Two- & Three-character solutions to MLDEs and Ramanujan-Eisenstein Identities for Fricke Groups

In this work we extend the study of arXiv:2210.07186 by investigating two- and three-character MLDEs for Fricke groups at prime levels. We have constructed these higher-character MLDEs by using a $\mathit{novel}$ Serre-Ramanujan type derivative operator which maps $k$-forms to $(k+2)$-forms in $\Gamma^{+}_0(p)$. We found that this $\mathit{novel}$ derivative construction enabled us to write down a general prescription for obtaining $\mathit{Ramanujan-Eisenstein}$ identities for these groups. We discovered several $\mathit{novel}$ single-, two-, and three-character admissible solutions for Fricke groups at levels $2$ and $3$ after solving the MLDEs among which we have realized some in terms of Mckay-Thompson series and others in terms of modular forms of the corresponding Hecke groups. Among these solutions, we have identified interesting non-trivial bilinear identities. Furthermore, we could construct $\mathit{putative}$ partition functions for these theories based on these bilinear pairings, which could have a range of lattice interpretations. We also present and discuss modular re-parameterization of MLDE and their solutions for Fricke groups of prime levels.Comment: 94 pages + 14 pages (appendices and references); 3 figures; 13 table

Effects of fluoxetine on functional outcomes after acute stroke (FOCUS): a pragmatic, double-blind, randomised, controlled trial

Background Results of small trials indicate that fluoxetine might improve functional outcomes after stroke. The FOCUS trial aimed to provide a precise estimate of these effects. Methods FOCUS was a pragmatic, multicentre, parallel group, double-blind, randomised, placebo-controlled trial done at 103 hospitals in the UK. Patients were eligible if they were aged 18 years or older, had a clinical stroke diagnosis, were enrolled and randomly assigned between 2 days and 15 days after onset, and had focal neurological deficits. Patients were randomly allocated fluoxetine 20 mg or matching placebo orally once daily for 6 months via a web-based system by use of a minimisation algorithm. The primary outcome was functional status, measured with the modified Rankin Scale (mRS), at 6 months. Patients, carers, health-care staff, and the trial team were masked to treatment allocation. Functional status was assessed at 6 months and 12 months after randomisation. Patients were analysed according to their treatment allocation. This trial is registered with the ISRCTN registry, number ISRCTN83290762. Findings Between Sept 10, 2012, and March 31, 2017, 3127 patients were recruited. 1564 patients were allocated fluoxetine and 1563 allocated placebo. mRS data at 6 months were available for 1553 (99·3%) patients in each treatment group. The distribution across mRS categories at 6 months was similar in the fluoxetine and placebo groups (common odds ratio adjusted for minimisation variables 0·951 [95% CI 0·839–1·079]; p=0·439). Patients allocated fluoxetine were less likely than those allocated placebo to develop new depression by 6 months (210 [13·43%] patients vs 269 [17·21%]; difference 3·78% [95% CI 1·26–6·30]; p=0·0033), but they had more bone fractures (45 [2·88%] vs 23 [1·47%]; difference 1·41% [95% CI 0·38–2·43]; p=0·0070). There were no significant differences in any other event at 6 or 12 months. Interpretation Fluoxetine 20 mg given daily for 6 months after acute stroke does not seem to improve functional outcomes. Although the treatment reduced the occurrence of depression, it increased the frequency of bone fractures. These results do not support the routine use of fluoxetine either for the prevention of post-stroke depression or to promote recovery of function. Funding UK Stroke Association and NIHR Health Technology Assessment Programme

Formation of the southern Bay of Bengal cold pool

A pool of relatively cooler water, called here as the southern Bay of Bengal cold pool, exists around Sri Lanka and southern tip of India during the summer monsoon. This cold pool is enveloped by the larger Indian Ocean warm pool and is believed to affect the intraseasonal variations of summer monsoon rainfall. In this study, we have investigated the mechanisms responsible for the formation of the cold pool using a combination of both satellite data sets and a general circulation model of the Indian Ocean. Sea surface temperature (SST) within the cold pool, after the steady increase during the February-April period, decreases first during a pre-monsoon spell in April and then with the monsoon onset during May. The onset cooling is stronger (similar to 1.8 ) than the pre-monsoon cooling (similar to 0.8 ) and culminates in the formation of the cold pool. Analysis of the model temperature equation shows that SST decrease during both events is primarily due to a decrease in incoming solar radiation and an increase in latent heat loss. These changes in the net heat flux are brought about by the arrival of cloud bands above the cold pool during both periods. During the pre-monsoon period, a cloud band originates in the western equatorial Indian Ocean and subsequently arrives above the cold pool. Similarly, during the monsoon onset, a band of clouds originating in the eastern equatorial Indian Ocean comes over the cold pool region. A lead-lag correlation calculation between daily SST and rainfall anomalies suggest that cooling in SST occurs in response to rainfall events with a lag of 5 days. These sequence of events occur every year with certain amount of interannual variability