In this paper, the general Cochrane sum is defined. The main purpose is to study the mean square value problem of general Cochrane sum by using the estimates for trigonometric sums and the properties of general Kloosterman sum, and finally give a sharp asymptotic formula.

We show that if two meromorphic functions sharing five one-point or two-point sets two points IM, then one of them is a Möbius transformation of the other.

In this paper, we point out some similarities between results on the existence and uniqueness of finite order entire solutions of the nonlinear differential equations and differential-difference equations of the form $$f^n+L(z,f)=h.$$ Here n is an integer $\geq 2$, h is a given non-vanishing meromorphic function of finite order, and L(z,f) is a linear differential-difference polynomial, with small meromorphic functions as the coefficients.

We give fibrewise mod p decompositions of the adjoint bundle of a principal G-bundle P when the topological group G has mod p decompositions by automorphisms as in [5], which imply mod p decompositions of the gauge group of P.

In this paper, we give an explicit description of 3-Hilbert class field of certain imaginary quadratic fields.