• Korean Journal of Mathematics
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• v.9 no.2
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• pp.123-128
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• 2001

Given a homomorphism ${\Pi}:X{\rightarrow}Y$, with Y minimal, we will introduce the concept of a relative (to ${\Pi}$) F-regular relation which generalize the notions of F-proximality, F-regularity and relative F-proximality, and will study its properties.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.543-554
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• 2007

In this paper, we prove the stability of a mixed type functional equation f( -x + y + z) + f(x - y + z) + f(x + y - z) = f(x - y) + f(y - z) + f(z - x) + f.(x) + f(y) + f(z).

• Journal of the Chungcheong Mathematical Society
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• v.32 no.3
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• pp.337-347
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• 2019

In this paper, we investigate the generalized Hyers-Ulam stability of the quadratic and quartic type functional equations $$f(kx+y)+f(kx-y)-k^2f(x+y)-k^2f(x-y)-2f(kx)\\{\hfill{67}}+2k^2f(x)+2(k^2-1)f(y)=0,\\f(x+5y)-5f(x+4y)+10f(x+3y)-10f(x+2y)+5f(x+y)\\{\hfill{67}}-f(-x)=0,\\f(kx+y)+f(kx-y)-k^2f(x+y)-k^2f(x-y)\\{\hfill{67}}-{\frac{k^2(k^2-1)}{6}}[f(2x)-4f(x)]+2(k^2-1)f(y)=0$$ by using the fixed point theory in the sense of L. $C{\breve{a}}dariu$ and V. Radu.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.1
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• pp.177-186
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• 2016

For a map $f:A{\rightarrow}X$, there are concepts of $H^f$-spaces, $T^f$-spaces, which are generalized ones of H-spaces [17,18]. In general, Any H-space is an $H^f$-space, any $H^f$-space is a $T^f$-space. For a principal fibration $E_k{\rightarrow}X$ induced by $k:X{\rightarrow}X^{\prime}$ from ${\epsilon}:PX^{\prime}{\rightarrow}X^{\prime}$, we obtain some sufficient conditions to having liftings $H^{\bar{f}}$-structures and $T^{\bar{f}}$-structures on $E_k$ of $H^f$-structures and $T^f$-structures on X respectively. We can also obtain some results about $H^f$-spaces and $T^f$-spaces in Postnikov systems for spaces, which are generalizations of Kahn's result about H-spaces.

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1171-1187
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• 2011

Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly regular pair. In this paper, we provide the same improvement for a jointly regular family: let h : ${\mathbb{P}}_{\mathbb{Q}}^n{\rightarrow}{{\mathbb{R}}$ be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is de ned in [10] and let {$f_1,{\ldots},f_k|f_l:\mathbb{A}^n{\rightarrow}\mathbb{A}^n$} bbe finite set of polynomial maps which is defined over a number field K. If the intersection of the indeterminacy loci of $f_1,{\ldots},f_k$ is empty, then there is a constant C such that $ \sum\limits_{l=1}^k\frac{1}{def\;f_\iota}h(f_\iota(P))>(1+\frac{1}{r})f(P)-C$ for all $P{\in}\mathbb{A}^n$ where r= $max_{\iota=1},{\ldots},k(r(f_l))$.

• Korean Journal of Mathematics
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• v.5 no.2
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• pp.119-125
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• 1997

We show that there exists an isomorphism between the basic construction $(M_f)_1$ for $N_f{\subset}M_f$ and the reduction $(M_1)_f$ of the basic construction $M_1$ for $N{\subset}M$, where $f$ is a nontrivial projection in N. For a nontrivial projection $f{\in}N^{\prime}{\cap}M$ we give the basic construction $(M_f)_1$ for $N_f{\subset}M_f$ and compare it with $(M_1)_f$.

• Nuclear Medicine and Molecular Imaging
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• v.40 no.4
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• pp.228-232
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• 2006

Purpose: electrophilic $^{18}F(T_{1/2}=110\;min)$ radionuclide in the form of $[^{18}F]F_2$ gas is of great significance for labeling radiopharmaceuticals for positron omission tomography (PET). However, its production In high yield and with high specific radioactivity is still a challenge to overcome several problems on targetry. The aim of the present study was to develop a method suitable for the routine production of $[^{18}F]F_2$ for the electrophilic substitution reaction. Materials and Methods: The target was designed water-cooled aluminum target chamber system with a conical bore shape. Production of the elemental fluorine was carried out via the $^{18}O(p,n)^{18}F$ reaction using a two-step irradiation protocol. In the first irradiation, the target filled with highly enriched $^{18}O_2$ was irradiated with protons for $^{18}F$ production, which were adsorbed on the inner surface of target body. In the second irradiation, the mixed gas ($1%[^{19}F]F_2/Ar$) was leaded into the target chamber, fellowing a short irradiation of proton for isotopic exchange between the carrier-fluorine and the radiofluorine absorbed in the target chamber. Optimization of production was performed as the function of irradiation time, the beam current and $^{18}O_2$ loading pressure. Results: Production runs was performed under the following optimum conditions: The 1st irradiation for the nuclear reaction (15.0 bar of 97% enriched $^{18}O_2$, 13.2 MeV protons, 30 ${\mu}A$, 60-90 min irradiation), the recovery of enriched oxygen via cryogenic pumping; The 2nd irradiation for the recovery of absorbed radiofluorine (12.0 bar of 1% $[^{19}F]fluorine/argon$ gas, 13.2 MeV protons, 30 ${\mu}A$, 20-30 min irradiation) the recovery of $[^{18}F]fluorine$ for synthesis. The yield of $[^{18}F]fluorine$ at EOB (end of bombardment) was achieved around $34{\pm}6.0$ GBq (n>10). Conclusion: The production of $^{18}F$ electrophilic agent via $^{18}O(p,n)^{18}F$ reaction was much under investigation. Especially, an aluminum gas target was very advantageous for routine production of $[^{18}F]fluorine$. These results suggest the possibility to use $[^{18}F]F_2$ gas as a electrophilic substitution agent.

• Kyungpook Mathematical Journal
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• v.52 no.1
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• pp.39-47
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• 2012

Let $\mathcal{F}$ be a family of meromorphic functions in a domain D, and let $k$, $n({\geq}2)$ be two positive integers, and let $S=\{a_1,a_2,{\ldots},a_n\}$, where $a_1$, $a_2$, ${\ldots}$, $a_n$ are distinct finite complex numbers. If for each $f{\in}\mathcal{F}$, all zeros of $f$ have multiplicity at least $k+1$, $f$ and $G(f)$ share the set $S$ in $D$, where $G(f)=P(f^{(k)})+H(f)$ is a differential polynomial of $f$, then$\mathcal{F}$ is normal in $D$.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.75-81
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• 2023

If a multiplicative function f is commutable with a quadratic form x² + xy + y², i.e., f(x² + xy + y²) = f(x)² + f(x) f(y) + f(y)², then f is the identity function. In other hand, if f is commutable with a quadratic form x² - xy + y², then f is one of three kinds of functions: the identity function, the constant function, and an indicator function for ℕ ＼ pℕ with a prime p.

• Asian-Australasian Journal of Animal Sciences
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• v.20 no.11
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• pp.1625-1630
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• 2007

A simulation study was conducted to evaluate the effect of reciprocal cross on the detection and characterization of Mendelian QTL in $F_2$ QTL swine populations. Data were simulated under two different mating designs. In the one-way cross design, six $F_0$ grand sires of one breed and 30 $F_0$ grand dams of another breed generated 10 $F_1$ offspring per dam. Sixteen $F_1$ sires and 64 $F_1$ dams were randomly chosen to produce a total of 640 $F_2$ offspring. In the reciprocal design, three $F_0$ grand sires of A breed and 15 $F_0$ grand dams of B breed were mated to generate 10 $F_1$ offspring per dam. Eight $F_1$ sires and 32 $F_1$ dams were randomly chosen to produce 10 $F_2$ offspring per $F_1$ dam, for a total of 320 $F_2$ offspring. Another mating set comprised three $F_0$ grand sires of B breed and 15 $F_0$ grand dams of A breed to produce the same number of $F_1$ and $F_2$ offspring. A chromosome of 100 cM was simulated with large, medium or small QTL with fixed, similar, or different allele frequencies in parental breeds. Tests between Mendelian models allowed QTL to be characterized as fixed (LC QTL), or segregating at similar (HS QTL) or different (CB QTL) frequencies in parental breeds. When alternate breed alleles segregated in parental breeds, a greater proportion of QTL were classified as CB QTL and estimates of QTL effects for the CB QTL were more unbiased and precise in the reciprocal cross than in the one-way cross. This result suggests that reciprocal cross design allows better characterization of Mendelian QTL in terms of allele frequencies in parental breeds.