Scattering theory plays the main role in the study of manifolds and the Laplacian spectrum. In this article, we process justifying the continuous Laplacian spectrum i g ∆ and i h ∆ on a complete Riemannian manifold. () , i M g is categorized by the use of bounded curvature of the metric. In particular, the covariant derivative is limitedly considered as an application in the geodesic distance from a fixed point.

In any completely close complex field C, generalized transcendental meromorphic functions may have some new properties. It is well known that a meromorphic function of characteristic zero is a rational function. This paper introduced some mathematical properties of the transcendental meromorphic function, which is generalized to the meromorphic function by multiplying and differentiating the generalized meromorphic function. The analysis shows that the difference between any non-zero constant and the derivative of the general meromorphic function has an infinite zero. In addition, for any natural number n, there are no practically exceptional values for the multiplication of the general meromorphic function and its derivative to the power of n.

The aims of this study, the researchers inspected that if X an algebraic variety over a field K and f is a f 1+2ε (P) = P for other ε >-1/2, then the aset of periods time is limited and the little one is 1+2ε. Also the researchers confirmed that if  be a separate estimate ring of characteristics zero with residue field k of characteristics (2+ ε) of integers in K and (A, m) be a local sub--algebra