• 대한수학회보
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• 제49권1호
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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 $\mathbb{F}_q[x]$.

• 대한수학회보
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• 제52권4호
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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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• 제9권2호
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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 $\{{\in}_n(x)\}$.

• 한국세라믹학회지
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• 제29권7호
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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.

• 한국세라믹학회지
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• 제28권5호
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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.

• 한국세라믹학회지
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• 제29권8호
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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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• 제17권3호
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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.

• 폴리머
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• 제31권3호
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• pp.206-214
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• 2007

3차원적 타원체 형태 ($a_1>a_2>a_3$)를 사용하여, 2가지의 종횡비 (1차 종횡비 $\rho_{\alpha}$와 2차 종횡비 $\rho_{\beta}$)를 갖는 배열된 3차원 형태의 충전제를 포함하는 복합체의 열팽창 변화를 이론적으로 예측하는 모델을 연구하였다. Eshelby의 등가 텐서를 이용하여 배열된 타원형 충전제를 기지재의 이론에 의해 열팽창 계수를 예측할 수 있었다. 종단방향 열팽창 계수 ${\alpha}_{11}$는 두 가지 종횡비 모두 증가하면, 감소하여 충전제의 열팽창에 접근한다. 수직방향 열팽창 계수 ${\alpha}_{33}$는 충전제 함유량에 따라 초기에는 증가하여 최대값을 갖는 경향을 보인다. 1차 종횡비 $\rho_{\alpha}$가 증가하면, 횡단방향 열팽창 계수는 감소하나, 2차 종횡비 $\rho_{\beta}$가 증가하면, 같이 증가한다. 또한, 수직방향 열팽창 계수 ${\alpha}_{33}$는 $\rho_{\alpha}$가 증가하면 증가하나, $\rho_{\beta}$가 증가하면, ${\alpha}_{33}$는 반대로 감소한다.

• 한국재료학회지
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• 제3권3호
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• pp.207-214
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• 1993

$Li_2O-Al_2O_3-SiO_3$계 결정화유리의 저온합성을 위하여 출방원료로서 각 해당 금속 알콕시드를 사용하였다. 알코올을 용매로 충분히 첨가하고, drying control chemical additive로 dimethy1 formamide를 적당량 첨가한 혼합용용액을 과잉의 물로 충분히 가수분해시킨 습윤겔을 저온으로 건고하여 균열이 없는 건조된 monolith겔을 합성하였다. 건조겔로부터 750-$950^{\circ}C$로 10시간 이상 소결하여 저열팽창성을 나타내는 ${\beta}$-eucrypytite(${\beta}$-quartz 고용체), $Li_2O\cdot Al_2O_3\cdot 3SiO_2$및 ${\beta}$-spodumene등의결정상을 석출시켰다.

• BMB Reports
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• 제45권1호
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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.