From the coarse geometry of warped cones to the measured coupling of groups

In this article, we prove that if two warped cones corresponding to two groups with free, isometric, measure-preserving, ergodic actions on two manifolds are quasi-isometric, then the corresponding groups are uniformly measured equivalent (UME). It was earlier known from the work of de Laat-Vigolo that if two warped cones are QI, then their stable products are QI. Our result strengthens this result and go further to prove UME of the groups. However, Fisher-Nguyen-Limbeek proves that if the warped cones corresponding to two finitely presented groups with no free abelian factors are QI, then there is an affine commensuration of the two actions. Our result can be seen as an extension of their result in the setting of infinite presentability under some extra assumptions.Comment: Comments welcome. arXiv admin note: text overlap with arXiv:1512.08828. Some typos have been corrected and some minor gaps of Proposition 4.2 have been filled u

On Ahn-Hendrey-Kim-Oum question for twin-width of graphs with 6 vertices

Twin-width is a recently introduced graph parameter for finite graphs. It is an open problem to determine whether there is an $n$-vertex graph having twin-width at least $n/2$ (due to J. Ahn, K. Hendrey, D. Kim and S. Oum). In an earlier paper, the author showed that such a graph with less than equal to 5 vertices does not exist. In this article, we show that such a graph with 6 vertices does not exist. More precisely, we prove that each graph with 6 vertices has twin-width less than equal to 2.Comment: Comments welcome! arXiv admin note: substantial text overlap with arXiv:2207.1433

On the information entropy of matter-waves in quasi-disorder potentials

We consider ultracold Bose gases in quasi-random potentials and quantify localization of matter waves by means of Shannon information entropy. We explicitly examine the role of quasi-random potentials in producing localized states in the linear and nonlinear regimes. It is seen that the information entropic-based approach can be more useful to quantify localization of different types of states observed in Bose-Einstein condensates.Comment: 8 pages, 6 figure

Perspective of self-help groups, Panchayati Raj institution and adolescents on implementing COVID appropriate behavior in the community in rural settings

The vaccination drive in India brought cheers to the country's people, only to be eroded by the unpredictable twitch of the COVID-19 second wave. The best way to tackle the pandemic has been to adopt appropriate practices, thereby breaking the chain of transmission. The second wave in India has already created havoc both on the people and health system, questioning and challenging our capabilities to handle any heave of sudden cases, particularly in rural areas. Time has arrived to inflate our horizon by including groups of society close to the people, particularly in the rural community. Like the previous wave, COVID appropriate behavior was very much instrumental in slowing the pace. Adolescents’ self-help groups in coalition with Panchayat Raj Institutions (PRI) will be the main stand in the fight against the dreadful virus. Their main role is to translate the prevailing preventive measures into action among the very people they live around. Passing the right message through IEC/BCC to the right people will bring about dramatic changes in the attitude, belief, and exercise they do in their day-to-day life. PRIs members would utilize their powers to execute optimum administrative services. A decentralized approach involving these stakeholders will spearhead the fight against the pandemic. This trifold power shall bear the torchbearer in preventing the disastrous consequences of any upcoming wave, if any, in a holistic manner