Proof of a Limited Version of Mao's Partition Rank Inequality using a Theta Function Identity

Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \cite{atkin}, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \cite{mao10} and for $M_2$ rank differences for partitions with no repeated odd parts modulo $6$ and $10$ \cite{maom2}. Alwaise et. al. proved four of Mao's conjectured inequalities \cite{swisher}, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.Comment: First draft. Comments are welcom

Mask Use pattern, Public Perception and Barriers to Effective Mask Usage-A battle far from over

Background & Objective: Proper mask wearing reduces the transmissibility and risk of Covid-19 infection still compliance to mask use is low. This study was conducted to observe the mask usage pattern, perception and barriers to effective mask usage (EMU). Settings and Design: This cross-sectional study was conducted in collaboration with district police administration to observe mask usage pattern in various crowded places. Methods and Material: Participants from general public (n=6995) in 10 preselected areas of city were observed and then interviewed about their perceptions about EMU. Indirect observation of EMU was also done by screening CCTV footage, followed by another set of observation at same places after conducting a health education campaign. Data was entered in Microsoft excel and analyzed using SPSS. P value <0.05 was considered significant. Results: EMU was seen in nearly 62% participants. Younger age, native population, education above matric, persons driving two & four wheelers, had significantly higher compliance. During CCTV observation, EMU was 57.5%, and was highest near public places, roads and intersections while lowest near bus stand and religious places. Some improvement in compliance was seen after health education at these places. Breathing difficulty, no use of wearing mask and feeling uncomfortable were the most common reasons given by participants for non-compliance.Conclusions: EMU was lower in elderly, migrants, less educated, unemployed, daily wage workers, so these groups should be especially sensitized about preventive measures. There is need for extensive public education to bring out behavioral change regarding proper mask wearing to control this pandemic and prevent further waves