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Soybean Oil 및 Flaxseed Oil 첨가 배양시 탄수화물 첨가수준에 의한 반추미생물의 Bio-hydrogenation과 CLA 생성에 미치는 효과 (Effect of the Level of Carbohydrates on Bio-hydrogenation and CLA Production by Rumen Bacteria When Incubated with Soybean Oil or Flaxseed Oil In vitro)

  • 최성호;임근우;김광림;송만강
    • Journal of Animal Science and Technology
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    • 제48권4호
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    • pp.521-532
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    • 2006
  • 본 실험은 대두유 또는 아마유를 in vitro 방법으로 배양 할 때, 탄수화물원의 첨가수준이 반추위 박테리아에 의한 bio-hydrogenation과 CLA 생성에 미치는 효과를 조사하였다. 4수준(0%, 0.3%, 0.6% 그리고 0.9%, w/v)의 혼합된 탄수화물원(glucose, cellobiose, soluble starch, 1:1:1, w/w/w)과 두 종류의 oil을 cellulose powder에 흡착시킨 형태로 각각 60mg씩 인공타액(120ml)과 반추위액(30ml)이 혼합된 배양액(150ml)에 넣은 다음 39℃에서 12시간동안 혐기적으로 배양하였다. 배양액의 pH와 암모니아 농도는 두 종류 oil을 첨가한 배양액 모두에서 탄수화물원의 첨가 수준이 높을수록 pH와 암모니아 농도가 낮았다(P<0.05). 탄수화물원의 첨가 수준이 증가할수록 total VFA 생성량 역시 증가되었으나(P<0.01) 첨가한 oil 간의 차이는 없었다. 배양시간이 경과됨에 따라 탄수화물원의 첨가수준이 높을수록 propionate의 조성비율이 증가된 반면(P<0.001) acetate와 butyrate의 조성비율은 감소되었다. 배양 후 3시간이 경과하였을 때 배양액 내 oleic acid의 조성비율은 대두유에 비하여 아마유를 첨가한 배양액에서 낮았으나(P<0.001) linoleic acid의 비율은 높았다(P<0.001). 이와는 달리 탄수화물원의 수준이 증가될수록 stearic acid(P<0.05), CLA(P<0.01) 및 cis-9, trans-11 CLA(P<0.001)의 조성비율은 감소되었으나, linoleic acid의 조성 비율은 증가되었다(P<0.05). Linolenic acid의 조성비율에 있어서는 첨가된 oil의 종류와 첨가된 탄수화물원의 수준간의 상호작용이 있는 것으로 나타났는데(P<0.001), 12시간의 배양종료 후 대두유 첨가구에 비해 아마유 첨가구에서 stearic acid(P<0.01), oleic acid(P<0.001), 그리고 trans-11C18:1(P<0.01)의 조성비율이 감소된 반면, linoleic acid(P<0.001)와 linolenic acid(P<0.01)의 조성비율은 증가되었다. 탄수화물원의 첨가수준이 증가될수록 stearic acid와 총 CLA의 조성비는 감소되었으나(P<0.01), trans-11-C18:1(P<0.05)와 linoleic acid(P<0.01)의 조성비율은 증가되었다. 배양 12시간 후 배양액 내의 oleic acid (P<0.05), linoleic acid(P<0.05) 및 linolenic acid(P<0.01)의 조성비율에 있어서는 첨가한 oil의 종류와 첨가한 탄수화물원의 수준간의 상호작용이 있었는데, 탄수화물원의 첨가수준이 감소됨에 따라 cis-9, trans-11 CLA와 trans-10, cis-12 CLA의 조성비 역시 감소되는 경향이었으나 첨가한 oil의 종류에 대한 영향은 거의 받지 않았다. 본 실험의 결과를 종합해 볼 때, 탄수화물의 첨가수준과 oil의 첨가는 반추미생물의 bio-hydrogenation 작용 및 CLA 생성에 영향할 수 있는 것으로 여겨진다.

대장균에서 Bacillus subtilis의 Mannanase 유전자 과잉발현 (High-Level Expression of A Bacillus subtilis Mannanase Gene in Escherichia coli.)

  • 권민아;손지영;윤기홍
    • 한국미생물·생명공학회지
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    • 제32권3호
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    • pp.212-217
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    • 2004
  • Glycosyl hydrolase family 26에 속하는 Bacillus subtilis WL-7 mannanase를 코드하는 유전자를 대장균에서 과잉 발현하였다. 아미노 말단의 signal peptide를 포함하거나 포함하지 않은 mannanase 유전자를 각각 pET24a(+)에 도입하여 재조합 플라스미드 pETMAN과 pENS7를 제조하였다. 이들 플라스미드를 함유하는 Escherichia coli BL21(DE3)에서 mannanase를 발현시킨 결과 signal peptide가 제거된 mannanase유전자의 발현량이 매우 높았다. 그러나 배양온도 $37^{\circ}C$에서 pENS7를 함유한 재조합 대장균에서 과잉 발현된 mannanase는 대부분이 불활성 형태로 존재하였으며, 배양온도를 $31^{\circ}C$이하로 하였을 때 수용화 형태의 효소량이 증가하면서 효소활성이 높아졌다. IPTG에 의해 발현된 재조합 대장균의 균체파쇄 상등액 중에 존재하는 mannanase 활성은 배양온도 $25^{\circ}C$~28$^{\circ}C$에서 가장 높았으며, 전체 단백질량을 기준으로 볼 때는 배양온도 $31^{\circ}C$에서 비활성이 가장 높은 것으로 확인되었다.

품종에 따른 돈육의 지방 함량, 지방산 조성 및 전자코에 의한 향기 패턴의 차이 분석

  • 강선문;강창기;이성기
    • 한국축산식품학회:학술대회논문집
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    • 한국축산식품학회 2006년도 정기총회 및 제37차 춘계 국제학술발표대회
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    • pp.107-111
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    • 2006
  • 본 연구는 돼지 품종에 따른 지방 함량, 지방산 조성 및 전자코에 의한 향기 패턴의 차이를 분석하고자 실시하였으며, 평균 생시체중이 111 kg인 멧돼지 5두, 64 kg인 재래흑돼지 5두 및 114 kg인 개량종 돼지($Landrace{\times}Yorkshire$) 5두의 등심(M. longissimus) 부위를 본 실험에 이용하였다. 근내 지방함량은 재래흑돼지가 멧돼지, 개량종 돼지보다 유의적으로 높았다(p<0.05). 지방산 조성에서 C14:0, C18:3n3, MUFA/SFA 비율은 재래흑돼지가 가장 유의적으로 높았으며(p<0.05), C18:1n9는 개량종 돼지가 멧돼지보다 높았다(p<0.05). C16:1n7, C18:1n7, UFA/SFA 비율은 멧돼지가 가장 높았으며(p<0.05), C18:0, C20:1n9는 멧돼지가 가장 낮았다(p<0.05). C20:4 n6은 멧돼지가 개량종 돼지보다 높았고(p<0.05), C22:4n6은 재래흑돼지가 가장 높았으며(p<0.05), C22:5n3은 멧돼지가 재래흑돼지보다 높았다(p<0.05). PUFA n6/n3 비율은 개량종 돼지가 가장 낮았으며(p<0.05), 멧돼지가 재래흑돼지보다 낮았다(p<0.05). 전자코의 PCA에 의한 향기패턴은 신선육에서 품종에 따른 차이가 명확히 나타났으며, 가열육에서는 멧돼지 및 재래흑돼지, 개량종 돼지간에 차이가 나타났다. 따라서 재래흑돼지가 가장 높은 근내 지방 함량 및 가장 낮은 MUFA/SFA 비율을 가졌고, 멧돼지가 가장 높은 UFA/SFA 비율을 가졌으며, 개량종 돼지가 가장 낮은 PUFA n6/n3 비율을 가졌다. 또한 전자코에 의해 돼지 품종에 따른 돈육의 향기 패턴 차이를 효과적으로 분별할 수 있었다.

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암환자를 위한 호스피스 케어에 관한 탐색적 연구 (An Exploratory Study of Hospice Care to Patients with Advanced Cancer)

  • 박혜자
    • 대한간호
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    • 제28권3호
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    • pp.52-67
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    • 1989
  • True nursing care means total nursing care which includes physical, emotional and spiritual care. The modern nursing care has tendency to focus toward physical care and needs attention toward emotional and spiritual care. The total nursing care is mandatory for patients with terminal cancer and for this purpose, hospice care became emerged. Hospice case originated from the place or shelter for the travellers to Jerusalem in medieval stage. However, the meaning of modem hospice care became changed to total nursing care for dying patients. Modern hospice care has been developed in England, and spreaded to U.S.A. and Canada for the patients with terminal cancer. Nowaday, it became a part of nursing care and the concept of hospice care extended to the palliative care of the cancer patients. Recently, it was introduced to Korea and received attention as model of total nursing care. This study was attempted to assess the efficacy of hospice care. The purpose of this study was to prove a difference in terms of physical, emotional a d spiritual aspect between the group who received hospice care and who didn't receive hospice care. The subject for this study were 113 patients with advanced cancer who were hospitalized in the S different hospitals. 67 patients received hospice care in 4 different hospitals, and 46 patients didn't receive hospice care in another 4 different hospitals. The method of this study was the questionaire which was made through the descriptive study. The descriptive study was made by individual contact with 102 patients cf advanced cancer for 9 months period. The measurement tool for questionaire was made by author through the descriptive study, and included the personal religious orientation obtained from chung(originated R. Fleck) and 5 emotional stages before dying from Kubler Ross. The content ol questionaire consisted in 67 items which included 11 for general characteristics, 10 for related condition with cancer, 13 for wishes far physical therapy, 13 for emotional reactions and 20 for personal religious orientation. Data for this study was collected from Aug. 25 to Oct. 6 by author and 4 other nurse's who received education and training by author for the collection of data. The collected data were ana lysed using descriptive statistics, $X^2-test$, t-test and pearson correlation coefficient. Results of the study were as follows: "H.C Group" means the group of patient with cancer who received hospice care. "Non H.C Group" means the group of patient with cancer who did not receive hospice care. 1. There is a difference between H.C Group and Non H.C Group in term of the number of physical symptoms, subjective degree of pain sensation and pain control, subjective beliefs in physical cure, emotional reaction, help of present emotional and spiritual care from other personal, needs of emotional and spiritual care in future, selection of treatment method by patients and personal religious orientation. 2. The comparison of H.C Group and Non H.C Group 1) There is no difference in wishes for physical therapy between two groups(p=.522). Among Non H.C Group, a group, who didn't receive traditional therapy and herb medicine was higher than a group who received these in degree of belief that the traditional therapy and herb medicine can cure their disease, and this result was higher in comparison to H.C Group(p=.025, p=.050). 2) Non H.C Group was higher than H.C Group in degree of emotional reaction(p=.050). H.C Group was higher than Non H.C Group in denial and acceptant stage among 5 different emotional stages before dying described by Kubler Ross, especially among the patient who had disease more than 13 months(p=.0069, p=.0198). 3) Non H.C Group was higher than H. C Group in demanding more emotional and spiritual care to doctor, nurse, family and pastor(p=. 010). 4) Non H.C Group was higher than H.C Group in demanding more emotional and spiritual care to each individual of doctor, nurse and family (p=.0110, p=.0029, P=. 0053). 5) H.C Group was higher th2.n Non H.C Group in degree of intrinsic behavior orientation and intrinsic belief orientation of personal religious orientation(p=.034, p=.026). 6) In H.C Group and Non H.C Group, the degree of emotional demanding of christians was significantly higher than non christians to doctor, nurse, family and pastor(p=. 000, p=.035). 7) In H.C Group there were significant positive correlations as following; (1) Between the degree of emotional demandings to doctor, nurse, family & pastor and: the degree of intrinsic behavior orientation in personal religious orientation(r=. 5512, p=.000). (2) Between the degree of emotional demandings to doctor, nurse. family & pastor and the degree of intrinsic belief orientation in personal religious orientation(r=.4795, p=.000). (3) Between the degree of intrinsic behavior orientation and the degree of intrinsic: belief orientation in personal religious orientation(r=.8986, p=.000). (4) Between the degree of extrinsic religious orientation and the degree of consensus religious orientation in personal religious orientation (r=. 2640, p=.015). In H.C. Group there were significant negative correlations as following; (1) Between the degree of intrinsic behavior orientation and extrinsic religious orientation in personal religious orientation (r=-.4218, p=.000). (2) Between the degree or intrinsic behavior orientation and consensus religious orientation in personal religious orientation(r=-. 4597, p=.000). (3) Between the degree of intrinsic belief orientations and the degree of extrinsic religious orientation in personal religious orientation(r=-.4388, p=.000). (4) Between the degree of intrinsic belief orientation and the degree of consensus religious orientation in personal religious orientation(r=-. 5424, p=.000). 8) In Non H.C Group there were significant positive correlation as following; (1) Between the degree of emotional demandings to doctor, nurse, family & pastor and the degree of intrinsic behavior orientation in personal religious orientation(r= .3566, p=.007). (2) Between the degree of emotional demandings to doctor, nurse, family & pastor and the degree of intrinsic belief orientation in personal religious orientation(r=.3430, p=.010). (3) Between the degree of intrinsic behavior orientation and the degree of intrinsic belief orientation in personal religious orientation(r=.9723, p=.000). In Non H.C Group there were significant negative correlation as following; (1) Between the degree of emotional demandings to doctor, nurse, family & pastor and the degree of extrinsic religious orientation in personal religious orientation(r= -.2862, p=.027). (2) Between the degree of intrinsic behavior orientation and the degree of extrinsic religious orientation in personal religious orientation(r=-. 5083, p=.000). (3) Between the degree of intrinsic belief orientation and the degree of extrinsic religious orientation in personal religious orientation(r=-. 5013, p=.000). In conclusion above datas suggest that hospice care provide effective total nursing care for the patients with terminal cancer, and hospice care is mandatory in all medical institutions.

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The Effects of Dietary Turkish Propolis and Vitamin C on Performance, Digestibility, Egg Production and Egg Quality in Laying Hens under Different Environmental Temperatures

  • Seven, Pinar Tatli
    • Asian-Australasian Journal of Animal Sciences
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    • 제21권8호
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    • pp.1164-1170
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    • 2008
  • In this study, the effects of propolis and vitamin C (L-ascorbic acid) supplementation in diets were investigated on feed intake (FI), body weight (BW), body weight gain (BWG), feed conversion rate (FCR) and digestibility and on egg production and qualities (weight, mortality, shell thickness) in laying hens exposed to heat stress. A total of 150 Hyline White Leghorn, aged 42 weeks, hens was divided into five groups of 30 hens. Chicks were randomly divided into 1 positive control, 1 control and 3 treatment groups. The chicks were kept in cages in temperature-controlled rooms at $22^{\circ}C$ for 24 h/d (positive control, Thermoneutral, TN group) or $34^{\circ}C$ for 9 h/d from 08.00-17.00 h followed by $22^{\circ}C$ for 15 h (control, heat stress, HS group) and fed a basal diet or basal diet supplemented with vitamin C (250 mg/kg of L- ascorbic acid/kg of diet) or two levels of propolis (2 and 5 g of ethanol extracted propolis/kg of diet). Increased FI (p<0.05) and improvement in FCR (p<0.05), hen day egg (p<0.05) and egg weight (p<0.05) were found in Vitamin C and propolis-supplemented laying hens reared under heat stress conditions. Mortality rate was higher in the control group than TN, vitamin C and propolis groups (p<0.05). Digestibility of dry matter, organic matter, crude protein and ether extract improved with increasing of both dietary vitamin C and propolis (p<0.05). Vitamin C or propolis supplementation did not affect either the percentage shape index, yolk index or haugh unit and albumen index (p>0.05). However, the egg shell thickness and egg shell weight appeared to be increased in Vitamin C and propolis groups in comparison to HS group birds (p<0.05). In conclusion, dietary supplementation of laying hens with anti-oxidants (vitamin C and propolis) can attenuate heat stress-induced oxidative damage. These positive effects were evidenced by increased growth performance and digestibility, improvement of egg shell thickness and egg weight in comparison to non-supplemented birds. Moreover, supplementation with propolis (5 g/kg diet) was the most efficient treatment.

농촌 성인 여성의 비만도와 혈청지질에 관한 연구 -무안군을 중심으로- (A Study of Serum Lipids and the Obesity of Housewives in Mooan, Chonnam)

  • 김현아;김현덕;남기선
    • 대한지역사회영양학회지
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    • 제2권3호
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    • pp.319-326
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    • 1997
  • This study was to investigate the level of serum lipids and obesity in housewives in Mooan, and relationships between them. The average age, height and weight were 51.1 years, 154.3cm and 58.3kg respectively. The level of obesity of subjects was higher than that of housewives in other areas investigated in other studies. BMI and RBW underestimated in the obese compared to BIA. The amounts of lean body mass and total body water of subjects above 60 years old were smaller than those of forties and their BMI and RBW were lower than those of forties. However, their body fat percentage was nearly same as that of forties. Serum HDL-cholesterol(HDL-C) of subjects was lower than 55mg/㎗, the lower limit of normal range. Triglyceride of subjects above 60years old was higher than the 95th percentile of those of the same age with reference to Mayo clinic. The serum total cholesterol(TC) showed positive correlation with LDL-cholesterol(LDL-C) and TC/HDL-C(p<0.001). LDL-C showed positive correlation with TC(p<0.001), but negative correlation with HDL-C(p<0.05). HDL-C showed negative correlation with triglyceride(p<0.05). TC/HDL-C showed a positive correlation with triglyceride(p<0.05). HDL-C was the serum lipid which showed the highest correlation with obesity and body composition. It showed a negative correlation with the percentage of body fat(p<0.001), total body fat(p<0.001), BMI(p< 0.05) and RBW(p<0.05). TC/HDL-C showed positive correlation with the percentage of body fat(p<0.01) and total body fat(p<0.05). As the result of above, it is needed to make more researches to find out the proper method to estimate the obesity of subjects and educate subjects in Mooan about nutritional information for obesity.

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Resolution of Molecular Species of the Triacylglycerol Containing Petroselinic Acid $(cis-C_{18:1{\omega}12)$ by Silver Ion-HPLC

  • Joh, Yong-Goe;Kim, Seong-Jin
    • 한국응용과학기술학회지
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    • 제22권4호
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    • pp.339-348
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    • 2005
  • On the analysis of triacylglycerol (TG) from the kernels of Acanthopanax sessiliflorus by reversed phase-HPLC, it was separated into three main fractions of PN 44, 46 and 48, according to partition number (PN). On the contrary, it could be clearly classified into seven fractions of SMM, MMM, SMD, MMD, SDD, MDD and MDT by silver ion-HPLC by the number of double bond in the acyl chains of TG species. But resolution of so-called critical pairs of TG molecular species such as molecular pairs of $P_eLL$ $[C_{18:1{\omega}12}/(C_{18:2{\omega}6)2}]$ and OLL $[C_{18:1{\omega}9}/(C_{18:2{\omega}6)2}]$ and OOL $[(C_{18:1{\omega}9)2}/C_{18:2{\omega}6]$, and $P_eP_eL$ $[(C_{18:2{\omega}12)2}/C_{18:1{\omega}6]$ was not achieved $(P_e;$ petroselinic acid, L; linoleic acid, O; oleic acid). On the other hand, TG extracted from Aralia continentalis kernels were also fractionated into seven groups of SSM, SMM, MMM, SMD, MMD, SDD and MDD (S; saturated acid, M; monoenoic acid, D; dienoic acid) by silver ion-HPLC, although it's were classified into three groups of PN 44, 46 and 48 by reversed phase-HPLC. The fractions of SMM, MMM, MMD and MDD were divided into two subfractions, respectively; the fractions of SMM, MMM, MMD and MDD were resolved into the subfraction of $PP_e/P_e$ and POO (critical pairs from each other), that of $P_e/P_e/P_e$ and OOO, that of $P_e/P_e/L$ and OOL, and that of $P_e/L/L$ and OLL.

P-3C 시뮬레이터를 활용한 해군 고정익조종사 UPRT 훈련 방안에 대한 연구 (A Study on the Upset Prevention & Recovery Training Method for Navy Fixed Wing Pilots Using P-3 Simulator)

  • 이중봉
    • 한국항행학회논문지
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    • 제27권3호
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    • pp.293-299
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    • 2023
  • UPRT(Upset Prevention And Recovery Training)은 2001년과 2011년 사이에 상업항공에서 발생한 항공기 사고 중 주요 원인이 LOC-I(비행 중 조종능력상실; Loss Of Control In flight)으로 분석된 이후 3년의 시간에 걸쳐 개발된 사고 예방 훈련 프로그램이다. 2014년 ICAO는 Doc.10011(Manual On Aeroplane Upset Prevention And Recovery Training)을 통해 고정익항공기에 대한 UPRT를 제시하였고 2019년 3월부터 체약국에 의무시행을 권고하였다. 해군 P-3C는 해상초계와 대잠전이 주요 임무이기 때문에 저고도(70~600 m)에서 비행하는 시간이 많고 P-3C 대다수의 조종사들이 비행착각을 경험한바 비정상 자세 예방 및 회복훈련은 해군 P-3C 조종사에게는 반드시 필요한 훈련이다. 이를 위해 본 연구에서는 해군이 보유하고 있는 P-3C 시뮬레이터를 이용하여 제한적인 상태에서의 비정상 자세 예방 및 회복훈련 방안을 제시하고자 한다.

식품소재로서의 Xylooligosaccharide의 물리화학적 특성 (Physicochemical Properties of Xylooligosaccharide as Food Material)

  • 박윤제;이지완;이창승;류보경;양창근
    • 한국식품과학회지
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    • 제33권1호
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    • pp.19-23
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    • 2001
  • Xylan과 xylanase의 효소반응 후 생성되는 xylooligosaccharide를 식품에 적용하기 위해 점도, 내열성 및 pH 안정성, 착색성, 수분보습력 등의 기본적인 물리화학적 특성을 조사하였다. 70bx xylooligosacchride의 점도는 $20^{\circ}C$에서 930cP, $60^{\circ}C$에서 90 cp로 설탕보다 높았으나, 다른 올리고당류보다는 약간 낮은 특성을 나타내었으며, pH $2.5{\sim}8.0$ 범위에서 $100^{\circ}C$로 1시간 동안 열처리한 후에도 전혀 분해되지 않고 안정하게 잔존하여 내열성과 pH 안정성을 가지고 있음을 알 수 있었다. Xylooligosacchride의 pH는 $5.0{\pm}0.1$로서 pH $2.5{\sim}4.0$에서는 그 착색성이 매우 낮았으며, pH가 증가할수록 착색성이 높아지는 경향을 보였으나, 식품가공에 영향을 미치는 수준은 아니었다. Glycine 1%가 함유된 아미노산 공존하에서의 착색성도 매우 낮았으며 pH에 따라 크게 차이를 보이지 않았다. 수분보습력을 조사하기 위하여 71일간 $25^{\circ}C$, 75% 상대습도에서 xylooligosacchride를 저장 후 측정한 결과 95% 이상의 중량을 유지하고 있어 28%가 감량된 설탕보다 수분손실이 훨씬 적어 수분보습력이 훨씬 큰 것을 알 수 있었다. 이러한 결과로 볼 때 xylooligosacchride는 식품에 사용하기에 적합한 물성을 가지고 있어 기능성 증진 효과를 나타내고자 하는 다양한 식품에 응용 가능할 것으로 판단된다.

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THE SPHERICAL NON-COMMUTATIVE TORI

  • Boo, Deok-Hoon;Oh, Sei-Qwon;Park, Chun-Gil
    • 대한수학회지
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    • 제35권2호
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    • pp.331-340
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    • 1998
  • We define the spherical non-commutative torus $L_{\omega}$/ as the crossed product obtained by an iteration of l crossed products by actions of, the first action on C( $S^{2n+l}$). Assume the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus $A_{p}$ with a matrix algebra $M_{m}$ ( ) (m > 1). We prove that $L_{\omega}$/ $M_{p}$ (C) is not isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{mp}$ (C), and that the tensor product of $L_{\omega}$/ with a UHF-algebra $M_{p{\infty}}$ of type $p^{\infty}$ is isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{m}$ (C) $M_{p{\infty}}$ if and only if the set of prime factors of m is a subset of the set of prime factors of p. Furthermore, it is shown that the tensor product of $L_{\omega}$/, with the C*-algebra K(H) of compact operators on a separable Hilbert space H is not isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{m}$ (C) K(H) if Prim( $L_{\omega}$/) is homeomorphic to $L^{k}$ (n)$\times$ $T^{l'}$ for k and l' non-negative integers (k > 1), where $L^{k}$ (n) is the lens space.$T^{l'}$ for k and l' non-negative integers (k > 1), where $L^{k}$ (n) is the lens space.e.

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