Diversity in substance use behaviour among street children of Delhi under Bayesian paradigm

Background Shannon’s index is one of the measures of biodiversity, which is intended to quantify both richness and evenness of the species/individuals in the ecosystem or community. However, application of Shannon’s index in the field of substance use among the street children has not been done till date. Methods This paper is concerned with methods of estimating Shannon’s diversity index (SDI), which can be used to capture the variation in the population due to certain characteristics. Under the consideration that the probability of abundance, based on certain characteristics in the population, is a random phenomenon, we derive a Bayesian estimate in connection with Shannon’s information measure and their properties (mean and variance), by using a probability matching prior, through simulation and compared it with those of the classical estimates of Shannon. The theoretical framework has been applied to the primary survey data of substance use among the street children in Delhi, collected during 2015. The measure of diversity was estimated across different age profiles and districts. Results The results unrevealing the diversity estimate for street children corresponding to each region of Delhi, under both the classical and Bayesian paradigms. Although the estimates were close to one another, a striking difference was noted in the age profile of children. Conclusions The Bayesian methodology provided evidence for a greater likelihood of finding substance-using street children, belonging to the lower age group (7-10, maximum Bayesian entropy-3.73), followed by the middle (11-14) and upper age group (15-18). Moreover, the estimated variance under the Bayesian paradigm was lesser than that of the classical estimate. There is ample scope for further refinement in these estimates, by considering more covariates that may have a possible role in initiating substance use among street children in developing countries like India.

A semi-classical estimate for the q-parameter and decay time with Tsallis entropy of black holes in quantum geometry

In this letter, using the non-extensive entropy of Tsallis, we study some properties of the Schwarzschild black holes (BHs), based on the loop quantum gravity (LQG), some novel characteristics and results of the Schwarzschild BH can be obtained in Mejrhit and Ennadifi (Phys Lett B 794:45–49, 2019). Here we find that these findings are strikingly identical to ones obtained by Hawking and Page in anti-de Sitter space within the original of the Boltzmann entropy formula. By using the semi-classical estimate analysis on the energy at this minimum $$M_{min}$$ M min , an approximate relationship between the q and $$\gamma $$ γ parameters of BHs can be found, ($$q\approx \frac{\sqrt{3}\gamma }{\pi \ln 2}+1$$ q ≈ 3 γ π ln 2 + 1 ), which is remarkable approaching to q-parameters of cosmic ray spectra and quarks coalescing to hadrons in high energy.

Modular invariants and isogenies

We provide explicit bounds on the difference of heights of the [Formula: see text]-invariants of isogenous elliptic curves defined over [Formula: see text]. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman’s inequality and isogeny estimates by Gaudron and Rémond. We give applications in the study of Vélu’s formulas and of modular polynomials.

INTEGRAL FORMULATION OF 3D NAVIER–STOKES AND LONGER TIME EXISTENCE OF SMOOTH SOLUTIONS

We extend Borel summability methods to the analysis of the 3D Navier–Stokes initial value problem, [Formula: see text] where [Formula: see text] is the Hodge projection to divergence-free vector fields. We assume that the Fourier transform norms [Formula: see text] and [Formula: see text] are finite. We prove that the integral equation obtained from (*) by Borel transform and Écalle acceleration, Û (k, q), is exponentially bounded for q in a sector centered on ℝ+, where q is the inverse Laplace dual to 1/tn for n ≥ 1. This implies in particular local existence of a classical solution to (*) for t ∈ (0, T), where T depends on [Formula: see text] and [Formula: see text]. Global existence of the solution to NS follows if ‖Û(⋅, q)‖l1 has subexponential bounds as q → ∞. If f = 0, then the converse is also true: if NS has global solution, then there exists n ≥ 1 for which ‖Û(⋅, q)‖ necessarily decays. More generally, if the exponential growth rate in q of Û is α, then a classical solution to NS exists for t ∈ (0, α-1/n). We show that α can be better estimated based on the values of Û on a finite interval [0, q0]. We also show how the integral equation can be solved numerically with controlled errors. Preliminary numerical calculations of the integral equation over a modest [0, 10], q-interval for n = 2 corresponding to Kida ([21]) initial conditions, though far from being optimized or rigorously controlled, suggest that this approach gives an existence time for 3D Navier–Stokes that substantially exceeds classical estimate.

Does glycosyl transfer involve an oxacarbenium intermediate? Computational simulation of the lifetime of the methoxymethyl cation in water

2D free-energy surfaces for transfer of the methoxymethyl cation between two water molecules are constructed from molecular dynamics (MD) simulations in which these atoms are treated quantum-mechanically within a box of 1030 classical solvent water molecules at 300 K. This provides a simple model for glycosyl transfer in water. The AM1/TIP3P surfaces with 2D-spline corrections at either MPWB1K/6-31+G(d,p) or MP2/6-31+G(d,p) contain a shallow free-energy well corresponding to an oxacarbenium ion intermediate in a DN*AN mechanism. MD analysis at three temperatures leads to a classical estimate of the lifetime of the methoxymethyl cation in water; when quantum corrections for vibrational zero-point energy are included, the lifetime is estimated to be about 1 ps, in agreement with the best experimental estimate. This suggests that computational simulation, with appropriate high-level correction, is a reliable tool to obtain detailed and reliable mechanistic descriptions for glycosidases. In view of the importance of developing improved anti-influenza drugs, simulations of sialidases that considered both sialyl oxacarbenium ion and covalent sialyl-enzyme as possible intermediates could provide particular insight.

An estimate for the Mellin transform of powers of Hardy's function.

International audience We show that a certain modified Mellin transform $\mathcal M(s)$ of Hardy's function is an entire function. There are reasons to connect $\mathcal M(s)$ with the function $\zeta(2s-1/2)$, and then the orders of $\mathcal M(s)$ and $\zeta(s)$ should be comparable on the critical line. Indeed, an estimate for $\mathcal M(s)$ is proved which in the particular case of the critical line coincides with the classical estimate of the zeta-function.

Approximation of Functions by Polynomials in C[-L, 1]

A pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].

“SUPERRADIANCE” EFFECT IN A GRAVITATIONAL ANTENNA

I show that the cross-section for absorption of a gravitational wave in a massive resonating antenna is greatly enhanced (about six orders of magnitude) upon its classical estimate if the atoms of the antenna perform coherent oscillations, as predicted by the new approach to “superradiance” in condensed matter.