The Bulk and the Extremes of Minimal Spanning Acycles and Persistence Diagrams of Random Complexes

Frieze showed that the expected weight of the minimum spanning tree (MST) of the uniformly weighted graph converges to $\zeta(3)$. Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue -- the Minimum Spanning Acycle (MSA). In this work, we go beyond and look at the histogram of the weights in this random MSA -- both in the bulk and in the extremes. In particular, we focus on the `incomplete' setting, where one has access only to a fraction of the potential face weights. Our first result is that the empirical distribution of the MSA weights asymptotically converges to a measure based on the shadow -- the complement of graph components in higher dimensions. As far as we know, this result is the first to explore the connection between the MSA weights and the shadow. Our second result is that the extremal weights converge to an inhomogeneous Poisson point process. A interesting consequence of our two results is that we can also state the distribution of the death times in the persistence diagram corresponding to the above weighted complex, a result of interest in applied topology.Comment: 15 pages, 5 figures, Corrected Typo

GAS CHROMATOGRAPHY MASS SPECTROMETRY (GC-MS) ANALYSIS AND DOCKING STUDIES OF ANDROGRAPHIS PANICULATA AGAINST DENGUE FEVER

Objective: To evaluate gas chromatography- mass spectrometry (GC-MS) analysis and molecular docking studies of ethanol extract of Andrographis paniculata (Burm.f.) against type 2 dengue virus (DEN2).Methods: The ethanol extract of A. paniculata compounds were identified by GC-MS analysis. These compounds were further analysed for their activities against NS2B/NS3 protease of DEN2 by molecular docking studies. Results: The nine compounds obtained from the ethanol extract of A. paniculata were characterized and docked. Among these nine compounds, Vor-diazepam,3-[[N-hydroxymethyl]aminocarbonyloxy], Dasycarpidan-1-methanol, acetate (ester), Ethanol, 2-(9, 12-octadecadienyloxy)-, (Z, Z), Gibb-2-ene-1,10-dicarboxylicacid4a,7-dihydroxy-1-methylene-1,4a-lactone,10 methylester, (1a,4aa,4ba,10a) were found to show better interaction energetically and also showed crucial interactions with the active site of NS2B/NS3 protease. Thus A. paniculata is suggested to be a medicinally important plant to cure dengue fever.Conclusions: From the result, it can be concluded that the crude extract of A. paniculata compounds positively inhibit the activities of NS2B/NSB protease of DEN2 thus by preventing the dengue viral infection. This strategy reflects a logical progression for an early stage drug discovery which can be used to identify new drug candidates.Â