Genericity aspects in gravitational collapse to black holes and naked singularities

We investigate here the genericity and stability aspects for naked singularities and black holes that arise as the final states for a complete gravitational collapse of a spherical massive matter cloud. The form of the matter considered is a general Type I matter field, which includes most of the physically reasonable matter fields such as dust, perfect fluids and such other physically interesting forms of matter widely used in gravitation theory. We first study here in some detail the effects of small pressure perturbations in an otherwise pressure-free collapse scenario, and examine how a collapse evolution that was going to the black hole endstate would be modified and go to a naked singularity, once small pressures are introduced in the initial data. This allows us to understand the distribution of black holes and naked singularities in the initial data space. Collapse is examined in terms of the evolutions allowed by Einstein equations, under suitable physical conditions and as evolving from a regular initial data. We then show that both black holes and naked singularities are generic outcomes of a complete collapse, when genericity is defined in a suitable sense in an appropriate space.Comment: 24 pages, 6 figures, some changes in text and figures to match the version accepted for publication by IJMP

Stability of Naked Singularity arising in gravitational collapse of Type I matter fields

Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary $C^{2}$- mass function $\textit{M}(r,v)$ and a $C^{1}$- function $h(r,v)$ (through the corresponding $C^{1}$- metric function $\nu(t,r)$), there exist infinitely many choices of energy distribution function $b(r)$ such that the `true' initial data ($\textit{M},h(r,v)$) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole and naked singularity form a "big" subset of the true initial data set, their occurrence is not generic. The terms `stability' and `genericity' are appropriately defined following the theory of dynamical systems. The particular case of radial pressure $p_{r}(r)$ has been illustrated in details to get clear picture of how naked singularity is formed and how, it is stable with respect to initial data.Comment: 16 pages, no figure, Latex, submitted to Praman

Singular normal form for the Painlev\'e equation P1

We show that there exists a rational change of coordinates of Painlev\'e's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlev\'e property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlev\'e property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures

Individual-level predictors of young children’s aspirations

Often young children already have some ideas about what they want to do in the future. Using data from a large UK cohort study, we investigated the individual determinants of seven-year-old children’s aspirations, controlling for parental socio-economic background and parental involvement in learning. At age 7, not all children’s aspirations were unrealistic (55.6% of children aspired to common occupations), few (1.2%) were fantasy, but most were gender-typical. White children had lower occupational aspirations and were more likely to have uncertain future orientations than other ethnic groups. The antecedents of fantasy aspirations, more typical of younger children, were difficult temperament and low school engagement. Uncertain aspirations were related to higher cognitive ability but also to lower school engagement. Higher occupational aspirations were directly related to higher family socio-economic status, and higher occupational and more intrinsic aspirations were associated with more school engagement (in turn, higher in girls and ethnic minority children). Boys, compared to girls, had lower, more extrinsic and more masculine aspirations, but were also more likely than girls to aspire to rare jobs and have fantasy or uncertain aspirations

The influence of grandparents’ social class on children’s aspirations

Social class mobility from grandparent to grandchild is a relatively neglected topic. Grandparents today are often healthier and more active, and have longer relationships with their grandchildren than in previous generations. We used data from the UK’s Millennium Cohort Study (n = 8570) to investigate the influence of maternal and paternal grandparents’ social class on the aspirations of children at age seven. Using path analysis and controlling for family income, mother’s and father’s education, lone motherhood, and child’s ethnicity and gender, we found very small direct effects from the paternal grandmother’s social class to the grandchild’s classed aspirations, and small, indirect effects, via parents’ class, of grandparents’ class on child’s classed aspirations. Multi-group analyses found few differences by ethnicity and gender. There was no evidence that, at this age, mixed-class parentage raises the aspirations of working-class children (the ‘sunken middle-class’ hypothesis)

A novel approach for laparoscopic retrieval of dermoid cyst

A dermoid cyst is a germ cell tumour, benign in nature and common in young women under the age of 30. They are usually asymptomatic and often detected incidentally on imaging. Resection of the cyst is important owing to complications like torsion or malignancy. Laparoscopic surgery is the modality of choice in today’s era. However, rupture of the cyst during laparoscopic retrieval is any surgeon’s nightmare. Hence, good surgical skill and a reliable method of retrieval are of prime importance here. Authors present a novel method of dermoid extraction done laparoscopically with a sterile urine collection (urobag) bag. This is an easy and simple method which decreases the rate of spillage owing to the strength of the bag