On colouring point visibility graphs

In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of $G$). In this paper, we prove that the maximum planar obstacle number of a planar graph of order $n$ is $n-3$, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least $3$ is $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Visibility based angular power spectrum estimation in low frequency radio interferometric observations

We present two estimators to quantify the angular power spectrum of the sky signal directly from the visibilities measured in radio interferometric observations. This is relevant for both the foregrounds and the cosmological 21-cm signal buried therein. The discussion here is restricted to the Galactic synchrotron radiation, the most dominant foreground component after point source removal. Our theoretical analysis is validated using simulations at 150 MHz, mainly for GMRT and also briefly for LOFAR. The Bare Estimator uses pairwise correlations of the measured visibilities, while the Tapered Gridded Estimator uses the visibilities after gridding in the uv plane. The former is very precise, but computationally expensive for large data. The latter has a lower precision, but takes less computation time which is proportional to the data volume. The latter also allows tapering of the sky response leading to sidelobe suppression, an useful ingredient for foreground removal. Both estimators avoid the positive bias that arises due to the system noise. We consider amplitude and phase errors of the gain, and the w-term as possible sources of errors . We find that the estimated angular power spectrum is exponentially sensitive to the variance of the phase errors but insensitive to amplitude errors. The statistical uncertainties of the estimators are affected by both amplitude and phase errors. The w-term does not have a significant effect at the angular scales of our interest. We propose the Tapered Gridded Estimator as an effective tool to observationally quantify both foregrounds and the cosmological 21-cm signal.Comment: 20 pages, 15 figures, 1 table.One typo corrected in Fig.13. Accepted for publication in MNRA

Reviving sub-TeV SU(2)LSU(2)_L lepton doublet Dark Matter

In this work we study the hybrid kind of dark matter(DM) production mechanism where both thermal and non-thermal contribution at two different epochs set the DM relic abundance. This hybrid set up in turn shifts the parameter space of DM in contrast to pure thermal DM scenario. We review such production mechanism in the context of the $SU(2)_L$ lepton doublet dark matter ($\Psi$) augmented with an additional singlet dark scalar ($S$). The neutral component of the dark doublet can serve as a stable DM candidate and in pure thermal scenario, it is under-abundant as well as excluded from direct detection constraints due to its strong gauge interactions in the sub-TeV mass regime. However, in addition to the thermal contribution, the late time non-thermal DM production from the decay of the long-lived dark scalar $S$ helps to fulfill the deficit in DM abundance. On the other hand, the strong gauge mediated direct detection constraint can be evaded with the help of a $SU(2)_L$ triplet scalar(with $Y=2$), resulting a pseudo-Dirac DM. To realize our proposed scenario we impose a discrete $\mathcal{Z}_2$ symmetry under which both $\Psi$ and $S$ are odd while rest of the fields are even. We find the lepton doublet pseudo-Dirac DM with mass $\sim 550-1200$ GeV, compatible with the observed relic density, direct, indirect, and existing collider search constraints.Comment: 15 page

CMB signature of non-thermal Dark Matter produced from self-interacting dark sector

The basic idea of this work is to achieve the observed relic density of a non-thermal dark matter(DM) and its connection with Cosmic Microwave Background (CMB) via additional relativistic degrees of freedom which are simultaneously generated during the period $T_{\rm BBN}~{\rm to}~T_{\rm CMB}$ from a long-lived dark sector particle. To realize this phenomena we minimally extend the type-I seesaw scenario with a Dirac fermion singlet($\chi$) and a complex scalar singlet ($\varphi$) which transform non-trivially under an unbroken symmetry $\mathcal{Z}_3$. $\chi$ being the lightest particle in the dark sector acts as a stable dark matter candidate while the next to lightest state $\varphi$ operates like a long lived dark scalar particle. The initial density of $\varphi$ can be thermally produced through either self-interacting number changing processes ($3 \varphi \to 2 \varphi$) within dark sector or the standard annihilation to SM particles ($2 \varphi \to 2~ {\rm SM}$). The late time (after neutrino decoupling) non-thermal decay of $\varphi$ can produce dark matter in association with active neutrinos. The presence of extra relativistic neutrino degrees of freedom at the time of CMB can have a significant impact on $\Delta \rm N_{eff}$. Thus the precise measurement of $\Delta \rm N_{ eff}$ by current PLANCK 2018 collaboration and future experiments like SPT-3G and CMB-S4 can indirectly probe this non-thermal dark matter scenario which is otherwise completely secluded due to its tiny coupling with the standard model.Comment: Accepted for publication in JCA

Performance Evaluation of Open-Ended ELSS Mutual Fund Schemes in India during Recession

This study examined the performance of the open-ended equity linked saving mutual fund schemes (ELSS) in India during the financial distress in 2008-2009. In this study, the month end net asset values of selected equity linked saving mutual fund schemes were considered and the data was obtained from the website of Association of Mutual Funds in India (AMFI). This paper empirically examined the risk-adjusted, selectivity, diversification and market-timing performances of the open-ended (ELSS) mutual fund schemes. Here, Sharpe, Treynor, Jensen and Treynor & Mazuy models were used to measure the above stated performances. It was observed that the Sharpe and Treynor ratios of the open-ended mutual fund schemes (ELSS) were negative during the recession. Similarly, the stock-selection and market-timing performances of the managers were statistically insignificant and finally, the diversification performances of the mutual fund schemes (ELSS) were found to be unsatisfactory. Therefore, it might be concluded that the overall performances of the open-ended mutual fund schemes (ELSS) were very unsatisfactory during the recession.Keywords: Mutual Fund, Performance, Recession, Sharpe model, Treynor mode

The prospects of measuring the angular power spectrum of the diffuse Galactic synchrotron emission with SKA1 Low

The diffuse Galactic synchrotron emission (DGSE) is the most important diffuse foreground component for future cosmological 21-cm observations. The DGSE is also an important probe of the cosmic ray electron and magnetic field distributions in the turbulent interstellar medium (ISM) of our Galaxy. In this paper we briefly review the Tapered Gridded Estimator (TGE) which can be used to quantify the angular power spectrum of the sky signal directly from the visibilities measured in radio-interferometric observations. The salient features of the TGE are (1.) it deals with the gridded data which makes it computationally very fast (2.) it avoids a positive noise bias which normally arises from the system noise inherent to the visibility data, and (3.) it allows us to taper the sky response and thereby suppresses the contribution from unsubtracted point sources in the outer parts and the sidelobes of the antenna beam pattern. We also summarize earlier work where the TGE was used to measure the C_l of the DGSE using 150 MHz GMRT data. Earlier measurements of the angular power spectrum are restricted to smaller angular multipole l ~ 10^3 for the DGSE, the signal at the larger l values is dominated by the residual point sources after source subtraction. The higher sensitivity of the upcoming SKA1 Low will allow the point sources to be subtracted to a fainter level than possible with existing telescopes. We predict that it will be possible to measure the angular power spectrum of the DGSE to larger values of l with SKA1 Low. Our results show that it should be possible to achieve l_{max} ~ 10^4 and ~ 10^5 with 2 minutes and 10 hours of observations respectively.Comment: 10 pages, 3 figures; Accepted for publication in Journal of Astrophysics and Astronomy (JOAA) special issue on "Science with the SKA: an Indian perspective