An 𝑛-dimensional pseudo-differential operator (p.d.o.) involving the 𝑛-dimensional Hankel transformation is defined. The symbol class $H^m$ is introduced. It is shown that p.d.o.'s associated with symbols belonging to this class are continuous linear mappings of the 𝑛-dimensional Zemanian space $H_\mu(I^n)$ into itself. An integral representation for the p.d.o. is obtained. Using the Hankel convolution, it is shown that the p.d.o. satisfies a certain $L^1$-norm inequality.

Formal orbifolds are defined in higher dimension to study wild ramification. Their étale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to approximate the étale fundamental groups of normal varieties. Étale site on formal orbifolds are also defined.This framework allows one to study wild ramification in an organized way. Brylinski–Kato filtration, Lefschetz theorem for fundamental groups and $l$-adic sheaves in these contexts are also studied.

In this article,we investigate the sign changes of the sequence of coefficientsat sum of two squareswhere the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group.We provide the quantitative result for the number of sign changes of the sequence in a small interval.