On Rational Sets in Euclidean Spaces and Spheres

IFor a positive rational $l$, we define the concept of an $l$-elliptic and an $l$-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite $l$-hyperbolic and $l$-ellitpic rationals subsets of the real line and unit circle. For the case of a circle, we prove that the existence of such sets depends on the positivity of ranks of certain associated elliptic curves. We also determine the closures of such sets which are maximal in case they are not dense. In higher dimensions, we show the existence of $l$-ellitpic and $l$-hyperbolic rational infinite sets in unit spheres and Euclidean spaces for certain values of $l$ which satisfy a weaker condition regarding the existence of elements of order more than two, than the positivity of the ranks of the same associated elliptic curves. We also determine their closures. A subset $T$ of the $k$-dimensional unit sphere $S^k$ has an antipodal pair if both $x,-x\in T$ for some $x\in S^k$. In this article, we prove that there does not exist a dense rational set $T\subset S^2$ which has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of general type. We actually show that the existence of such a dense rational set in $S^k$ is equivalent to the existence of a dense $2$-hyperbolic rational set in $S^k$ which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space $\mathbb{R}^k$.Comment: 20 page

Friedmann model with viscous cosmology in modified f(R,T)f(R,T) gravity theory

In this paper, we introduce bulk viscosity in the formalism of modified gravity theory in which the gravitational action contains a general function $f(R,T)$, where $R$ and $T$ denote the curvature scalar and the trace of the energy-momentum tensor, respectively within the framework of a flat Friedmann-Robertson-Walker model. As an equation of state for prefect fluid, we take $p=(\gamma-1)\rho$, where $0 \leq \gamma \leq 2$ and viscous term as a bulk viscosity due to isotropic model, of the form $\zeta =\zeta_{0}+\zeta_{1}H$, where $\zeta_{0}$ and $\zeta_{1}$ are constants, and $H$ is the Hubble parameter. The exact non-singular solutions to the corresponding field equations are obtained with non- viscous and viscous fluids, respectively by assuming a simplest particular model of the form of $f(R,T) = R+2f(T)$, where $f(T)=\alpha T$ ( $\alpha$ is a constant). A big-rip singularity is also observed for $\gamma<0$ at a finite value of cosmic time under certain constraints. We study all possible scenarios with the possible positive and negative ranges of $\alpha$ to analyze the expansion history of the universe. It is observed that the universe accelerates or exhibits transition from decelerated phase to accelerated phase under certain constraints of $\zeta_0$ and $\zeta_1$. We compare the viscous models with the non-viscous one through the graph plotted between scale factor and cosmic time and find that bulk viscosity plays the major role in the expansion of the universe. A similar graph is plotted for deceleration parameter with non-viscous and viscous fluids and find a transition from decelerated to accelerated phase with some form of bulk viscosity.Comment: 19 pages, 3 figures, the whole paper has been revised to improve the quality of paper. Some references added. arXiv admin note: text overlap with arXiv:1307.4262 by other author

Dark sector interaction: a remedy of the tensions between CMB and LSS data

The well-known tensions on the cosmological parameters $H_0$ and $\sigma_8$ within the $\Lambda$CDM cosmology shown by the Planck-CMB and LSS data are possibly due to the systematics in the data or our ignorance of some new physics beyond the $\Lambda$CDM model. In this letter, we focus on the second possibility, and investigate a minimal extension of the $\Lambda$CDM model by allowing a coupling between its dark sector components (dark energy and dark matter). We analyze this scenario with Planck-CMB, KiDS and HST data, and find that the $H_0$ and $\sigma_8$ tensions disappear at 68\% CL. In the joint analyzes with Planck, HST and KiDS data, we find non-zero coupling in the dark sector up to 99\% CL. Thus, we find a strong statistical support from the observational data for an interaction in the dark sector of the Universe while solving the $H_0$ and $\sigma_8$ tensions simultaneously.Comment: 5 pages, 3 figure

You Manage What You Measure: Using Mobile Phones to Strengthen Outcome Monitoring in Rural Sanitation

This paper addresses the sanitation challenge in India, where it is home to the majority of people defecating in the open in the world and also one of the top rapidly growing emerging economies. The paper focuses on the need for a reliable and timely monitoring system to ensure investments in sanitation lead to commensurate outcomes