In this paper we determine automorphism groups of cyclic algebraic curves defined over finite fields of any characteristic.

In this note we discuss explicitly the structure of some simple sets of values on the critical line (the “trivial critical values”) which are associated with the mean staircases emerging from the Zeta function. They are given as solutions of an equation involving the Lambert W-function. The argument of the latter function may then be set equal to a special $N\times N$ (classical) matrix (for every $N$) related to the Hamiltonian of the Mehta-Dyson model. In this way we specify a function of an hermitean operator whose eigenvalues are exactly these values. In the general case, the sets of such trivial critical values (zeros) are special solutions of a parametric equation involving a linear combination of $\text{Re}\zeta (s)$ and $\text{Im}\zeta (s)$ on the critical line.

In this paper, we introduce and consider a new class of variational inequalities, which is called the general nonconvex variational inequality. We establish the equivalence between the general nonconvex variational inequalities and the fixed point problems as well as the Wiener-Hopf equations using the projection method. This alternative equivalent formulation is used to study the existence of a solution of the general convex variational inequalities. We also use this equivalence formulation to suggest some iterative methods. Convergence criteria of these new iterative is also discussed under suitable conditions. Our method of proofs is very simple as compared with other techniques.

Integers with two prime factors occur in the RSA cryptosystem. In this paper, we provide density estimates for such integers occuring in the RSA cryptosystem satisfying various conditions. Cryptographic applications are given as a consequence of the estimates obtained.