• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.127-133
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• 2012

It is well known that if the ${\beta}$-expansion of any nonnegative integer is finite, then ${\beta}$ is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the ${\beta}$-expansion of the polynomial part of ${\beta}$ is finite if and only if ${\beta}$ is a Pisot series. Consequently we give an other proof of Scheiche theorem about finiteness property in $\mathbb{F}_q((x^{-1}))$. Finally we show that if the base ${\beta}$ is a Pisot series, then there is a bound of the length of the fractional part of ${\beta}$-expansion of any polynomial P in $\mathbb{F}_q[x]$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1047-1057
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• 2015

In [6], it is proved that the lengths of periods occurring in the ${\beta}$-expansion of a rational series r noted by $Per_{\beta}(r)$ depend only on the denominator of the reduced form of r for quadratic Pisot unit series. In this paper, we will show first that every rational r in the unit disk has strictly periodic ${\beta}$-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if $r=\frac{P}{Q}$ is written in reduced form with |P| < |Q|, we will generalize the curious property "$Per_{\beta}(\frac{P}{Q})=Per_{\beta}(\frac{1}{Q})$".

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.2
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• pp.91-97
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• 2005

If ${\beta}\;>\;1$, then every non-negative number x has a ${\beta}-expansion$, i.e., $$x\;=\;{\epsilon}_0(x)\;+\;{\frac{\epsilon_1(x)}{\beta}}\;+\;{\frac{\epsilon_2(x)}{\beta}}\;+\;{\cdots}$$ where ${\epsilon}_0(x)\;=\;[x],\;{\epsilon}_1(x)\;=\;[\beta(x)],\;{\epsilon}_2(x)\;=\;[\beta(({\beta}x))]$, and so on ([x] denotes the integral part and (x) the fractional part of the real number x). Let T be a transformation on [0,1) defined by $x\;{\rightarrow}\;({\beta}x)$. It is well known that the relative frequency of $k\;{\in}\;\{0,\;1,\;{\cdots},\;[\beta]\}$ in ${\beta}-expansion$ of x is described by the T-invariant absolutely continuous measure ${\mu}_{\beta}$. In this paper, we show the mod M normality of the sequence $\{{\in}_n(x)\}$.

• Journal of the Korean Ceramic Society
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• v.29 no.7
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• pp.533-538
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• 1992

Effect of Li2O.4B2O frit on the microstructural and thermal expansion behaviour of $\beta$-spodumene ceramics was investigated. With 4.0~8.0 wt.% frit addition, sintering temperature range was enlarged and densification was enhanced through liquid phase sintering. As the amount of frit addition was increased, thermal expansion coefficient was increased to 1.72$\times$10-6$^{\circ}C$-1 of its highest value, which indicated that the increment of thermal expansion coefficient due to the second phase was constrained.

• Journal of the Korean Ceramic Society
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• v.28 no.5
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• pp.415-421
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• 1991

Influence of Li2O.4B2O3 frit addition on the microstructure and thermal expansion properties of $\beta$-eucryptite ceramics was studied. With 1.0~10.0wt% frit addition, sintering temperature range was enlarged and densified body was obtained. Thermal expansion was increased from -6.67$\times$10-6 $^{\circ}C$-1 of pure sintered body to 1.03$\times$10-6$^{\circ}C$-1 of 6 wt% added body, but this increasement was accompanied by inhibition of thermal expansion anisotropy, so that improved microcracking property.

• Journal of the Korean Ceramic Society
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• v.29 no.8
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• pp.646-650
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• 1992

Effect of Li2O.4B2O3 frit addition on microstructural and thermal expansion behaviour of the $\beta$-spodumene ceramics made from kaolinite was investigated. With 2.0~4.0 wt% frit addition, the densification by liquid phase sintering caused low water absorption below 3% and enhancement of beding strength. As the amount of frit addition was increased, thermal expansion coefficient was increased to large scale, but in 2 wt% frit addition, its thermal expansion coefficient was constrained to 1.92$\times$10-6$^{\circ}C$-1.

• Asian-Australasian Journal of Animal Sciences
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• v.17 no.3
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• pp.317-324
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• 2004

The present study was conducted to investigate the involvement of nitric oxide (NO) in cumulus expansion, oocyte mortality and meiotic maturation of porcine cumulus enclosed oocytes (CEOs) cultured in two different models when gonadotropins, including follicle-stimulating hormone (FSH) and human chorionic gonadotropin (hCG) were presented or not. And the interaction between NO and $\beta$-mercaptoethanol ($\beta$-ME), a free radical scavenger was also investigated. Two models refer to spontaneous maturation model and hypoxanthine (HX) medium model. All the 3,433 eligible CEOs were incubated at $39^{\circ}C$ and the cumulus expansion, oocyte morphology and nuclear phase were evaluated 44 h after incubation. (1) In spontaneous maturation model, NO stimulates the cumulus expansion and $\beta$-ME delayed it. NO doesn't affect the oocyte meiotic resumption but inhibits the oocytes to develop to metaphase II. (2) In HX medium model, NO or $\beta$-ME doesn't affect the expansion in the absence of gonadotropins, but in the presence of gonadotropins, NO or $\beta$-ME inhibits the expansion. In the presence of gonadotropins, NO inhibits the oocyte meiotic resumption and it especially inhibits the oocyte to develop to metaphase II, and $\beta$-ME reverses such inhibitory effects. The cooperation of gonadotropins and $\beta$-ME stimulates the meiotic resumption and especially, promotes the CEOs to develop to metaphase II in both models. Moreover, HX might contribute to the fragility of oocyte zona pellucida and gonadotropins, nitric oxide and $\beta$-ME could alleviate it separately, and cooperatively. It is concluded that NO exerts different functions in two models and $\beta$-ME affected the functions of NO in different models.

• Polymer(Korea)
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• v.31 no.3
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• pp.206-214
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• 2007

The theoretical study is developed for predicting the thermal expansion changes of composites which include complex inclusion, which is used three-dimensional ellipsoid model ($a_1>a_2>a_3$), which has two aspect ratios (the primary aspect ratio, $\rho_{\alpha}=a_1/a_3$ and the secondary aspect ratio, $\rho_{\beta}=a_1/a_2$). We can predict the feature of general thermal expansion factors by theoretical approach of matrix with aligned ellipsoidal inclusion using the Eshelby's equivalent tensor. The coefficients of longitudinal linear thermal expansion ${\alpha}_{11}$ decrease to those of inclusions, ${\alpha}_f$, as both aspect ratios increase. The coefficients of transverse linear thermal expansion of composites ${\alpha}_{33}$ initially increase and show the parabolic corves with maximum values, as the concentrations of filler increase. The coefficient of thermal expansion, ${\alpha}_{22}$ in the transverse direction decreases, as $\rho_{\alpha}$ increases, however, ${\alpha}_{22}$ increases as $\rho_{\beta}$ increases. The coefficient of linear thermal expansion of composites, ${\alpha}_{33}$ in the normal direction increases, as $\rho_{\alpha}$ increases, while ${\alpha}_{33}$ decreases as $\rho_{\beta}$ increases.

• Korean Journal of Materials Research
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• v.3 no.3
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• pp.207-214
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• 1993

Glass-ceramic monoliths with an ultra-low thermal expansion coefficient have been synthesized by the sol-gel technique using metal alkoxides as starting materials and dimethyl formamide as a drying control chemical additive. The ternary gels: $Li_2O\cdot Al_2O_3\cdot 2, 4 or $6SiO_2$ were obtained by hydrolysis and polycondensation reactions of metal alkoxides of silicon, aluminum and lithium. To produce cylindrical crack-free gel monoliths, excess water was used to the starting solutions and drying rates were controlled precisely to prevent cracking. In conversion process ,${\beta}$-eucryptite, $Li_2O\cdot Al_2O_3\cdot 3SiO_2$ and P-spodumene with ,${\beta}$-quartz solid solution phase were obtained by heating at the range of 750 ~$1000^{\circ}C$. Above $800^{\circ}C$, the ,${\beta}$-spodumene phase increased while ,${\beta}$-eucryptite phase decreased. The thermal expansion coefficient of the crystallized specimens were -15~ $+5{\times}{10^{-7}}/{\circ}C$ over the temperature range from room temperature to $600^{\circ}C$.

• BMB Reports
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• v.45 no.1
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• pp.51-56
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• 2012

The purpose of this study was to determine the effects of duration and timing of glucocorticoid treatment on the expansion and differentiation of porcine neonatal pancreas cell clusters (NPCCs) into ${\beta}$-cells. After transplantation of NPCCs, the ductal cyst area and ${\beta}$-cell mass in the grafts both showed positive and negative correlations with duration of dexamethasone (Dx) treatment. Pdx-1 and HNF-3${\beta}$ gene expression was significantly downregulated following Dx treatment, whereas PGC-1${\alpha}$ expression increased. Pancreatic duct cell apoptosis significantly increased following Dx treatment, whereas proliferation did not change. Altogether, transdifferentiation of porcine NPCCs into ${\beta}$-cells was influenced by the duration of Dx treatment, which might have been due to the suppression of key pancreatic transcription factors. PGC-1${\alpha}$ plays an important role in the expansion and transdifferentiation of porcine NPCCs, and the initial 2 weeks following transplantation of porcine NPCCs is a critical period in determining the final ${\beta}$-cell mass in grafts.