Here, we are concerned with a new, highly precise, numerical solution to the one-dimensional neutron transport equation based on Case’s analytical, singular eigenfunction expansion (SEE). While considerable number of solutions currently exist, understandably, because its complexity even in one dimension, there only few truly analytical equation. In 1960, Case introduced consistent theory SEE fo...

On any compact Riemannian manifold (M,g) of dimension n, the Lnormalized eigenfunctions {φλ} satisfy ||φλ||∞ ≤ Cλ n−1 2 where −∆φλ = λ 2φλ. The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere S. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori R/Γ. We say that S, but not R/Γ, is a Riemannian manifol...

In this short note, we prove the convexity of the first eigenfunction of the drifting Laplacian operator with zero Dirichlet boundary value provided a suitable assumption to the drifting term is added. After giving a gradient estimate, we then use the convexity of the first eigenfunction to get a lower bound of the difference of the first and second eigenvalues of the drifting Laplacian.

We consider the co-axial cylindrical structure as a composite submerged solid cylinder above special bottom undulation, i.e., circular plate at impermeable horizontal bottom. diffraction problem of proposed in water finite depth. This can be expressed wave energy converter. The variables separation and eigenfunction expansion methods are utilized to determine analytical solutions for their iden...