Double bubbles in the three-torus

13 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200113 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200113 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200113 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200

Evaluation by ultrasound of the velocity time integral in carotid stenosis

Abstract Funding Acknowledgements Type of funding sources: None. Introduction  Atherosclerotic disease is one of the main causes of cerebrovascular accident. Doppler ultrasound is one of the most important tools in the diagnosis and quantification of such issues. The classics variables to assess through this method are peak systolic velocity (PSV) at the highest stenosis carotid level, diastolic velocity at such level and the PSV rate between the PSV at the stenosis level of the internal carotid artery (ICA) respect the PSV at the common carotid artery (CCA). Objectives Compare the correlation of the velocity time integral (VTI) and the ratio between VTI of ICA and CCA to identify carotid stenosis by carotid ultrasound. Material and methods We included patients that assist at the institution for a carotid ultrasound to identify a carotid stenosis, between 2017 and 2020. Correlation tests between continuous variables were applied according to the type of distribution. Pearson"s test for continuous variables with normal distribution and Spearman"s for those with abnormal distribution.Fig. 1 Results Three hundred and forty two patients were included, 78% male, mean age 67 ± 8. According to its distribution, the VTI ACI / CCA ratio obtained as the median is 5 cm (1.27-8). By dividing the VTI ratio by quartiles, we observe that the percentage of individuals with a higher degree of carotid obstruction, increase significantly as the quartile increase: 0%, 36.5%, 94.2%, 98.8 %. p: 0.0001. Likewise, a high correlation value between the classical quantitative values and VTI and ratio of VTI at different grade of carotid stenosis was seen by carotid ultrasound (Table 1). Conclusions VTI and VTI ratio are parameters that show good correlation with the classical variables to assess a carotid stenosis by carotid ultrasound. In future investigations, the evaluation of these parameters could lead the estimation of the stenosis area, considering the principles of the continuity equation. Abstract Figure. Fig 1  Abstract Table. 1 </jats:sec

Experiences with HPTN 067/ADAPT Study-Provided Open-Label PrEP Among Women in Cape Town: Facilitators and Barriers Within a Mutuality Framework

Placebo-controlled trials of pre-exposure prophylaxis (PrEP) have reported challenges with study-product uptake and use, with the greatest challenges reported in studies with young women in sub-Saharan Africa. We conducted a qualitative sub-study to explore experiences with open-label PrEP among young women in Cape Town, South Africa participating in HTPN 067/Alternative Dosing to Augment Pre-Exposure Prophylaxis Pill Taking (ADAPT). HPTN 067/ADAPT provided open label oral FTC/TDF PrEP to young women in Cape Town, South Africa who were randomized to daily and non-daily PrEP regimens. Following completion of study participation, women were invited into a qualitative sub-study including focus groups and in-depth interviews. Interviews and groups followed a semi-structured guide, were recorded, transcribed, and translated to English from isiXhosa, and coded using framework analysis. Sixty of the 179 women enrolled in HPTN 067/ADAPT participated in either a focus group (six groups for a total of 42 participants) or an in-depth interview (n = 18). This sample of mostly young, unmarried women identified facilitators of and barriers to PrEP use, as well as factors influencing study participation. Cross-cutting themes characterizing discourse suggested that women placed high value on contributing to the well-being of one’s community (Ubuntu), experienced a degree of skepticism towards PrEP and the study more generally, and reported a wide range of approaches towards PrEP (ranging from active avoidance to high levels of persistence and adherence). A Mutuality Framework is proposed that identifies four dynamics (distrust, uncertainty, alignment, and mutuality) that represent distinct interactions between self, community and study and serve to contextualize women’s experiences. Implications for better understanding PrEP use, and non-use, and intervention opportunities are discussed. In this sample of women, PrEP use in the context of an open-label research trial was heavily influenced by underlying beliefs about safety, reciprocity of contributions to community, and trust in transparency and integrity of the research. Greater attention to factors positioning women in the different dynamics of the proposed Mutuality Framework could direct intervention approaches in clinical trials, as well as open-label PrEP scale-up