Perturbations of planar algebras

We analyze the effect of pivotal structures (on a 2-category) on the planar algebra associated to a 1-cell as in \cite{Gho08} and come up with the notion of {\em perturbations of planar algebras by weights} (a concept that appeared earlier in Michael Burns' thesis \cite{Bur03}); we establish a one-to-one correspondence between weights and pivotal structures. Using the construction of \cite{Gho08}, to each bifinite bimodule over $II_1$-factors, we associate a {\em bimodule planar algebra} in such a way that extremality of the bimodule corresponds to sphericality of the planar algebra. As a consequence of this, we reproduce an extension of Jones' theorem (\cite{Jon}) (of associating `subfactor planar algebras' to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with, using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. The perturbation technique helps us to construct an example of a family of non-spherical planar algebras starting from a particular spherical one; we also show that this family is associated to a known family of subfactors constructed by Jones.Comment: 28 page

Lactation curve modelling using Legendre Polynomial in Sahiwal cattle

Present study aimed at modelling lactation curve in Sahiwal cattle using normalized Legendre polynomials functions of standardized units of time and understanding the variation of lactation curve in different lactations. Among the first three lactations, third lactation recorded highest peak yield as well other test day yields except the last test day yield. Further, first lactation curve was flatter compared to second and third lactation curve. Among the six different Legendre polynomial model considered higher degree Legendre polynomial model LEG5 and LEG6 fitted lactation curve better. Overall fifth degree Legendre polynomial lactation curve model (LEG5) can be used for accurate fitting for lactation curve in Sahiwal cattle as it gave almost similar accuracy like LEG6 with less parameter fitted

Early Secreted Antigen ESAT-6 of Mycobacterium tuberculosis Promotes Protective T Helper 17 Cell Responses in a Toll-Like Receptor-2-dependent Manner

Despite its relatively poor efficacy, Bacillus Calmette-Guérin (BCG) has been used as a tuberculosis (TB) vaccine since its development in 1921. BCG induces robust T helper 1 (Th1) immune responses but, for many individuals, this is not sufficient for host resistance against Mycobacterium tuberculosis (M. tb) infection. Here we provide evidence that early secreted antigenic target protein 6 (ESAT-6), expressed by the virulent M. tb strain H37Rv but not by BCG, promotes vaccine-enhancing Th17 cell responses. These activities of ESAT-6 were dependent on TLR-2/MyD88 signalling and involved IL-6 and TGF-β production by dendritic cells. Thus, animals that were previously infected with H37Rv or recombinant BCG containing the RD1 region (BCG::RD1) exhibited improved protection upon re-challenge with virulent H37Rv compared with mice previously infected with BCG or RD1-deficient H37Rv (H37RvΔRD1). However, TLR-2 knockout (TLR-2-/-) animals neither showed Th17 responses nor exhibited improved protection in response to immunization with H37Rv. Furthermore, H37Rv and BCG::RD1 infection had little effect on the expression of the anti-inflammatory microRNA-146a (miR146a) in dendritic cells (DCs), whereas BCG and H37RvΔRD1 profoundly induced its expression in DCs. Consistent with these findings, ESAT-6 had no effect on miR146a expression in uninfected DCs, but dramatically inhibited its upregulation in BCG-infected or LPS-treated DCs. Collectively, our findings indicate that, in addition to Th1 immunity induced by BCG, RD1/ESAT-6-induced Th17 immune responses are essential for optimal vaccine efficacy