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Uniform distributions on curves and quantization
Joseph Rosenblatt,Mrinal Kanti Roychowdhury 대한수학회 2023 대한수학회논문집 Vol.38 No.2
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise \cite{BW}, which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.
Lee, Ji-Hoon,Atherton, Timothy J.,Barna, Valentin,De Luca, Antonio,Bruno, Emanuela,Petschek, Rolfe G.,Rosenblatt, Charles American Physical Society 2009 Physical Review Letters Vol.102 No.16
<P>The spatial and temperature dependence of the surface-induced orientational order parameter S(z,T) was determined in the isotropic phase. An optical fiber was immersed in a thin liquid crystal layer and the retardation was measured as a function of the fiber's height above the surface, from which the model-independent S(z,T) was deduced with resolution <or=2 nm. It was found that (i) S(z=0) <or=0.12 close to the nematic transition temperature, (ii) the susceptibility is mean-field-like, and (iii) S(z,T) deviates significantly from exponential spatial decay. The results are discussed in terms of a nonlocal potential.</P>
임한혁,송웅주,김구환,김유미,장미영,길홍량,김숙자,Lim, Han Hyuk,Song, Wung Joo,Kim, Gu-Hwan,Watkins, David,Rosenblatt, David S.,Kim, Yoo-Mi,Chang, Mea Young,Kil, Hong Ryang,Kim, Sook Za The Korean Society of Inherited Metabolic Disease 2019 대한유전성대사질환학회지 Vol.19 No.1
목적: 코발라민(Cobalamin)과 동반되지 않은 독립형 메틸말론산혈증(methylmalonic acidemia)은 프로피오네이트 대사 질환으로 상염색체 열성으로 유전된다. Methylmalonyl-CoA mutase (MCM)효소발현에 관련된 유전자인 MMUT에는 유전자 결함에는 두 가지 아형이 있다. $Mut^0$은 효소 활성도가 완전히 없는 것이고 Mut-형은 효소활성도가 저하되어 있지만 hydroxocobalamin (OHCbl) 보충으로 잔여효소의 활성도가 증가될 수 있는 형이다. 본 연구의 목적은 한국인 MMA 환아에서 코발라민의 반응성과 돌연변이를 조사하는 것이다. 방법: 최적의 치료를 위해 MCM 활성도와 비타민 $B_{12}$ 반응성을 측정하기 위하여 섬유 아세포의 체세포 보완 분석을 사용하여 10명의 MMA 환자를 평가했다. MMUT 유전자는 MMA 돌연변이의 염기서열을 확인하였다. 결과: $^{14}C-propionate$의 첨가는 OHCbl에 반응이 없는 모든 환자에서 낮게 나타났다. $^{14}C-methyltetrahydrofolate$와 $^{57}Co-cyanocobalamin$의 투여 후 모두 정상범위 내에 있었다. 아데노 실 코발라민의 합성은 낮지 만 메틸 코발라민의 합성은 적절하였다. 보완 분석 결과 모든 환자들은 $mut^0$ 유형이었다. DNA 염기서열분석결과에서 2개의 새로운 돌연변이, p.Gln267Ter 및 p.Ile697Phe를 포함하여 12개의 상이한 MMUT 돌연변이를 확인하였다. 신생아에서 증상이 나타나며 $mut^0$ 형인 MMA 환자 10례 모두에서 코발라민 반응을 보이지 않았다. 결론: 본 연구에서는 모든 한국 MMA 환자에서 코발라민 반응을 시험한 결과 음성이었다. Purpose: Isolated methylmalonic acidemia (MMA) is an autosomal recessive inherited disorder of propionate metabolism. There are two subtypes of MMUT gene defects. $Mut^0$ represents complete loss of methylmalonyl-CoA mutase (MCM) activity while mut- is associated with residual MCM activity, which can be stimulated by hydroxocobalamin (OHCbl) supplementation. The objective of this study is to investigate cobalamin responsiveness and mutations present in Korean MMA population. Methods: We evaluated 10 MMA patients using somatic cell complementation analysis on their fibroblasts to measure MCM activity and vitamin B12 responsiveness for the optimal treatment. MMUT gene was sequenced to identify the MMA mutations. Results: For all patients, the incorporation of $[^{14}C]-propionate$ was low, and there was no response to OHCbl. The incorporation of $[^{14}C]-methyltetrahydrofolate$ and $[^{57}Co]-CNCbl$ fell within the normal range. There was adequate synthesis of methylcobalamin while the synthesis of adenosylcobalamin was low. The complementation analysis showed all patients were $mut^0$. The sequence analysis identified 12 different MMUT mutations, including 2 novel mutations, p.Gln267Ter and p.Ile697Phe, were identified. All the patients in this study had neonatal onset of symptoms, belonged to $mut^0$ complementation class, and as a result, showed no cobalamin responsiveness. Conclusion: No Korean MMA patient showed cobalamin responsiveness.