Z-버퍼만을 사용하여 3차원 객체를 표현하는 방법이다. 기존에 앤티앨리어싱 효과를 높이기 위하여 전체적으로 멀티 Z-버퍼를 사용하였는데 사용되는 Z-버퍼의 개수를 줄이면서 멀티 Z-버퍼를 사용한 것과 비슷한 효과를 보이는 효율적인 방법이다.