Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis Keeler) have been incorporated. The formulation of some results has been improve

Non-Extensive Bose-Einstein Condensation Model

The imperfect Boson gas supplemented with a gentle repulsive interaction is completely solved. In particular it is proved that it has non-extensive Bose-Einstein condensation, i.e., there is condensation without macroscopic occupation of the ground state (k=0) level

Some Global Characteristics of the Galactic Globular Cluster System

The relations between the luminosities $M_{V}$, the metallicities $[Fe/H]$, the Galactocentric radii $R$, and the central concentration indices $c$ of Galactic globular clusters are discussed. It is found that the most luminous clusters rarely have collapsed cores. The reason for this might be that the core collapse time scales for such populous clusters are greater than the age of the Galaxy. Among those clusters, for which the structure has not been modified by core collapse, there is a correlation between central concentration and integrated luminosity, in the sense that the most luminous clusters have the strongest central concentration. The outermost region of the Galaxy with $R>10$ kpc was apparently not able to form metal-rich $([Fe/H]>-1.0)$ globular clusters, whereas such clusters (of which Ter 7 is the prototype) were able to form in some nearby dwarf spheroidal galaxies. It is not yet clear how the popular hypothesis that globular clusters were initially formed with a single power law mass spectrum can be reconciled with the observation that both (1) Galactic globular clusters with $R>80$ kpc, and (2) the globulars associated with the Sagittarius dwarf, appear to have bi-modal luminosity functions.Comment: 15 pages, 1 figur

Non-home prepared foods : contribution to energy and nutrient intake of consumers living in two low-income areas in Nairobi

Objective: To determine the nutritional importance of non-home prepared foods for men, women and schoolchildren living in two low-income residential areas of Nairobi, and the sources of these non-home prepared foods. Design, setting and subjects: A survey was conducted in Korogocho, a slum area, and Dandora, a low-middle-income residential area. Some 241 men, 254 women and 146 children aged 9 to 14 years were included in the study. Food intake was measured using three 24-hour recalls per individual, with special attention on the sources of all foods consumed. Results: The median proportion of daily energy intake of consumers provided by non-home prepared foods ranged from 13% for schoolchildren in Korogocho to 36% for men in Dandora. The median contribution to fat intake was higher than to energy, but the contributions to iron and vitamin A intakes were lower than to energy intake. Men consumed more non-home prepared foods on weekdays than at the weekend. Intakes of energy and most nutrients were below Kenyan Recommended Daily Intakes in all groups, but similar for consumers and non-consumers. In Korogocho, street foods were the main source of non-home prepared foods. In Dandora, both kiosks and street foods were major sources. Conclusions: Non-home prepared foods are an important source of energy and nutients for men, women and schoolchildren in Nairobi. In Korogocho, street foods, and in Dandora, both kiosks and street foods are the main sources of non-home prepared foods. The adequacy of energy and nutrient intakes does not differ between consumers and non-consumers of non-home prepared foods

Intra-cavitary uterine pathology in women with abnormal uterine bleeding: a prospective study of 1220 women.

OBJECTIVES: Our primary aim was to assess how patients characteristics, bleeding pattern, sonographic endometrial thickness (ET) and additional features at unenhanced ultrasound examination (UTVS) and at fluid instillation sonography (FIS) contribute to the diagnosis of intracavitary uterine pathology in women presenting with abnormal uterine bleeding (AUB). We further aimed to report the prevalence of pathology in women presenting with AUB. METHODS: 1220 consecutive women presenting with AUB underwent UTVS, colour Doppler imaging (CDI) and FIS. Most women (n = 1042) had histological diagnosis. RESULTS: Mean age was 50 years and 37% were postmenopausal. Of 1220 women 54% were normal, polyps were diagnosed in 26%, intracavitary fibroids in 11%, hyperplasia without atypia in 4% and cancer in 3%. All cancers were diagnosed in postmenopausal (7%) or perimenopausal (1%) women. ET had a low predictive value in premenopausal women (LR+ and LR- of 1.34 and 0.74, respectively), while FIS had a LR+ and LR- of 6.20 and 0.24, respectively. After menopause, ET outperformed all patient characteristics for the prediction of endometrial pathology (LR+ and LR- of 3.13 and 0.24). The corresponding LR+ and LR- were 10.85 and 0.71 for CDI and 8.23 and 0.26 for FIS. CONCLUSION: About half of the women presenting to a bleeding clinic will have pathology. In premenopausal women, benign lesions are often the cause of AUB. For the prediction of intracavitary pathology ET is of little value in premenopausal women. CDI and FIS substantially improve the diagnostic accuracy