Lie-Yamaguti Algebra Bundle

We introduce the notion of Lie-Yamaguti algebra bundle, define its cohomology groups with coefficients in a representation and show that such bundles appeared naturally from geometric considerations in the work of M. Kikkawa, which motivates us to introduce this object in the proper mathematical framework. We also study abelian extensions of Lie-Yamaguti algebra bundles and investigate their relationship with suitable cohomology group.Comment: 26 page

On Hausdorff Metric Spaces

An {\it expansive mapping of Lipschitz type} is introduced. A map, induced by a given map $T$ between two metric spaces $X$ and $Y$, from the power set of $X$ to the power set of $Y$ is considered. It is proved that the induced map preserves continuity, Lipschitz continuity and expansiveness of Lipschitz type. A nonempty intersection property in a metric space is achieved which also provides a partial generalization of the classical Cantor's Intersection Theorem. Using this nonempty intersection property and the considered induced map, it is shown that the converse of Henrikson's result (i.e. a Hausdorff metric space is complete if its underlying space is complete) also holds.Comment: 14 page

Menger convex metric spaces and fixed point theorems

Takahashi et al have shown that every decreasing sequence of nonempty, bounded, closed, convex subsets of a complete, uniformly (Takahashi's) $W-$ convex metric space has nonempty intersection. This article presents an extension of this nonempty intersection property of $W-$ convex metric space to a Menger convex metric space. We study some characteristics of the Menger convex metric spaces. Using these results, fixed point results for $(\alpha, \beta)-$ generalized hybrid mapping on Menger convex metric space are discussed. As an application, it is discussed how these results are applicable to a metrizable topological real vector space with some specific Takahashi's $W-$ convex structure on it.Comment: 17 page

Identification of internalin-A-like virulent proteins in Leishmania donovani

Abstract Background An active immune surveillance and a range of barriers to infection allow the host to effectively eliminate microbial pathogens. However, pathogens may use diverse strategies to subdue such host defences. For instance, one such mechanism is the use of leucine-rich repeat (LRR) proteins by pathogens (microbial) to cause infection. In this study, we aimed at identifying novel virulence factor(s) in Leishmania donovani, based on the possibility of lateral gene transfers of bacterial virulence factor(s) to L. donovani. Methods Rigorous homology searching protocols including Hidden Markov Model (HMM) and BLASTp based searches were employed to detect remote but significant similarities between L. donovani proteins and bacterial virulence factors. Results We found that some L. donovani proteins are similar to internalin-A (Inl-A) protein of Listeria monocytogenes, a surface LRR protein that helps mediate host cell invasion by interacting with E-cadherin on the cell membrane. However, to date, no such invasion mechanism has been reported in Leishmania donovani, the causative agent of visceral leishmaniasis. Moreover, a comparative LRR motif analysis of L. donovani Inl-A-like proteins against the Inl-A protein of L. monocytogenes revealed existence of characteristic consensus LRR regions, suggesting a reliable evolutionary relationship between them. Further, through rigorous three dimensional (3D) modeling of L. donovani Inl-A-like proteins and subsequent molecular docking studies we suggest the probability of human E-cadherin binding with the L. donovani Inl-A-like proteins. Conclusions We have identified three potential candidates (UniProt ID: E9B7L9, E9BMT7 and E9BUL5) of Inl-A-like LRR containing proteins in L. donovani with the help of systematic whole genome sequence analysis. Thus, herein we propose the existence of a novel class of Inl-A-like virulence factor proteins in L. donovani and other Leishmania species based on sequence similarity, phylogenetic analysis and molecular modelling studies in L. donovani