Testing Isotropy of Cosmic Microwave Background Radiation

We introduce new symmetry-based methods to test for isotropy in cosmic microwave background radiation. Each angular multipole is factored into unique products of power eigenvectors, related multipoles and singular values that provide 2 new rotationally invariant measures mode by mode. The power entropy and directional entropy are new tests of randomness that are independent of the usual CMB power. Simulated galactic plane contamination is readily identified, and the new procedures mesh perfectly with linear transformations employed for windowed-sky analysis. The ILC -WMAP data maps show 7 axes well aligned with one another and the direction Virgo. Parameter free statistics find 12 independent cases of extraordinary axial alignment, low power entropy, or both having 5% probability or lower in an isotropic distribution. Isotropy of the ILC maps is ruled out to confidence levels of better than 99.9%, whether or not coincidences with other puzzles coming from the Virgo axis are included. Our work shows that anisotropy is not confined to the low l region, but extends over a much larger l range.Comment: 40 pages 15 figure

Precocious puberty: a clinical review

Precocious puberty is defined as pubertal development occurring more than 2.5 standard deviations earlier than the average age. It may comprise of central or gonadotropin-dependent precocious puberty and peripheral or gonadotropin-independent precocious puberty. Variants of precocious puberty include premature thelarche, premature pubarche and isolated premature menarche which principally implies onset of menstruation without any other signs of sexual development. Precocious puberty may have long-term consequences including short stature later on in adulthood owing to premature epiphyseal fusion as also psychosocial problems. Evaluation includes a detailed history, physical examination, biochemical tests and imaging directed towards detecting the cause. Gonadotropin Releasing Hormone (GnRH) analogues are effective for treatment of central precocious puberty. Treatment of peripheral precocious puberty should be based on the cause. Isolated variants are usually normal but should be closely monitored. Multi-speciality consultation with involvement of pediatricians and enocrinologists may improve treatment outcomes in these children, who otherwise pose significant challenges to the gynaecologist

Foreign body in vagina: a cause of persistent vaginal discharge in children

Vulvovaginitis and vaginal discharge in pediatric patients, while not uncommon, is commonly believed to be due to such causes as absence of the protective effect on the vaginal mucosa. However, other causes need also to be kept in mind. We report a case of chronic vaginal discharge in a 5 yr old, who had retained a foreign body in her vagina for 6-7 months

Comparative analysis of two discretizations of Ricci curvature for complex networks

We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.Comment: Published version. New results added in this version. Supplementary tables can be freely downloaded from the publisher websit