Abstract. We use tools from n-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M . On one hand we extend a theorem of Lieb (see [L]) and prove that any nodal domain Ω λ almost fully contains a ball of radius ∼, which is made precise by Theorem 1.6 below. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains ([Man2]). On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside M (this is Theorem 1.5).

Let Ω ⊂ R n be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as "obstacle". We are interested in the behaviour of the first Dirichlet eigenvalue λ 1 (Ω \ (x + D)).First, we prove an upper bound on λ 1 (Ω \ (x + D)) in terms of the distance of the set x + D to the set of maximum points x 0 of the first Dirichlet ground state φ λ 1 > 0 of Ω. In short, a direct corollary is that ifis large enough in terms of λ 1 (Ω), then all maximizer sets x+D of µ Ω are close to each maximum point x 0 of φ λ 1 . Second, we discuss the distribution of φ λ 1 (Ω) and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if µ Ω is sufficiently large with respect to λ 1 (Ω), then all maximizers x + D of µ Ω contain all maximum points x 0 of φ λ 1 (Ω) .

We consider a Laplace eigenfunction ϕ λ on a smooth closed Riemannian manifold, that is, satisfying −∆ϕ λ = λϕ λ . We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains.First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of ϕ λ . This extends previous work of Jakobson and Mangoubi in case (M, g) is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf.[L1]).Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains Ω λ on which ϕ λ satisfies exponentially small L ∞ bounds, we give some quantitative estimates for radii of inscribed balls. 1 We use the analyst's sign convention, namely, −∆ is positive semidefinite.

We consider mass concentration properties of Laplace eigenfunctions ϕ λ , that is, smooth functions satisfying the equation −∆ϕ λ = λϕ λ , on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" part and "thin" elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.

Butterfly diversity in and around Neora Valley National Park (NVNP), West Bengal, India was studied from three different habitat types that included thick vegetation assemblage with closed canopy cover, edges of forest and areas of human intervention during AprilMay 2010. A total of 30 butterfly species belonging to the families of Hespeririidae (3.33%), Papilionidae (16.65%), Pieriidae (13.32%), Nymphalidae (53.28%) and Lycaenidae (13.32%) were identified in the present investigation. Highest butterfly diversity and abundance was recorded from areas of forest edges (54.83% of individuals represented by 16 different species), while dense forest (30.64 % of individuals represented by 11 different species) and areas with human habitats (14.52 % of individuals represented by 8 different species) showed lower butterfly diversity and abundance. Accordingly highest Shannon Weiner diversity score of 2.32 was recorded from areas of forest edges. The butterflies that showed high occurrences were Indian Tortoise Shell (Aglais cashmiriensis), Yellow Coster (Acraea issoria) and Himalayan Five Ring (Ypthima sakra). Only 1 butterfly species, Yellow Coster (A. issoria) was found to co-occur in all the three sites. Accelerating human civilizations has lead to destruction of much of the global natural habitats while it has often been found to exert adverse effects on biodiversity. Findings made during this study also indicate negative influence of anthropogenic intervention on overall butterfly diversity from the present location.