Curvature, Harnack's Inequality, and a Spectral Characterization of Nilmanifolds

For closed n-dimensional Riemannian manifolds M with almostnonnegative Ricci curvature, the Laplacian on one-forms is known toadmit at most n small eigenvalues. If there are n small eigenvalues, or if M is orientable and has n – 1 small eigenvalues, then M isdiffeomorphic to a nilmanifold, and the metric is almost left invariant.We show that our results are optimal for n 4.     

Molecular docking studies,biological evaluation and synthesis of novel 3-mercapto-1,2,4-triazole derivatives

Molecular Diversity - Synthesis of bioactive heterocyclic compounds having effective biological activity is an essential research area for wide-ranging applications. In this study, a conventional...