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For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.
Pattern analysis has emerged as a tool to depict the role of multiple nutrients/foods in relation to health outcomes. The present study aimed at extracting nutrient patterns with respect to breast cancer (BC) aetiology.
Design
Nutrient patterns were derived with treelet transform (TT) and related to BC risk. TT was applied to twenty-three log-transformed nutrient densities from dietary questionnaires. Hazard ratios (HR) and 95 % confidence intervals computed using Cox proportional hazards models quantified the association between quintiles of nutrient pattern scores and risk of overall BC, and by hormonal receptor and menopausal status. Principal component analysis was applied for comparison.
Setting
The European Prospective Investigation into Cancer and Nutrition (EPIC).
Subjects
Women (n 334 850) from the EPIC study.
Results
The first TT component (TC1) highlighted a pattern rich in nutrients found in animal foods loading on cholesterol, protein, retinol, vitamins B12 and D, while the second TT component (TC2) reflected a diet rich in β-carotene, riboflavin, thiamin, vitamins C and B6, fibre, Fe, Ca, K, Mg, P and folate. While TC1 was not associated with BC risk, TC2 was inversely associated with BC risk overall (HRQ5 v. Q1=0·89, 95 % CI 0·83, 0·95, Ptrend<0·01) and showed a significantly lower risk in oestrogen receptor-positive (HRQ5 v. Q1=0·89, 95 % CI 0·81, 0·98, Ptrend=0·02) and progesterone receptor-positive tumours (HRQ5 v. Q1=0·87, 95 % CI 0·77, 0·98, Ptrend<0·01).
Conclusions
TT produces readily interpretable sparse components explaining similar amounts of variation as principal component analysis. Our results suggest that participants with a nutrient pattern high in micronutrients found in vegetables, fruits and cereals had a lower risk of BC.
Using two complementary X-ray galaxy cluster studies we present new
cosmological constraints on dark energy. Using Chandra measurements
of the X-ray gas mass fraction, fgas, we obtain a detection
of the effects of dark energy comparable in significance to recent
type Ia supernovae (SNIa) studies. Using X-ray luminosity function
(XLF) data from galaxy cluster surveys, we obtain the first
interesting determination of the dark energy equation of state from
measurements of the growth of cosmic structure in clusters. Both of
our experiments provide strong, independent support to the
ΛCDM paradigm.
The Supernova Working Group was re-established at the IAU XXV General Assembly in Sydney, 21 July 2003, sponsored by Commissions 28 (Galaxies) and 47 (Cosmology). Here we report on some of its activities since 2005.
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